diff options
| author |  Jason Gross <jagro@google.com> | 2018-08-22 17:17:02 -0400 |
|---|---|---|
| committer |  Jason Gross <jasongross9@gmail.com> | 2018-08-23 21:12:36 -0700 |
| commit | 5cab97ed8f17e294f4e7e66010147a518c45f3a6 (patch) | |
| tree | 0746d590811eb94045aa37b01965563a13ec1f4c /src | |
| parent | a24640e12177576b4c7fcc299f19df09e6b36d81 (diff) | |
Add Z.land, Z.lor bounds stuff to zutil, also split up ZUtil
The file src/Util/ZUtil.v no longer exports any lemmas, nor does it
contain any lemmas. Instead, it pulls in all of the various ZUtil files
so that `Search` will pick up the relevant lemma, and allow users to
`Require Import` the relevant file.
This allows more parallelization in the build. It also prevents
needlessly rebuilding lots of files whenever we change anything anywhere
in ZUtil.
From this point forward, no file in the development should `Require`
`Crypto.Util.ZUtil` itself.
After | File Name | Before || Change | % Change
-----------------------------------------------------------------------------------------------------------------------
73m47.61s | Total | 73m49.49s || -0m01.87s | -0.04%
-----------------------------------------------------------------------------------------------------------------------
0m14.49s | Util/ZUtil/LandLorBounds | N/A || +0m14.49s | ∞
0m00.42s | Util/ZUtil | 0m11.07s || -0m10.65s | -96.20%
0m03.54s | Util/ZUtil/LandLorShiftBounds | N/A || +0m03.54s | ∞
0m03.49s | Util/ZUtil/Shift | N/A || +0m03.49s | ∞
4m09.67s | Experiments/NewPipeline/RewriterRulesGood | 4m07.61s || +0m02.05s | +0.83%
1m22.68s | Experiments/NewPipeline/RewriterWf2 | 1m20.22s || +0m02.46s | +3.06%
1m21.09s | Compilers/Named/MapCastInterp | 1m23.14s || -0m02.04s | -2.46%
0m17.79s | Arithmetic/MontgomeryReduction/WordByWord/Abstract/Dependent/Proofs | 0m20.61s || -0m02.82s | -13.68%
0m08.00s | Arithmetic/MontgomeryReduction/Proofs | 0m10.90s || -0m02.90s | -26.60%
0m05.46s | LegacyArithmetic/Double/Proofs/SpreadLeftImmediate | 0m08.07s || -0m02.61s | -32.34%
8m39.34s | Experiments/SimplyTypedArithmetic | 8m38.00s || +0m01.34s | +0.25%
1m30.51s | Spec/Test/X25519 | 1m28.92s || +0m01.59s | +1.78%
1m16.98s | Experiments/NewPipeline/Rewriter | 1m18.02s || -0m01.03s | -1.33%
0m25.51s | Experiments/NewPipeline/UnderLetsProofs | 0m27.02s || -0m01.50s | -5.58%
0m10.75s | Arithmetic/MontgomeryReduction/WordByWord/Abstract/Proofs | 0m12.00s || -0m01.25s | -10.41%
0m09.28s | LegacyArithmetic/Double/Proofs/Multiply | 0m10.97s || -0m01.69s | -15.40%
0m05.70s | LegacyArithmetic/Double/Proofs/RippleCarryAddSub | 0m07.01s || -0m01.30s | -18.68%
0m03.36s | LegacyArithmetic/InterfaceProofs | 0m04.67s || -0m01.31s | -28.05%
0m01.43s | Util/ZUtil/Ones | N/A || +0m01.42s | ∞
0m01.37s | Arithmetic/BarrettReduction/Wikipedia | 0m02.38s || -0m01.00s | -42.43%
6m02.75s | Experiments/NewPipeline/SlowPrimeSynthesisExamples | 6m03.58s || -0m00.82s | -0.22%
4m45.52s | Experiments/NewPipeline/Toplevel1 | 4m45.02s || +0m00.50s | +0.17%
3m46.16s | Curves/Montgomery/XZProofs | 3m45.20s || +0m00.96s | +0.42%
2m11.24s | Specific/X25519/C64/ladderstep | 2m11.78s || -0m00.53s | -0.40%
1m52.27s | Specific/NISTP256/AMD64/femul | 1m52.36s || -0m00.08s | -0.08%
1m43.21s | Experiments/NewPipeline/Toplevel2 | 1m43.70s || -0m00.48s | -0.47%
1m30.09s | Experiments/NewPipeline/Arithmetic | 1m29.59s || +0m00.50s | +0.55%
1m18.58s | Specific/X2448/Karatsuba/C64/femul | 1m19.45s || -0m00.87s | -1.09%
0m59.72s | Specific/X25519/C32/femul | 1m00.00s || -0m00.28s | -0.46%
0m52.94s | Demo | 0m52.24s || +0m00.69s | +1.33%
0m48.98s | Compilers/Z/Named/RewriteAddToAdcInterp | 0m49.12s || -0m00.14s | -0.28%
0m47.34s | Compilers/Z/ArithmeticSimplifierInterp | 0m47.24s || +0m00.10s | +0.21%
0m43.02s | Specific/X25519/C32/fesquare | 0m43.06s || -0m00.03s | -0.09%
0m42.02s | Arithmetic/Karatsuba | 0m42.08s || -0m00.05s | -0.14%
0m41.76s | Experiments/NewPipeline/AbstractInterpretationWf | 0m42.33s || -0m00.57s | -1.34%
0m38.51s | p521_32.c | 0m38.75s || -0m00.24s | -0.61%
0m37.14s | Experiments/NewPipeline/ExtractionOCaml/word_by_word_montgomery | 0m37.22s || -0m00.07s | -0.21%
0m36.10s | Spec/Ed25519 | 0m36.23s || -0m00.12s | -0.35%
0m35.83s | Experiments/NewPipeline/LanguageInversion | 0m35.71s || +0m00.11s | +0.33%
0m34.38s | Experiments/NewPipeline/ExtractionHaskell/word_by_word_montgomery | 0m34.91s || -0m00.52s | -1.51%
0m33.32s | Specific/X25519/C32/freeze | 0m33.32s || +0m00.00s | +0.00%
0m31.92s | p521_64.c | 0m32.20s || -0m00.28s | -0.86%
0m31.32s | Compilers/Z/ArithmeticSimplifierWf | 0m30.82s || +0m00.50s | +1.62%
0m28.96s | Compilers/CommonSubexpressionEliminationWf | 0m29.19s || -0m00.23s | -0.78%
0m27.73s | Specific/NISTP256/AMD128/femul | 0m27.70s || +0m00.03s | +0.10%
0m26.79s | Primitives/EdDSARepChange | 0m26.90s || -0m00.10s | -0.40%
0m25.27s | Specific/X25519/C32/fecarry | 0m25.34s || -0m00.07s | -0.27%
0m24.87s | Experiments/NewPipeline/AbstractInterpretationZRangeProofs | 0m25.26s || -0m00.39s | -1.54%
0m23.71s | p384_32.c | 0m23.60s || +0m00.10s | +0.46%
0m22.37s | Experiments/NewPipeline/LanguageWf | 0m22.49s || -0m00.11s | -0.53%
0m21.43s | Arithmetic/Core | 0m21.60s || -0m00.17s | -0.78%
0m21.41s | Specific/X25519/C32/fesub | 0m21.28s || +0m00.12s | +0.61%
0m20.86s | Experiments/NewPipeline/ExtractionHaskell/unsaturated_solinas | 0m20.26s || +0m00.59s | +2.96%
0m20.86s | Specific/NISTP256/AMD64/fesub | 0m21.69s || -0m00.83s | -3.82%
0m20.53s | Specific/X25519/C64/femul | 0m20.54s || -0m00.00s | -0.04%
0m19.67s | Specific/X25519/C32/Synthesis | 0m19.54s || +0m00.13s | +0.66%
0m19.44s | Curves/Edwards/XYZT/Basic | 0m19.05s || +0m00.39s | +2.04%
0m19.30s | Specific/NISTP256/AMD64/feadd | 0m19.35s || -0m00.05s | -0.25%
0m19.23s | Specific/X25519/C32/feadd | 0m19.20s || +0m00.03s | +0.15%
0m18.84s | Experiments/NewPipeline/ExtractionOCaml/unsaturated_solinas | 0m18.60s || +0m00.23s | +1.29%
0m18.07s | Compilers/Named/MapCastWf | 0m17.87s || +0m00.19s | +1.11%
0m18.01s | Compilers/Z/CNotations | 0m18.15s || -0m00.13s | -0.77%
0m17.50s | Specific/X25519/C64/freeze | 0m17.52s || -0m00.01s | -0.11%
0m17.03s | Specific/X25519/C64/fesquare | 0m17.14s || -0m00.10s | -0.64%
0m16.10s | Curves/Edwards/AffineProofs | 0m15.80s || +0m00.30s | +1.89%
0m15.71s | Specific/NISTP256/AMD64/feopp | 0m15.97s || -0m00.25s | -1.62%
0m15.62s | Compilers/Named/ContextProperties/SmartMap | 0m15.57s || +0m00.04s | +0.32%
0m15.35s | Compilers/Named/ContextProperties/NameUtil | 0m15.30s || +0m00.04s | +0.32%
0m15.04s | Specific/NISTP256/AMD128/fesub | 0m15.16s || -0m00.12s | -0.79%
0m14.88s | Specific/NISTP256/AMD128/feadd | 0m14.83s || +0m00.05s | +0.33%
0m14.20s | Specific/NISTP256/AMD64/fenz | 0m14.56s || -0m00.36s | -2.47%
0m14.16s | Specific/X25519/C64/fecarry | 0m14.13s || +0m00.02s | +0.21%
0m13.78s | Experiments/NewPipeline/ExtractionHaskell/saturated_solinas | 0m13.54s || +0m00.24s | +1.77%
0m13.69s | Arithmetic/Saturated/AddSub | 0m13.66s || +0m00.02s | +0.21%
0m13.67s | Specific/NISTP256/AMD128/fenz | 0m13.69s || -0m00.01s | -0.14%
0m13.16s | Experiments/NewPipeline/CStringification | 0m13.08s || +0m00.08s | +0.61%
0m13.10s | Specific/X25519/C64/fesub | 0m13.16s || -0m00.06s | -0.45%
0m12.34s | Experiments/NewPipeline/AbstractInterpretationProofs | 0m12.47s || -0m00.13s | -1.04%
0m12.29s | Specific/NISTP256/AMD128/feopp | 0m12.34s || -0m00.05s | -0.40%
0m12.26s | Compilers/Z/Syntax/Equality | 0m12.69s || -0m00.42s | -3.38%
0m11.94s | Specific/X25519/C64/feadd | 0m11.90s || +0m00.03s | +0.33%
0m11.87s | Primitives/MxDHRepChange | 0m11.76s || +0m00.10s | +0.93%
0m11.85s | Experiments/NewPipeline/GENERATEDIdentifiersWithoutTypesProofs | 0m11.66s || +0m00.18s | +1.62%
0m11.42s | Arithmetic/Saturated/MontgomeryAPI | 0m11.45s || -0m00.02s | -0.26%
0m10.69s | Experiments/NewPipeline/ExtractionOCaml/saturated_solinas | 0m10.30s || +0m00.38s | +3.78%
0m10.61s | Arithmetic/Saturated/Core | 0m10.66s || -0m00.05s | -0.46%
0m09.89s | Specific/X2448/Karatsuba/C64/Synthesis | 0m09.79s || +0m00.10s | +1.02%
0m09.35s | Util/ZRange/CornersMonotoneBounds | 0m09.90s || -0m00.55s | -5.55%
0m08.79s | Experiments/NewPipeline/ExtractionOCaml/word_by_word_montgomery.ml | 0m08.60s || +0m00.18s | +2.20%
0m08.64s | p384_64.c | 0m08.50s || +0m00.14s | +1.64%
0m08.63s | LegacyArithmetic/ArchitectureToZLikeProofs | 0m08.61s || +0m00.02s | +0.23%
0m08.49s | LegacyArithmetic/Double/Proofs/ShiftRightDoubleWordImmediate | 0m08.38s || +0m00.10s | +1.31%
0m08.19s | Compilers/Named/CompileInterpSideConditions | 0m07.78s || +0m00.40s | +5.26%
0m08.15s | Experiments/NewPipeline/GENERATEDIdentifiersWithoutTypes | 0m08.20s || -0m00.04s | -0.60%
0m07.98s | Compilers/Named/RegisterAssignInterp | 0m08.11s || -0m00.12s | -1.60%
0m07.81s | Arithmetic/BarrettReduction/RidiculousFish | 0m08.20s || -0m00.38s | -4.75%
0m07.45s | Compilers/InlineConstAndOpWf | 0m07.40s || +0m00.04s | +0.67%
0m07.01s | Specific/NISTP256/AMD64/Synthesis | 0m06.98s || +0m00.02s | +0.42%
0m06.55s | Compilers/Z/Bounds/InterpretationLemmas/PullCast | 0m06.44s || +0m00.10s | +1.70%
0m06.51s | Arithmetic/Saturated/MulSplit | 0m06.45s || +0m00.05s | +0.93%
0m06.40s | Util/FixedWordSizesEquality | 0m06.46s || -0m00.05s | -0.92%
0m06.28s | Arithmetic/BarrettReduction/Generalized | 0m06.64s || -0m00.35s | -5.42%
0m06.08s | Util/ZUtil/Modulo | 0m05.58s || +0m00.50s | +8.96%
0m06.07s | Compilers/InlineWf | 0m06.20s || -0m00.12s | -2.09%
0m06.04s | Util/ZUtil/Morphisms | 0m06.09s || -0m00.04s | -0.82%
0m05.94s | Specific/X25519/C64/Synthesis | 0m05.95s || -0m00.00s | -0.16%
0m05.66s | Compilers/LinearizeWf | 0m05.73s || -0m00.07s | -1.22%
0m05.64s | LegacyArithmetic/Pow2BaseProofs | 0m06.27s || -0m00.62s | -10.04%
0m05.60s | Experiments/NewPipeline/ExtractionHaskell/word_by_word_montgomery.hs | 0m05.45s || +0m00.14s | +2.75%
0m05.45s | Experiments/NewPipeline/ExtractionOCaml/unsaturated_solinas.ml | 0m05.54s || -0m00.08s | -1.62%
0m05.29s | Compilers/Z/HexNotationConstants | 0m05.44s || -0m00.15s | -2.75%
0m05.02s | Compilers/WfProofs | 0m04.98s || +0m00.03s | +0.80%
0m04.76s | Experiments/NewPipeline/RewriterWf1 | 0m04.74s || +0m00.01s | +0.42%
0m04.69s | Specific/Framework/ArithmeticSynthesis/Montgomery | 0m04.71s || -0m00.01s | -0.42%
0m04.63s | Arithmetic/BarrettReduction/HAC | 0m05.14s || -0m00.50s | -9.92%
0m04.48s | Compilers/Z/Bounds/Pipeline/Definition | 0m04.68s || -0m00.19s | -4.27%
0m04.34s | Compilers/Z/BinaryNotationConstants | 0m04.39s || -0m00.04s | -1.13%
0m04.14s | Compilers/Named/CompileWf | 0m04.19s || -0m00.05s | -1.19%
0m04.02s | Experiments/NewPipeline/MiscCompilerPassesProofs | 0m03.98s || +0m00.03s | +1.00%
0m03.99s | Experiments/NewPipeline/ExtractionHaskell/unsaturated_solinas.hs | 0m03.87s || +0m00.12s | +3.10%
0m03.90s | secp256k1_32.c | 0m03.91s || -0m00.01s | -0.25%
0m03.87s | Experiments/NewPipeline/ExtractionOCaml/saturated_solinas.ml | 0m03.98s || -0m00.10s | -2.76%
0m03.83s | Arithmetic/MontgomeryReduction/WordByWord/Proofs | 0m03.72s || +0m00.10s | +2.95%
0m03.76s | p256_32.c | 0m03.81s || -0m00.05s | -1.31%
0m03.67s | LegacyArithmetic/Double/Proofs/ShiftRight | 0m03.59s || +0m00.08s | +2.22%
0m03.58s | Compilers/Z/ArithmeticSimplifier | 0m03.60s || -0m00.02s | -0.55%
0m03.48s | Compilers/Z/Bounds/InterpretationLemmas/IsBoundedBy | 0m03.46s || +0m00.02s | +0.57%
0m03.41s | Experiments/NewPipeline/ExtractionHaskell/saturated_solinas.hs | 0m03.32s || +0m00.09s | +2.71%
0m03.28s | Specific/NISTP256/AMD128/Synthesis | 0m03.30s || -0m00.02s | -0.60%
0m03.24s | Util/ZUtil/Div | 0m02.78s || +0m00.46s | +16.54%
0m03.12s | LegacyArithmetic/Double/Proofs/ShiftLeft | 0m03.13s || -0m00.00s | -0.31%
0m03.05s | Compilers/InlineInterp | 0m02.91s || +0m00.13s | +4.81%
0m03.04s | LegacyArithmetic/Double/Proofs/Decode | 0m03.48s || -0m00.43s | -12.64%
0m02.98s | Compilers/Named/ContextProperties | 0m03.00s || -0m00.02s | -0.66%
0m02.90s | Compilers/TestCase | 0m02.89s || +0m00.00s | +0.34%
0m02.88s | LegacyArithmetic/ZBoundedZ | 0m03.85s || -0m00.97s | -25.19%
0m02.88s | Util/WordUtil | 0m02.90s || -0m00.02s | -0.68%
0m02.72s | Compilers/Named/CompileInterp | 0m02.74s || -0m00.02s | -0.72%
0m02.58s | Arithmetic/Saturated/Freeze | 0m02.56s || +0m00.02s | +0.78%
0m02.58s | LegacyArithmetic/BarretReduction | 0m02.72s || -0m00.14s | -5.14%
0m02.54s | Compilers/Named/ContextProperties/Proper | 0m02.61s || -0m00.06s | -2.68%
0m02.40s | Arithmetic/ModularArithmeticTheorems | 0m03.33s || -0m00.93s | -27.92%
0m02.38s | Specific/NISTP256/FancyMachine256/Montgomery | 0m02.33s || +0m00.04s | +2.14%
0m02.30s | Specific/NISTP256/FancyMachine256/Core | 0m02.28s || +0m00.02s | +0.87%
0m02.27s | Compilers/CommonSubexpressionEliminationProperties | 0m02.20s || +0m00.06s | +3.18%
0m02.22s | Compilers/Z/Bounds/Relax | 0m02.21s || +0m00.01s | +0.45%
0m02.22s | Specific/NISTP256/FancyMachine256/Barrett | 0m02.27s || -0m00.04s | -2.20%
0m02.16s | Util/ZUtil/Quot | 0m02.08s || +0m00.08s | +3.84%
0m02.10s | Compilers/Named/NameUtilProperties | 0m02.04s || +0m00.06s | +2.94%
0m02.10s | Compilers/Z/RewriteAddToAdcInterp | 0m02.14s || -0m00.04s | -1.86%
0m02.10s | p224_32.c | 0m02.11s || -0m00.00s | -0.47%
0m02.09s | Specific/Framework/ArithmeticSynthesis/Defaults | 0m02.11s || -0m00.02s | -0.94%
0m02.07s | Util/ZRange/SplitBounds | 0m02.01s || +0m00.06s | +2.98%
0m02.01s | curve25519_32.c | 0m02.02s || -0m00.01s | -0.49%
0m01.90s | Compilers/Z/JavaNotations | 0m01.96s || -0m00.06s | -3.06%
0m01.84s | LegacyArithmetic/MontgomeryReduction | 0m01.82s || +0m00.02s | +1.09%
0m01.84s | Util/ZUtil/AddGetCarry | 0m01.87s || -0m00.03s | -1.60%
0m01.73s | Util/ZUtil/Pow2Mod | 0m01.48s || +0m00.25s | +16.89%
0m01.68s | Compilers/Named/WfFromUnit | 0m01.70s || -0m00.02s | -1.17%
0m01.67s | Util/Tuple | 0m01.63s || +0m00.04s | +2.45%
0m01.66s | Arithmetic/CoreUnfolder | 0m01.66s || +0m00.00s | +0.00%
0m01.64s | secp256k1_64.c | 0m01.50s || +0m00.13s | +9.33%
0m01.63s | p256_64.c | 0m01.53s || +0m00.09s | +6.53%
0m01.56s | Specific/Framework/ReificationTypes | 0m01.71s || -0m00.14s | -8.77%
0m01.53s | p224_64.c | 0m01.52s || +0m00.01s | +0.65%
0m01.52s | Specific/Framework/ArithmeticSynthesis/Base | 0m01.49s || +0m00.03s | +2.01%
0m01.50s | Compilers/Relations | 0m01.48s || +0m00.02s | +1.35%
0m01.47s | Specific/Framework/OutputType | 0m01.48s || -0m00.01s | -0.67%
0m01.45s | Compilers/Named/InterpretToPHOASWf | 0m01.48s || -0m00.03s | -2.02%
0m01.45s | Experiments/NewPipeline/CLI | 0m01.41s || +0m00.04s | +2.83%
0m01.40s | LegacyArithmetic/Double/Proofs/BitwiseOr | 0m01.28s || +0m00.11s | +9.37%
0m01.39s | curve25519_64.c | 0m01.38s || +0m00.01s | +0.72%
0m01.37s | Arithmetic/PrimeFieldTheorems | 0m01.30s || +0m00.07s | +5.38%
0m01.32s | Curves/Edwards/XYZT/Precomputed | 0m01.28s || +0m00.04s | +3.12%
0m01.30s | Specific/Framework/ArithmeticSynthesis/Karatsuba | 0m01.34s || -0m00.04s | -2.98%
0m01.28s | Util/QUtil | 0m01.52s || -0m00.24s | -15.78%
0m01.25s | Util/ZUtil/Testbit | 0m01.06s || +0m00.18s | +17.92%
0m01.23s | Experiments/NewPipeline/StandaloneOCamlMain | 0m01.23s || +0m00.00s | +0.00%
0m01.22s | Arithmetic/Saturated/CoreUnfolder | 0m01.12s || +0m00.09s | +8.92%
0m01.20s | Experiments/NewPipeline/Language | 0m01.22s || -0m00.02s | -1.63%
0m01.20s | Experiments/NewPipeline/StandaloneHaskellMain | 0m01.30s || -0m00.10s | -7.69%
0m01.19s | LegacyArithmetic/Double/Proofs/LoadImmediate | 0m01.40s || -0m00.20s | -15.00%
0m01.18s | LegacyArithmetic/BaseSystemProofs | 0m01.24s || -0m00.06s | -4.83%
0m01.18s | Util/ZRange/BasicLemmas | 0m01.18s || +0m00.00s | +0.00%
0m01.17s | Compilers/LinearizeInterp | 0m01.19s || -0m00.02s | -1.68%
0m01.17s | Experiments/NewPipeline/CompilersTestCases | 0m01.07s || +0m00.09s | +9.34%
0m01.14s | Compilers/Z/Syntax/Util | 0m00.84s || +0m00.29s | +35.71%
0m01.11s | Compilers/MultiSizeTest | 0m01.13s || -0m00.01s | -1.76%
0m01.08s | Compilers/Z/RewriteAddToAdcWf | 0m01.08s || +0m00.00s | +0.00%
0m01.06s | Experiments/NewPipeline/AbstractInterpretation | 0m01.07s || -0m00.01s | -0.93%
0m01.00s | Experiments/NewPipeline/RewriterProofs | 0m00.90s || +0m00.09s | +11.11%
0m01.00s | Specific/X25519/C32/CurveParameters | 0m00.95s || +0m00.05s | +5.26%
0m00.96s | Arithmetic/Saturated/WrappersUnfolder | 0m01.08s || -0m00.12s | -11.11%
0m00.96s | Compilers/Named/InterpretToPHOASInterp | 0m01.02s || -0m00.06s | -5.88%
0m00.96s | Specific/Framework/SynthesisFramework | 0m01.10s || -0m00.14s | -12.72%
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0m00.24s | Util/ZUtil/Zselect | 0m00.25s || -0m00.01s | -4.00%
Diffstat (limited to 'src')
89 files changed, 2334 insertions, 1858 deletions
diff --git a/src/Arithmetic/BarrettReduction/Generalized.v b/src/Arithmetic/BarrettReduction/Generalized.v index 9fa37721a..c2885bc77 100644 --- a/src/Arithmetic/BarrettReduction/Generalized.v +++ b/src/Arithmetic/BarrettReduction/Generalized.v @@ -9,7 +9,15 @@ base ([b]), exponent ([k]), and the [offset] than those given in the HAC. *) Require Import Coq.ZArith.ZArith Coq.micromega.Psatz. -Require Import Crypto.Util.ZUtil Crypto.Util.Tactics.BreakMatch. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Pow. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.SimplifyFractionsLe. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.ZSimplify. +Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z_scope. @@ -94,17 +102,17 @@ Section barrett. : q * n <= a. Proof using a_nonneg a_small base_good k_big_enough m_good n_pos n_reasonable offset_nonneg. subst q r m. - assert (0 < b^(k-offset)). zero_bounds. - assert (0 < b^(k+offset)) by zero_bounds. - assert (0 < b^(2 * k)) by zero_bounds. + assert (0 < b^(k-offset)). Z.zero_bounds. + assert (0 < b^(k+offset)) by Z.zero_bounds. + assert (0 < b^(2 * k)) by Z.zero_bounds. Z.simplify_fractions_le. autorewrite with pull_Zpow pull_Zdiv zsimplify; reflexivity. Qed. Lemma q_nice : { b : bool * bool | q = a / n + (if fst b then -1 else 0) + (if snd b then -1 else 0) }. Proof using a_nonneg a_small base_good k_big_enough m_good n_large n_pos n_reasonable offset_nonneg. - assert (0 < b^(k+offset)) by zero_bounds. - assert (0 < b^(k-offset)) by zero_bounds. + assert (0 < b^(k+offset)) by Z.zero_bounds. + assert (0 < b^(k-offset)) by Z.zero_bounds. assert (a / b^(k-offset) <= b^(2*k) / b^(k-offset)) by auto with zarith lia. assert (a / b^(k-offset) <= b^(k+offset)) by (autorewrite with pull_Zpow zsimplify in *; assumption). subst q r m. diff --git a/src/Arithmetic/BarrettReduction/HAC.v b/src/Arithmetic/BarrettReduction/HAC.v index 70661ee96..c9fb2f16f 100644 --- a/src/Arithmetic/BarrettReduction/HAC.v +++ b/src/Arithmetic/BarrettReduction/HAC.v @@ -9,7 +9,13 @@ have to carry around extra precision), but requires more stringint conditions on the base ([b]), exponent ([k]), and the [offset]. *) Require Import Coq.ZArith.ZArith Coq.micromega.Psatz. -Require Import Crypto.Util.ZUtil Crypto.Util.Tactics.BreakMatch. +Require Import Crypto.Util.Tactics.BreakMatch. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Hints. +Require Import Crypto.Util.ZUtil.ZSimplify. Local Open Scope Z_scope. @@ -72,8 +78,8 @@ Section barrett. Let R := x mod m. Lemma q3_nice : { b : bool * bool | q3 = Q + (if fst b then -1 else 0) + (if snd b then -1 else 0) }. Proof using base_pos k_big_enough m_large m_pos m_small offset_nonneg x_nonneg x_small μ_good. - assert (0 < b^(k+offset)) by zero_bounds. - assert (0 < b^(k-offset)) by zero_bounds. + assert (0 < b^(k+offset)) by Z.zero_bounds. + assert (0 < b^(k-offset)) by Z.zero_bounds. assert (x / b^(k-offset) <= b^(2*k) / b^(k-offset)) by auto with zarith lia. assert (x / b^(k-offset) <= b^(k+offset)) by (autorewrite with pull_Zpow zsimplify in *; assumption). subst q1 q2 q3 Q r_mod_3m r_mod_3m_orig r1 r2 R μ. diff --git a/src/Arithmetic/BarrettReduction/RidiculousFish.v b/src/Arithmetic/BarrettReduction/RidiculousFish.v index b9697839d..af030dcfe 100644 --- a/src/Arithmetic/BarrettReduction/RidiculousFish.v +++ b/src/Arithmetic/BarrettReduction/RidiculousFish.v @@ -1,5 +1,5 @@ Require Import Crypto.Util.Notations. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Hints.ZArith. Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. Require Import Coq.ZArith.ZArith. Require Import Coq.micromega.Lia. diff --git a/src/Arithmetic/BarrettReduction/Wikipedia.v b/src/Arithmetic/BarrettReduction/Wikipedia.v index 69ce10c4b..46f831281 100644 --- a/src/Arithmetic/BarrettReduction/Wikipedia.v +++ b/src/Arithmetic/BarrettReduction/Wikipedia.v @@ -1,7 +1,11 @@ (*** Barrett Reduction *) (** This file implements Barrett Reduction on [Z]. We follow Wikipedia. *) Require Import Coq.ZArith.ZArith Coq.micromega.Psatz. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.SimplifyFractionsLe. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Modulo. Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z_scope. @@ -80,7 +84,7 @@ Section barrett. : q * n <= a. Proof using a_nonneg k_good m_good n_pos n_reasonable. pose proof k_nonnegative; subst q r m. - assert (0 <= 2^(k-1)) by zero_bounds. + assert (0 <= 2^(k-1)) by Z.zero_bounds. Z.simplify_fractions_le. Qed. diff --git a/src/Arithmetic/Core.v b/src/Arithmetic/Core.v index 98600d9a3..48046d7e3 100644 --- a/src/Arithmetic/Core.v +++ b/src/Arithmetic/Core.v @@ -246,8 +246,9 @@ Local Open Scope Z_scope. Require Import Crypto.Algebra.Nsatz. Require Import Crypto.Util.Decidable Crypto.Util.LetIn. -Require Import Crypto.Util.ZUtil Crypto.Util.ListUtil Crypto.Util.Sigma. +Require Import Crypto.Util.ListUtil Crypto.Util.Sigma. Require Import Crypto.Util.CPSUtil Crypto.Util.Prod. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. Require Import Crypto.Util.ZUtil.Zselect. Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.ZUtil.Definitions. @@ -257,6 +258,7 @@ Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.UniquePose. Require Import Crypto.Util.Tactics.VM. Require Import Crypto.Util.IdfunWithAlt. +Require Import Crypto.Util.Notations. Require Import Coq.Lists.List. Import ListNotations. Require Crypto.Util.Tuple. Local Notation tuple := Tuple.tuple. diff --git a/src/Arithmetic/Karatsuba.v b/src/Arithmetic/Karatsuba.v index ad5e25be8..1873e5ef1 100644 --- a/src/Arithmetic/Karatsuba.v +++ b/src/Arithmetic/Karatsuba.v @@ -1,10 +1,11 @@ Require Import Coq.ZArith.ZArith. Require Import Coq.micromega.Lia. Require Import Crypto.Algebra.Nsatz. -Require Import Crypto.Util.ZUtil Crypto.Util.LetIn Crypto.Util.CPSUtil. +Require Import Crypto.Util.LetIn Crypto.Util.CPSUtil. Require Import Crypto.Arithmetic.Core. Import B. Import Positional. Require Import Crypto.Util.Tuple. Require Import Crypto.Util.IdfunWithAlt. +Require Import Crypto.Util.ZUtil.EquivModulo. Local Open Scope Z_scope. Section Karatsuba. diff --git a/src/Arithmetic/ModularArithmeticTheorems.v b/src/Arithmetic/ModularArithmeticTheorems.v index 666dd54eb..d1fcd80b9 100644 --- a/src/Arithmetic/ModularArithmeticTheorems.v +++ b/src/Arithmetic/ModularArithmeticTheorems.v @@ -8,7 +8,9 @@ Require Export Coq.setoid_ring.Ring_theory Coq.setoid_ring.Ring_tac. Require Import Crypto.Algebra.Hierarchy Crypto.Algebra.ScalarMult. Require Crypto.Algebra.Ring Crypto.Algebra.Field. -Require Import Crypto.Util.Decidable Crypto.Util.ZUtil. +Require Import Crypto.Util.Decidable. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. Require Export Crypto.Util.FixCoqMistakes. Module F. diff --git a/src/Arithmetic/MontgomeryReduction/Definition.v b/src/Arithmetic/MontgomeryReduction/Definition.v index 04c097460..e5cdcc603 100644 --- a/src/Arithmetic/MontgomeryReduction/Definition.v +++ b/src/Arithmetic/MontgomeryReduction/Definition.v @@ -2,7 +2,7 @@ (** This file implements Montgomery Form, Montgomery Reduction, and Montgomery Multiplication on [Z]. We follow Wikipedia. *) Require Import Coq.ZArith.ZArith. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.EquivModulo. Require Import Crypto.Util.Notations. Local Open Scope Z_scope. diff --git a/src/Arithmetic/MontgomeryReduction/Proofs.v b/src/Arithmetic/MontgomeryReduction/Proofs.v index 5f459e52d..e6be440fb 100644 --- a/src/Arithmetic/MontgomeryReduction/Proofs.v +++ b/src/Arithmetic/MontgomeryReduction/Proofs.v @@ -4,7 +4,9 @@ Wikipedia. *) Require Import Coq.ZArith.ZArith Coq.micromega.Psatz Coq.Structures.Equalities. Require Import Crypto.Arithmetic.MontgomeryReduction.Definition. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.EquivModulo. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.SimplifyRepeatedIfs. Require Import Crypto.Util.Notations. diff --git a/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Dependent/Proofs.v b/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Dependent/Proofs.v index 266fc1f6f..3dd7fc0b3 100644 --- a/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Dependent/Proofs.v +++ b/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Dependent/Proofs.v @@ -4,11 +4,19 @@ Require Import Coq.ZArith.BinInt Coq.ZArith.ZArith Coq.ZArith.Zdiv Coq.micromega Require Import Crypto.Util.LetIn. Require Import Crypto.Util.Prod. Require Import Crypto.Util.NatUtil. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Arithmetic.ModularArithmeticTheorems Crypto.Spec.ModularArithmetic. Require Import Crypto.Arithmetic.MontgomeryReduction.WordByWord.Abstract.Dependent.Definition. Require Import Crypto.Algebra.Ring. Require Import Crypto.Util.ZUtil.MulSplit. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.EquivModulo. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. +Require Import Crypto.Util.ZUtil.Tactics.PeelLe. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.Sigma. Require Import Crypto.Util.Tactics.SetEvars. Require Import Crypto.Util.Tactics.SubstEvars. diff --git a/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Proofs.v b/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Proofs.v index 9ca2cf4e9..9eabc5ce4 100644 --- a/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Proofs.v +++ b/src/Arithmetic/MontgomeryReduction/WordByWord/Abstract/Proofs.v @@ -4,11 +4,19 @@ Require Import Coq.ZArith.BinInt Coq.ZArith.ZArith Coq.ZArith.Zdiv Coq.micromega Require Import Crypto.Util.LetIn. Require Import Crypto.Util.Prod. Require Import Crypto.Util.NatUtil. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Arithmetic.ModularArithmeticTheorems Crypto.Spec.ModularArithmetic. Require Import Crypto.Arithmetic.MontgomeryReduction.WordByWord.Abstract.Definition. Require Import Crypto.Algebra.Ring. Require Import Crypto.Util.ZUtil.MulSplit. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.EquivModulo. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. +Require Import Crypto.Util.ZUtil.Tactics.PeelLe. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.Sigma. Require Import Crypto.Util.Tactics.SetEvars. Require Import Crypto.Util.Tactics.SubstEvars. diff --git a/src/Arithmetic/MontgomeryReduction/WordByWord/Proofs.v b/src/Arithmetic/MontgomeryReduction/WordByWord/Proofs.v index 6d2925d6b..35c9e377b 100644 --- a/src/Arithmetic/MontgomeryReduction/WordByWord/Proofs.v +++ b/src/Arithmetic/MontgomeryReduction/WordByWord/Proofs.v @@ -6,7 +6,6 @@ Require Import Crypto.Arithmetic.Saturated.MontgomeryAPI. Require Import Crypto.Arithmetic.MontgomeryReduction.WordByWord.Abstract.Dependent.Definition. Require Import Crypto.Arithmetic.MontgomeryReduction.WordByWord.Abstract.Dependent.Proofs. Require Import Crypto.Arithmetic.MontgomeryReduction.WordByWord.Definition. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z_scope. diff --git a/src/Arithmetic/PrimeFieldTheorems.v b/src/Arithmetic/PrimeFieldTheorems.v index a4305a849..47ea89e31 100644 --- a/src/Arithmetic/PrimeFieldTheorems.v +++ b/src/Arithmetic/PrimeFieldTheorems.v @@ -8,7 +8,10 @@ Require Import Crypto.Util.NumTheoryUtil. Require Import Coq.Classes.Morphisms Coq.Setoids.Setoid. Require Import Coq.ZArith.BinInt Coq.NArith.BinNat Coq.ZArith.ZArith Coq.ZArith.Znumtheory Coq.NArith.NArith. (* import Zdiv before Znumtheory *) Require Import Coq.Logic.Eqdep_dec. -Require Import Crypto.Util.NumTheoryUtil Crypto.Util.ZUtil. +Require Import Crypto.Util.NumTheoryUtil. +Require Import Crypto.Util.ZUtil.Odd. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. Require Import Crypto.Util.Tactics.SpecializeBy. Require Import Crypto.Util.Decidable. Require Export Crypto.Util.FixCoqMistakes. @@ -105,7 +108,7 @@ Module F. repeat match goal with | |- _ => progress subst | |- _ => progress rewrite ?F.pow_0_l, <-?F.pow_add_r - | |- _ => progress rewrite <-?Z2N.inj_0, <-?Z2N.inj_add by zero_bounds + | |- _ => progress rewrite <-?Z2N.inj_0, <-?Z2N.inj_add by Z.zero_bounds | |- _ => rewrite <-@euler_criterion by auto | |- ?x ^ (?f _) = ?a <-> ?x ^ (?f _) = ?a => do 3 f_equiv; [ ] | |- _ => rewrite !Zmod_odd in *; repeat (break_match; break_match_hyps); omega @@ -114,10 +117,10 @@ Module F. | |- (?x ^ Z.to_N ?a = 1) <-> _ => transitivity (x ^ Z.to_N a * x ^ Z.to_N 1 = x); [ rewrite F.pow_1_r, Algebra.Field.mul_cancel_l_iff by auto; reflexivity | ] - | |- (_ <> _)%N => rewrite Z2N.inj_iff by zero_bounds - | |- (?a <> 0)%Z => assert (0 < a) by zero_bounds; omega + | |- (_ <> _)%N => rewrite Z2N.inj_iff by Z.zero_bounds + | |- (?a <> 0)%Z => assert (0 < a) by Z.zero_bounds; omega | |- (_ = _)%Z => replace 4 with (2 * 2)%Z in * by ring; - rewrite <-Z.div_div by zero_bounds; + rewrite <-Z.div_div by Z.zero_bounds; rewrite Z.add_diag, Z.mul_add_distr_l, Z.mul_div_eq by omega end. Qed. @@ -166,8 +169,8 @@ Module F. replace (Z.to_N (q / 8 + 1) * (2*2))%N with (Z.to_N (q / 2 + 2))%N. Focus 2. { (* this is a boring but gnarly proof :/ *) change (2*2)%N with (Z.to_N 4). - rewrite <- Z2N.inj_mul by zero_bounds. - apply Z2N.inj_iff; try zero_bounds. + rewrite <- Z2N.inj_mul by Z.zero_bounds. + apply Z2N.inj_iff; try Z.zero_bounds. rewrite <- Z.mul_cancel_l with (p := 2) by omega. ring_simplify. rewrite Z.mul_div_eq by omega. @@ -179,7 +182,7 @@ Module F. ring. } Unfocus. - rewrite Z2N.inj_add, F.pow_add_r by zero_bounds. + rewrite Z2N.inj_add, F.pow_add_r by Z.zero_bounds. replace (x ^ Z.to_N (q / 2)) with (F.of_Z q 1) by (symmetry; apply @euler_criterion; eauto). change (Z.to_N 2) with 2%N; ring. @@ -215,8 +218,8 @@ Module F. repeat match goal with | |- _ => progress subst | |- _ => progress rewrite ?F.pow_0_l - | |- (_ <> _)%N => rewrite <-Z2N.inj_0, Z2N.inj_iff by zero_bounds - | |- (?a <> 0)%Z => assert (0 < a) by zero_bounds; omega + | |- (_ <> _)%N => rewrite <-Z2N.inj_0, Z2N.inj_iff by Z.zero_bounds + | |- (?a <> 0)%Z => assert (0 < a) by Z.zero_bounds; omega | |- _ => congruence end. break_match; diff --git a/src/Arithmetic/Saturated/Core.v b/src/Arithmetic/Saturated/Core.v index 189e76113..9c796ad57 100644 --- a/src/Arithmetic/Saturated/Core.v +++ b/src/Arithmetic/Saturated/Core.v @@ -9,7 +9,12 @@ Require Import Crypto.Arithmetic.Core. Require Import Crypto.Util.LetIn Crypto.Util.CPSUtil. Require Import Crypto.Util.Tuple Crypto.Util.ListUtil. Require Import Crypto.Util.Tactics.BreakMatch. -Require Import Crypto.Util.Decidable Crypto.Util.ZUtil. +Require Import Crypto.Util.Decidable. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. +Require Import Crypto.Util.ZUtil.Le. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. Require Import Crypto.Util.NatUtil. Require Import Crypto.Util.Tactics.SpecializeBy. Local Notation "A ^ n" := (tuple A n) : type_scope. @@ -273,7 +278,7 @@ Module Columns. rewrite <-Z.mul_mod_distr_l with (c:=a) by omega. rewrite Z.mul_add_distr_l, Z.mul_div_eq, <-Z.add_mod_full by omega. f_equal; ring. } - { split; [zero_bounds|]. + { split; [Z.zero_bounds|]. apply Z.lt_le_trans with (m:=a*(b-1)+a); [|ring_simplify; omega]. apply Z.add_le_lt_mono; try apply Z.mul_le_mono_nonneg_l; omega. } Qed. diff --git a/src/Arithmetic/Saturated/Freeze.v b/src/Arithmetic/Saturated/Freeze.v index 78a86bc73..d8e7f4b5e 100644 --- a/src/Arithmetic/Saturated/Freeze.v +++ b/src/Arithmetic/Saturated/Freeze.v @@ -7,9 +7,11 @@ Require Import Crypto.Arithmetic.Saturated.Core. Require Import Crypto.Arithmetic.Saturated.Wrappers. Require Import Crypto.Util.ZUtil.AddGetCarry. Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. +Require Import Crypto.Util.ZUtil.Le. Require Import Crypto.Util.ZUtil.CPS. Require Import Crypto.Util.Tactics.BreakMatch. -Require Import Crypto.Util.Decidable Crypto.Util.ZUtil. +Require Import Crypto.Util.Decidable. Require Import Crypto.Util.Tuple Crypto.Util.LetIn. Local Notation "A ^ n" := (tuple A n) : type_scope. diff --git a/src/Arithmetic/Saturated/MontgomeryAPI.v b/src/Arithmetic/Saturated/MontgomeryAPI.v index 6dee1d22b..d08fe7a8b 100644 --- a/src/Arithmetic/Saturated/MontgomeryAPI.v +++ b/src/Arithmetic/Saturated/MontgomeryAPI.v @@ -11,12 +11,18 @@ Require Import Crypto.Arithmetic.Saturated.AddSub. Require Import Crypto.Util.LetIn Crypto.Util.CPSUtil. Require Import Crypto.Util.Tuple Crypto.Util.LetIn. Require Import Crypto.Util.Decidable. -Require Import Crypto.Util.ZUtil Crypto.Util.ListUtil. +Require Import Crypto.Util.ListUtil. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.Modulo. Require Import Crypto.Util.ZUtil.Definitions. Require Import Crypto.Util.ZUtil.CPS. Require Import Crypto.Util.ZUtil.Zselect. Require Import Crypto.Util.ZUtil.AddGetCarry. Require Import Crypto.Util.ZUtil.MulSplit. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Opp. Require Import Crypto.Util.Tactics.UniquePose. Local Notation "A ^ n" := (tuple A n) : type_scope. diff --git a/src/Arithmetic/Saturated/UniformWeight.v b/src/Arithmetic/Saturated/UniformWeight.v index bd351b6cd..bf069f2d6 100644 --- a/src/Arithmetic/Saturated/UniformWeight.v +++ b/src/Arithmetic/Saturated/UniformWeight.v @@ -4,7 +4,9 @@ Local Open Scope Z_scope. Require Import Crypto.Arithmetic.Core. Require Import Crypto.Arithmetic.Saturated.Core. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Le. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Tactics.PeelLe. Require Import Crypto.Util.LetIn Crypto.Util.Tuple. Local Notation "A ^ n" := (tuple A n) : type_scope. @@ -88,4 +90,4 @@ Section UniformWeight. Definition small {n} (p : Z^n) : Prop := forall x, In x (to_list _ p) -> 0 <= x < bound. -End UniformWeight.
\ No newline at end of file +End UniformWeight. diff --git a/src/Compilers/Named/MapCastInterp.v b/src/Compilers/Named/MapCastInterp.v index e6dbb01ed..c38eadd9c 100644 --- a/src/Compilers/Named/MapCastInterp.v +++ b/src/Compilers/Named/MapCastInterp.v @@ -11,7 +11,6 @@ Require Import Crypto.Compilers.Named.ContextProperties.SmartMap. Require Import Crypto.Compilers.Named.InterpSideConditions. Require Import Crypto.Compilers.Named.InterpSideConditionsInterp. Require Import Crypto.Compilers.Named.MapCast. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.Bool. Require Import Crypto.Util.Option. Require Import Crypto.Util.Sigma. diff --git a/src/Compilers/Named/MapCastWf.v b/src/Compilers/Named/MapCastWf.v index b04613fa7..eb141cad1 100644 --- a/src/Compilers/Named/MapCastWf.v +++ b/src/Compilers/Named/MapCastWf.v @@ -11,7 +11,6 @@ Require Import Crypto.Compilers.Named.ContextProperties.SmartMap. Require Import Crypto.Compilers.Named.Wf. Require Import Crypto.Compilers.Named.MapCast. Require Import Crypto.Util.PointedProp. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.Bool. Require Import Crypto.Util.Option. Require Import Crypto.Util.Prod. diff --git a/src/Compilers/Z/ArithmeticSimplifierInterp.v b/src/Compilers/Z/ArithmeticSimplifierInterp.v index c34089a60..c1c841c9f 100644 --- a/src/Compilers/Z/ArithmeticSimplifierInterp.v +++ b/src/Compilers/Z/ArithmeticSimplifierInterp.v @@ -10,8 +10,17 @@ Require Import Crypto.Compilers.Z.OpInversion. Require Import Crypto.Compilers.Z.ArithmeticSimplifier. Require Import Crypto.Compilers.Z.ArithmeticSimplifierUtil. Require Import Crypto.Compilers.Z.Syntax.Equality. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Hints. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.ZSimplify.Core. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Z2Nat. Require Import Crypto.Util.ZUtil.AddGetCarry. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. Require Import Crypto.Util.Option. Require Import Crypto.Util.Prod. Require Import Crypto.Util.Sum. diff --git a/src/Compilers/Z/ArithmeticSimplifierWf.v b/src/Compilers/Z/ArithmeticSimplifierWf.v index ff690688a..4efa7445a 100644 --- a/src/Compilers/Z/ArithmeticSimplifierWf.v +++ b/src/Compilers/Z/ArithmeticSimplifierWf.v @@ -10,7 +10,6 @@ Require Import Crypto.Compilers.Z.OpInversion. Require Import Crypto.Compilers.Z.ArithmeticSimplifier. Require Import Crypto.Compilers.Z.Syntax.Equality. Require Import Crypto.Compilers.Z.Syntax.Util. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.Option. Require Import Crypto.Util.Sum. Require Import Crypto.Util.Prod. diff --git a/src/Compilers/Z/Bounds/InterpretationLemmas/IsBoundedBy.v b/src/Compilers/Z/Bounds/InterpretationLemmas/IsBoundedBy.v index 45566839a..b23e0ff1b 100644 --- a/src/Compilers/Z/Bounds/InterpretationLemmas/IsBoundedBy.v +++ b/src/Compilers/Z/Bounds/InterpretationLemmas/IsBoundedBy.v @@ -8,9 +8,13 @@ Require Import Crypto.Compilers.Z.Bounds.Interpretation. Require Import Crypto.Compilers.Z.Bounds.InterpretationLemmas.Tactics. Require Import Crypto.Compilers.SmartMap. Require Import Crypto.Util.ZRange.CornersMonotoneBounds. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.ZUtil.Stabilization. Require Import Crypto.Util.ZUtil.MulSplit. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.LandLorShiftBounds. +Require Import Crypto.Util.ZUtil.Morphisms. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.PointedProp. Require Import Crypto.Util.Bool. Require Import Crypto.Util.FixedWordSizesEquality. diff --git a/src/Compilers/Z/Bounds/InterpretationLemmas/PullCast.v b/src/Compilers/Z/Bounds/InterpretationLemmas/PullCast.v index 1701b4688..7ffe0beb1 100644 --- a/src/Compilers/Z/Bounds/InterpretationLemmas/PullCast.v +++ b/src/Compilers/Z/Bounds/InterpretationLemmas/PullCast.v @@ -7,7 +7,12 @@ Require Import Crypto.Compilers.Syntax. Require Import Crypto.Compilers.Z.Bounds.Interpretation. Require Import Crypto.Compilers.Z.Bounds.InterpretationLemmas.Tactics. Require Import Crypto.Compilers.SmartMap. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Morphisms. +Require Import Crypto.Util.ZUtil.Log2. +Require Import Crypto.Util.ZUtil.Pow2. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.PeelLe. Require Import Crypto.Util.Bool. Require Import Crypto.Util.FixedWordSizesEquality. Require Import Crypto.Util.Option. diff --git a/src/Compilers/Z/Bounds/InterpretationLemmas/Tactics.v b/src/Compilers/Z/Bounds/InterpretationLemmas/Tactics.v index 71f4b758b..6486b2e00 100644 --- a/src/Compilers/Z/Bounds/InterpretationLemmas/Tactics.v +++ b/src/Compilers/Z/Bounds/InterpretationLemmas/Tactics.v @@ -1,7 +1,14 @@ Require Import Coq.ZArith.ZArith. Require Import Coq.micromega.Psatz. Require Import Crypto.Compilers.Z.Bounds.Interpretation. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Hints. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.ZSimplify.Core. +Require Import Crypto.Util.ZUtil.Log2. +Require Import Crypto.Util.ZUtil.Shift. +Require Import Crypto.Util.ZUtil.LandLorShiftBounds. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Notations. Require Import Crypto.Util.ZRange.Operations. Require Import Crypto.Util.Bool. Require Import Crypto.Util.FixedWordSizesEquality. diff --git a/src/Compilers/Z/Bounds/Relax.v b/src/Compilers/Z/Bounds/Relax.v index 328ea5f48..8178592ed 100644 --- a/src/Compilers/Z/Bounds/Relax.v +++ b/src/Compilers/Z/Bounds/Relax.v @@ -15,7 +15,8 @@ Require Import Crypto.Util.Tactics.SpecializeBy. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.SplitInContext. Require Import Crypto.Util.Option. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Log2. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.Bool. Local Lemma helper logsz v diff --git a/src/Compilers/Z/Syntax/Util.v b/src/Compilers/Z/Syntax/Util.v index a4f5506fe..110e6b816 100644 --- a/src/Compilers/Z/Syntax/Util.v +++ b/src/Compilers/Z/Syntax/Util.v @@ -1,4 +1,5 @@ Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. Require Import Crypto.Compilers.Syntax. Require Import Crypto.Compilers.SmartMap. Require Import Crypto.Compilers.Wf. @@ -6,9 +7,7 @@ Require Import Crypto.Compilers.TypeUtil. Require Import Crypto.Compilers.TypeInversion. Require Import Crypto.Compilers.Z.Syntax. Require Import Crypto.Util.FixedWordSizesEquality. -Require Import Crypto.Util.NatUtil. Require Import Crypto.Util.HProp. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.DestructHead. Require Import Crypto.Util.Notations. @@ -113,8 +112,8 @@ Proof. | [ H : ?leb _ _ = false |- _ ] => apply Compare_dec.leb_iff_conv in H | [ H : TWord _ = TWord _ |- _ ] => inversion H; clear H end - | rewrite ZToWord_wordToZ_ZToWord by omega * - | rewrite wordToZ_ZToWord_wordToZ by omega * + | rewrite ZToWord_wordToZ_ZToWord by lia + | rewrite wordToZ_ZToWord_wordToZ by lia | rewrite wordToZ_ZToWord by assumption | rewrite ZToWord_wordToZ_ZToWord_small by omega ]. Qed. @@ -182,8 +181,8 @@ Proof. | [ H : ?leb _ _ = false |- _ ] => apply Compare_dec.leb_iff_conv in H | [ H : TWord _ = TWord _ |- _ ] => inversion H; clear H end - | rewrite ZToWord_wordToZ_ZToWord by omega * - | rewrite wordToZ_ZToWord_wordToZ by omega * ]. + | rewrite ZToWord_wordToZ_ZToWord by lia + | rewrite wordToZ_ZToWord_wordToZ by lia ]. Qed. Lemma ZToInterp_eq_inj {a} x y diff --git a/src/Curves/Montgomery/XZProofs.v b/src/Curves/Montgomery/XZProofs.v index 650ed6920..632ec7bd9 100644 --- a/src/Curves/Montgomery/XZProofs.v +++ b/src/Curves/Montgomery/XZProofs.v @@ -3,7 +3,9 @@ Require Import Crypto.Algebra.ScalarMult. Require Import Crypto.Util.Sum Crypto.Util.Prod Crypto.Util.LetIn. Require Import Crypto.Util.Decidable. Require Import Crypto.Util.Tuple. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Shift. Require Import Crypto.Util.ZUtil.Peano. Require Import Crypto.Util.Tactics.SetoidSubst. Require Import Crypto.Util.Tactics.SpecializeBy. @@ -248,7 +250,7 @@ Module M. Proof. t. Qed. Lemma transitive_eq {p} q {r} : projective q -> eq p q -> eq q r -> eq p r. Proof. t. Qed. - + Lemma projective_to_xz Q : projective (to_xz Q). Proof. t. Qed. diff --git a/src/Experiments/NewPipeline/Arithmetic.v b/src/Experiments/NewPipeline/Arithmetic.v index 410cb8dfe..56a7051af 100644 --- a/src/Experiments/NewPipeline/Arithmetic.v +++ b/src/Experiments/NewPipeline/Arithmetic.v @@ -21,8 +21,15 @@ Require Import Crypto.Util.Prod. Require Import Crypto.Util.Sum. Require Import Crypto.Util.Bool. Require Import Crypto.Util.Sigma. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.ZUtil.Modulo Crypto.Util.ZUtil.Div Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.PeelLe. +Require Import Crypto.Util.ZUtil.Tactics.LinearSubstitute. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. +Require Import Crypto.Util.ZUtil.Opp. +Require Import Crypto.Util.ZUtil.Log2. +Require Import Crypto.Util.ZUtil.Le. Require Import Crypto.Util.ZUtil.Hints.PullPush. Require Import Crypto.Util.ZUtil.AddGetCarry Crypto.Util.ZUtil.MulSplit. Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. @@ -37,7 +44,7 @@ Require Import Crypto.Util.ZUtil.Sorting. Require Import Crypto.Util.ZUtil.CC Crypto.Util.ZUtil.Rshi. Require Import Crypto.Util.ZUtil.Zselect Crypto.Util.ZUtil.AddModulo. Require Import Crypto.Util.ZUtil.AddGetCarry Crypto.Util.ZUtil.MulSplit. -Require Import Crypto.Util.ZUtil Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Hints.Core. Require Import Crypto.Util.ZUtil.Modulo Crypto.Util.ZUtil.Div. Require Import Crypto.Util.ZUtil.Hints.PullPush. Require Import Crypto.Util.ZUtil.EquivModulo. diff --git a/src/Experiments/NewPipeline/Toplevel1.v b/src/Experiments/NewPipeline/Toplevel1.v index 869abea8e..01018c3e2 100644 --- a/src/Experiments/NewPipeline/Toplevel1.v +++ b/src/Experiments/NewPipeline/Toplevel1.v @@ -30,10 +30,17 @@ Require Import Crypto.Util.ZUtil.Rshi. Require Import Crypto.Util.Option. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.SpecializeBy. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Log2. Require Import Crypto.Util.ZUtil.Zselect. Require Import Crypto.Util.ZUtil.AddModulo. Require Import Crypto.Util.ZUtil.CC. +Require Import Crypto.Util.ZUtil.EquivModulo. +Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Modulo. Require Import Crypto.Arithmetic.MontgomeryReduction.Definition. Require Import Crypto.Arithmetic.MontgomeryReduction.Proofs. Require Import Crypto.Util.ZUtil.ModInv. diff --git a/src/Experiments/NewPipeline/Toplevel2.v b/src/Experiments/NewPipeline/Toplevel2.v index e35e14616..ffb6b0215 100644 --- a/src/Experiments/NewPipeline/Toplevel2.v +++ b/src/Experiments/NewPipeline/Toplevel2.v @@ -16,6 +16,7 @@ Require Import Crypto.Util.LetIn. Require Import Crypto.Arithmetic.PrimeFieldTheorems. Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. Require Import Crypto.Util.Tactics.SplitInContext. Require Import Crypto.Util.Tactics.SubstEvars. Require Import Crypto.Util.Tactics.DestructHead. @@ -29,10 +30,14 @@ Require Import Crypto.Util.ZUtil.Rshi. Require Import Crypto.Util.Option. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.SpecializeBy. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.ZUtil.Zselect. Require Import Crypto.Util.ZUtil.AddModulo. Require Import Crypto.Util.ZUtil.CC. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.EquivModulo. Require Import Crypto.Arithmetic.MontgomeryReduction.Definition. Require Import Crypto.Arithmetic.MontgomeryReduction.Proofs. Require Import Crypto.Util.ErrorT. @@ -2087,24 +2092,13 @@ Module ProdEquiv. let new_ctx := fun n => if reg_eqb n rd then result mod wordmax else ctx n in interp256 cont new_cc new_ctx. Proof. reflexivity. Qed. - (* TODO : move *) - Lemma tuple_map_ext {A B} (f g : A -> B) n (t : tuple A n) : - (forall x : A, f x = g x) -> - Tuple.map f t = Tuple.map g t. - Proof. - destruct n; [reflexivity|]; cbn in *. - induction n; cbn in *; intro H; auto; [ ]. - rewrite IHn by assumption. - rewrite H; reflexivity. - Qed. - Lemma interp_state_equiv e : forall cc ctx cc' ctx', cc = cc' -> (forall r, ctx r = ctx' r) -> interp256 e cc ctx = interp256 e cc' ctx'. Proof. induction e; intros; subst; cbn; [solve[auto]|]. - apply IHe; rewrite tuple_map_ext with (g:=ctx') by auto; + apply IHe; rewrite Tuple.map_ext with (g:=ctx') by auto; [reflexivity|]. intros; break_match; auto. Qed. @@ -2116,17 +2110,6 @@ Module ProdEquiv. reflexivity. Qed. - Lemma tuple_map_ext_In {A B} (f g : A -> B) n (t : tuple A n) : - (forall x, In x (to_list n t) -> f x = g x) -> - Tuple.map f t = Tuple.map g t. - Proof. - destruct n; [reflexivity|]; cbn in *. - induction n; cbn in *; intro H; auto; [ ]. - destruct t. - rewrite IHn by auto using in_cons. - rewrite H; auto using in_eq. - Qed. - Definition value_unused r e : Prop := forall x cc ctx, interp256 e cc ctx = interp256 e cc (fun r' => if reg_eqb r' r then x else ctx r'). @@ -2141,7 +2124,7 @@ Module ProdEquiv. match goal with |- ?lhs = ?rhs => match lhs with context [Tuple.map ?f ?t] => match rhs with context [Tuple.map ?g ?t] => - rewrite (tuple_map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) + rewrite (Tuple.map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) end end end. apply interp_state_equiv; [ congruence | ]. { intros; cbv [reg_eqb] in *; break_match; congruence. } @@ -2155,7 +2138,7 @@ Module ProdEquiv. match goal with |- ?lhs = ?rhs => match lhs with context [Tuple.map ?f ?t] => match rhs with context [Tuple.map ?g ?t] => - rewrite (tuple_map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) + rewrite (Tuple.map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) end end end. apply interp_state_equiv; [ congruence | ]. { intros; cbv [reg_eqb] in *; break_match; congruence. } @@ -2649,19 +2632,6 @@ Module Barrett256. => apply interp_equivZZ_256; [ simplify_op_equiv ctx | simplify_op_equiv ctx | generalize_result] end. - (* TODO: move this lemma to ZUtil *) - Lemma testbit_neg_eq_if x n : - 0 <= n -> - - (2 ^ n) <= x < 2 ^ n -> - Z.b2z (if x <? 0 then true else Z.testbit x n) = - (x / 2 ^ n) mod 2. - Proof. - intros. break_match; Z.ltb_to_lt. - { autorewrite with zsimplify. reflexivity. } - { autorewrite with zsimplify. - rewrite Z.bits_above_pow2 by omega. - reflexivity. } - Qed. - Lemma prod_barrett_red256_correct : forall (cc_start_state : Fancy.CC.state) (* starting carry flags *) (start_context : register -> Z) (* starting register values *) @@ -2755,7 +2725,7 @@ Module Barrett256. { reflexivity. } { autorewrite with zsimplify_fast. match goal with |- context [?x mod ?m] => pose proof (Z.mod_pos_bound x m ltac:(omega)) end. - rewrite <-testbit_neg_eq_if with (n:=256) by (cbn; omega). + rewrite <-Z.testbit_neg_eq_if with (n:=256) by (cbn; omega). reflexivity. } step start_context. { reflexivity. } @@ -2763,7 +2733,7 @@ Module Barrett256. rewrite Z.mod_small with (a:=(if (if _ <? 0 then true else _) then _ else _)) (b:=2) by (break_innermost_match; omega). match goal with |- context [?a - ?b - ?c] => replace (a - b - c) with (a - (b + c)) by ring end. match goal with |- context [?x mod ?m] => pose proof (Z.mod_pos_bound x m ltac:(omega)) end. - rewrite <-testbit_neg_eq_if with (n:=256) by (break_innermost_match; cbn; omega). + rewrite <-Z.testbit_neg_eq_if with (n:=256) by (break_innermost_match; cbn; omega). reflexivity. } step start_context. { rewrite Z.bit0_eqb. @@ -2782,7 +2752,7 @@ Module Barrett256. { reflexivity. } { autorewrite with zsimplify_fast. repeat match goal with |- context [?x mod ?m] => unique pose proof (Z.mod_pos_bound x m ltac:(omega)) end. - rewrite <-testbit_neg_eq_if with (n:=256) by (cbn; omega). + rewrite <-Z.testbit_neg_eq_if with (n:=256) by (cbn; omega). reflexivity. } step start_context; [ break_innermost_match; Z.ltb_to_lt; omega | ]. reflexivity. @@ -3169,19 +3139,6 @@ Module Montgomery256. => apply interp_equivZZ_256; [ simplify_op_equiv ctx | simplify_op_equiv ctx | generalize_result] end. - (* TODO: move this lemma to ZUtil *) - Lemma testbit_neg_eq_if x y n : - 0 <= n -> - 0 <= x < 2 ^ n -> - 0 <= y < 2 ^ n -> - Z.b2z (if (x - y) <? 0 then true else Z.testbit (x - y) n) = - ((x - y) / 2 ^ n) mod 2. - Proof. - intros. rewrite Z.sub_pos_bound_div_eq by omega. - break_innermost_match; Z.ltb_to_lt; try lia; try reflexivity; [ ]. - rewrite Z.testbit_eqb, Z.div_between_0_if by omega. - break_innermost_match; Z.ltb_to_lt; try lia; reflexivity. - Qed. - Local Ltac break_ifs := repeat (break_innermost_match_step; Z.ltb_to_lt; try (exfalso; omega); []). @@ -3228,7 +3185,7 @@ Module Montgomery256. { let r := eval cbv in (2^256) in replace (2^256) with r by reflexivity. rewrite !Z.shiftl_0_r, !Z.mod_mod by omega. - apply testbit_neg_eq_if; + apply Z.testbit_neg_eq_if; let r := eval cbv in (2^256) in replace (2^256) with r by reflexivity; auto using Z.mod_pos_bound with omega. } step start_context; [ break_innermost_match; Z.ltb_to_lt; omega | ]. diff --git a/src/Experiments/SimplyTypedArithmetic.v b/src/Experiments/SimplyTypedArithmetic.v index 3745e59ff..c0c8dbdeb 100644 --- a/src/Experiments/SimplyTypedArithmetic.v +++ b/src/Experiments/SimplyTypedArithmetic.v @@ -23,13 +23,21 @@ Require Import Crypto.Util.Tactics.Head. Require Import Crypto.Util.Option. Require Import Crypto.Util.OptionList. Require Import Crypto.Util.Sum. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.ZUtil.Modulo Crypto.Util.ZUtil.Div Crypto.Util.ZUtil.Hints.Core. Require Import Crypto.Util.ZUtil.Hints.PullPush. Require Import Crypto.Util.ZUtil.AddGetCarry Crypto.Util.ZUtil.MulSplit. Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.Le. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Log2. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Shift. +Require Import Crypto.Util.ZUtil.LandLorShiftBounds. +Require Import Crypto.Util.ZUtil.Testbit. +Require Import Crypto.Util.ZUtil.Notations. Require Import Crypto.Util.Tactics.SpecializeBy. Require Import Crypto.Util.Tactics.SplitInContext. Require Import Crypto.Util.Tactics.SubstEvars. @@ -41,7 +49,7 @@ Require Import Crypto.Util.ZUtil.Definitions. Require Import Crypto.Util.ZUtil.CC Crypto.Util.ZUtil.Rshi. Require Import Crypto.Util.ZUtil.Zselect Crypto.Util.ZUtil.AddModulo. Require Import Crypto.Util.ZUtil.AddGetCarry Crypto.Util.ZUtil.MulSplit. -Require Import Crypto.Util.ZUtil Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Hints.Core. Require Import Crypto.Util.ZUtil.Modulo Crypto.Util.ZUtil.Div. Require Import Crypto.Util.ZUtil.Hints.PullPush. Require Import Crypto.Util.ZUtil.EquivModulo. @@ -1245,13 +1253,6 @@ Module Rows. rewrite Columns.length_from_associational in *; auto. Qed. - (* TODO : move *) - Lemma max_0_iff a b : Nat.max a b = 0%nat <-> (a = 0%nat /\ b = 0%nat). - Proof. - destruct a, b; try tauto. - rewrite <-Nat.succ_max_distr. - split; [ | destruct 1]; congruence. - Qed. Lemma max_column_size_zero_iff x : max_column_size x = 0%nat <-> (forall c, In c x -> c = nil). Proof. @@ -10402,24 +10403,13 @@ Module ProdEquiv. let new_ctx := fun n => if reg_eqb n rd then result mod wordmax else ctx n in interp256 cont new_cc new_ctx. Proof. reflexivity. Qed. - (* TODO : move *) - Lemma tuple_map_ext {A B} (f g : A -> B) n (t : tuple A n) : - (forall x : A, f x = g x) -> - Tuple.map f t = Tuple.map g t. - Proof. - destruct n; [reflexivity|]; cbn in *. - induction n; cbn in *; intro H; auto; [ ]. - rewrite IHn by assumption. - rewrite H; reflexivity. - Qed. - Lemma interp_state_equiv e : forall cc ctx cc' ctx', cc = cc' -> (forall r, ctx r = ctx' r) -> interp256 e cc ctx = interp256 e cc' ctx'. Proof. induction e; intros; subst; cbn; [solve[auto]|]. - apply IHe; rewrite tuple_map_ext with (g:=ctx') by auto; + apply IHe; rewrite Tuple.map_ext with (g:=ctx') by auto; [reflexivity|]. intros; break_match; auto. Qed. @@ -10431,17 +10421,6 @@ Module ProdEquiv. reflexivity. Qed. - Lemma tuple_map_ext_In {A B} (f g : A -> B) n (t : tuple A n) : - (forall x, In x (to_list n t) -> f x = g x) -> - Tuple.map f t = Tuple.map g t. - Proof. - destruct n; [reflexivity|]; cbn in *. - induction n; cbn in *; intro H; auto; [ ]. - destruct t. - rewrite IHn by auto using in_cons. - rewrite H; auto using in_eq. - Qed. - Definition value_unused r e : Prop := forall x cc ctx, interp256 e cc ctx = interp256 e cc (fun r' => if reg_eqb r' r then x else ctx r'). @@ -10456,7 +10435,7 @@ Module ProdEquiv. match goal with |- ?lhs = ?rhs => match lhs with context [Tuple.map ?f ?t] => match rhs with context [Tuple.map ?g ?t] => - rewrite (tuple_map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) + rewrite (Tuple.map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) end end end. apply interp_state_equiv; [ congruence | ]. { intros; cbv [reg_eqb] in *; break_match; congruence. } @@ -10470,7 +10449,7 @@ Module ProdEquiv. match goal with |- ?lhs = ?rhs => match lhs with context [Tuple.map ?f ?t] => match rhs with context [Tuple.map ?g ?t] => - rewrite (tuple_map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) + rewrite (Tuple.map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence) end end end. apply interp_state_equiv; [ congruence | ]. { intros; cbv [reg_eqb] in *; break_match; congruence. } @@ -11525,19 +11504,6 @@ Module Barrett256. | _ => apply interp_equivZZ_256; [ simplify_op_equiv ctx | simplify_op_equiv ctx | generalize_result] end. - (* TODO: move this lemma to ZUtil *) - Lemma testbit_neg_eq_if x n : - 0 <= n -> - - (2 ^ n) <= x < 2 ^ n -> - Z.b2z (if x <? 0 then true else Z.testbit x n) = - (x / 2 ^ n) mod 2. - Proof. - intros. break_match; Z.ltb_to_lt. - { autorewrite with zsimplify. reflexivity. } - { autorewrite with zsimplify. - rewrite Z.bits_above_pow2 by omega. - reflexivity. } - Qed. - Lemma prod_barrett_red256_correct : forall (cc_start_state : Fancy.CC.state) (* starting carry flags *) (start_context : register -> Z) (* starting register values *) @@ -11631,7 +11597,7 @@ Module Barrett256. { reflexivity. } { autorewrite with zsimplify_fast. match goal with |- context [?x mod ?m] => pose proof (Z.mod_pos_bound x m ltac:(omega)) end. - rewrite <-testbit_neg_eq_if with (n:=256) by (cbn; omega). + rewrite <-Z.testbit_neg_eq_if with (n:=256) by (cbn; omega). reflexivity. } step start_context. { reflexivity. } @@ -11639,7 +11605,7 @@ Module Barrett256. rewrite Z.mod_small with (a:=(if (if _ <? 0 then true else _) then _ else _)) (b:=2) by (break_innermost_match; omega). match goal with |- context [?a - ?b - ?c] => replace (a - b - c) with (a - (b + c)) by ring end. match goal with |- context [?x mod ?m] => pose proof (Z.mod_pos_bound x m ltac:(omega)) end. - rewrite <-testbit_neg_eq_if with (n:=256) by (break_innermost_match; cbn; omega). + rewrite <-Z.testbit_neg_eq_if with (n:=256) by (break_innermost_match; cbn; omega). reflexivity. } step start_context. { rewrite Z.bit0_eqb. @@ -11658,7 +11624,7 @@ Module Barrett256. { reflexivity. } { autorewrite with zsimplify_fast. repeat match goal with |- context [?x mod ?m] => unique pose proof (Z.mod_pos_bound x m ltac:(omega)) end. - rewrite <-testbit_neg_eq_if with (n:=256) by (cbn; omega). + rewrite <-Z.testbit_neg_eq_if with (n:=256) by (cbn; omega). reflexivity. } step start_context; [ break_innermost_match; Z.ltb_to_lt; omega | ]. reflexivity. @@ -12567,19 +12533,6 @@ Module Montgomery256. | _ => apply interp_equivZZ_256; [ simplify_op_equiv ctx | simplify_op_equiv ctx | generalize_result] end. - (* TODO: move this lemma to ZUtil *) - Lemma testbit_neg_eq_if x y n : - 0 <= n -> - 0 <= x < 2 ^ n -> - 0 <= y < 2 ^ n -> - Z.b2z (if (x - y) <? 0 then true else Z.testbit (x - y) n) = - ((x - y) / 2 ^ n) mod 2. - Proof. - intros. rewrite Z.sub_pos_bound_div_eq by omega. - break_innermost_match; Z.ltb_to_lt; try lia; try reflexivity; [ ]. - rewrite Z.testbit_eqb, Z.div_between_0_if by omega. - break_innermost_match; Z.ltb_to_lt; try lia; reflexivity. - Qed. - Lemma prod_montred256_correct : forall (cc_start_state : Fancy.CC.state) (* starting carry flags can be anything *) (start_context : register -> Z) (* starting register values *) @@ -12622,9 +12575,12 @@ Module Montgomery256. { let r := eval cbv in (2^256) in replace (2^256) with r by reflexivity. rewrite !Z.shiftl_0_r, !Z.mod_mod by omega. - apply testbit_neg_eq_if; + repeat match goal with + | |- context [?a mod ?b] => unique pose proof (Z.mod_pos_bound a b ltac:(omega)) + end. + apply Z.testbit_neg_eq_if; let r := eval cbv in (2^256) in replace (2^256) with r by reflexivity; - auto using Z.mod_pos_bound with omega. } + omega. } step start_context; [ break_innermost_match; Z.ltb_to_lt; omega | ]. reflexivity. Qed. diff --git a/src/LegacyArithmetic/ArchitectureToZLikeProofs.v b/src/LegacyArithmetic/ArchitectureToZLikeProofs.v index 8d4b59ceb..5ff67d7ec 100644 --- a/src/LegacyArithmetic/ArchitectureToZLikeProofs.v +++ b/src/LegacyArithmetic/ArchitectureToZLikeProofs.v @@ -8,7 +8,8 @@ Require Import Crypto.LegacyArithmetic.Double.Proofs.Multiply. Require Import Crypto.LegacyArithmetic.ArchitectureToZLike. Require Import Crypto.LegacyArithmetic.ZBounded. Require Import Crypto.Util.Tuple. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. Require Import Crypto.Util.Tactics.UniquePose. Require Import Crypto.Util.LetIn. diff --git a/src/LegacyArithmetic/BarretReduction.v b/src/LegacyArithmetic/BarretReduction.v index e278dc082..37c4d4915 100644 --- a/src/LegacyArithmetic/BarretReduction.v +++ b/src/LegacyArithmetic/BarretReduction.v @@ -4,7 +4,6 @@ Require Import Coq.ZArith.ZArith Coq.Lists.List Coq.Classes.Morphisms Coq.micromega.Psatz. Require Import Crypto.Arithmetic.BarrettReduction.HAC. Require Import Crypto.LegacyArithmetic.ZBounded. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.Notations. Local Open Scope small_zlike_scope. diff --git a/src/LegacyArithmetic/BaseSystem.v b/src/LegacyArithmetic/BaseSystem.v index 3f426e98b..359b4313b 100644 --- a/src/LegacyArithmetic/BaseSystem.v +++ b/src/LegacyArithmetic/BaseSystem.v @@ -1,7 +1,7 @@ Require Import Coq.Lists.List. Require Import Coq.ZArith.ZArith Coq.ZArith.Zdiv. Require Import Coq.omega.Omega Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Arith. -Require Import Crypto.Util.ListUtil Crypto.Util.ZUtil. +Require Import Crypto.Util.ListUtil. Require Import Crypto.Util.Notations. Require Export Crypto.Util.FixCoqMistakes. Import Nat. diff --git a/src/LegacyArithmetic/BaseSystemProofs.v b/src/LegacyArithmetic/BaseSystemProofs.v index f0f0a80d2..042fdb270 100644 --- a/src/LegacyArithmetic/BaseSystemProofs.v +++ b/src/LegacyArithmetic/BaseSystemProofs.v @@ -1,5 +1,5 @@ Require Import Coq.Lists.List Coq.micromega.Psatz. -Require Import Crypto.Util.ListUtil Crypto.Util.ZUtil. +Require Import Crypto.Util.ListUtil. Require Import Coq.ZArith.ZArith Coq.ZArith.Zdiv. Require Import Coq.omega.Omega Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Arith. Require Import Crypto.LegacyArithmetic.BaseSystem. diff --git a/src/LegacyArithmetic/Double/Proofs/BitwiseOr.v b/src/LegacyArithmetic/Double/Proofs/BitwiseOr.v index 0f07c6299..8588836ee 100644 --- a/src/LegacyArithmetic/Double/Proofs/BitwiseOr.v +++ b/src/LegacyArithmetic/Double/Proofs/BitwiseOr.v @@ -2,7 +2,8 @@ Require Import Coq.ZArith.ZArith. Require Import Crypto.LegacyArithmetic.Interface. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.LandLorShiftBounds. Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z_scope. diff --git a/src/LegacyArithmetic/Double/Proofs/Decode.v b/src/LegacyArithmetic/Double/Proofs/Decode.v index 1cd5bf06d..c0748cf8e 100644 --- a/src/LegacyArithmetic/Double/Proofs/Decode.v +++ b/src/LegacyArithmetic/Double/Proofs/Decode.v @@ -3,7 +3,7 @@ Require Import Crypto.LegacyArithmetic.Interface. Require Import Crypto.LegacyArithmetic.InterfaceProofs. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.Util.Tuple. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. Require Import Crypto.Util.ListUtil. Require Import Crypto.Util.Notations. diff --git a/src/LegacyArithmetic/Double/Proofs/LoadImmediate.v b/src/LegacyArithmetic/Double/Proofs/LoadImmediate.v index 2c7f87dd7..13847b50a 100644 --- a/src/LegacyArithmetic/Double/Proofs/LoadImmediate.v +++ b/src/LegacyArithmetic/Double/Proofs/LoadImmediate.v @@ -3,7 +3,7 @@ Require Import Crypto.LegacyArithmetic.Interface. Require Import Crypto.LegacyArithmetic.InterfaceProofs. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. Local Open Scope Z_scope. Local Opaque tuple_decoder. diff --git a/src/LegacyArithmetic/Double/Proofs/Multiply.v b/src/LegacyArithmetic/Double/Proofs/Multiply.v index 98692ed7f..ebe19cc46 100644 --- a/src/LegacyArithmetic/Double/Proofs/Multiply.v +++ b/src/LegacyArithmetic/Double/Proofs/Multiply.v @@ -5,7 +5,10 @@ Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. Require Import Crypto.LegacyArithmetic.Double.Proofs.SpreadLeftImmediate. Require Import Crypto.LegacyArithmetic.Double.Proofs.RippleCarryAddSub. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. +Require Import Crypto.Util.ZUtil.Modulo. Require Import Crypto.Util.Tactics.SimplifyProjections. Require Import Crypto.Util.Notations. Require Import Crypto.Util.LetIn. diff --git a/src/LegacyArithmetic/Double/Proofs/RippleCarryAddSub.v b/src/LegacyArithmetic/Double/Proofs/RippleCarryAddSub.v index 4f6a79e8d..b2d53a33a 100644 --- a/src/LegacyArithmetic/Double/Proofs/RippleCarryAddSub.v +++ b/src/LegacyArithmetic/Double/Proofs/RippleCarryAddSub.v @@ -4,7 +4,10 @@ Require Import Crypto.LegacyArithmetic.InterfaceProofs. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. Require Import Crypto.Util.Tuple. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.SimplifyProjections. Require Import Crypto.Util.Notations. diff --git a/src/LegacyArithmetic/Double/Proofs/ShiftLeft.v b/src/LegacyArithmetic/Double/Proofs/ShiftLeft.v index 2230e36b6..1944f99b2 100644 --- a/src/LegacyArithmetic/Double/Proofs/ShiftLeft.v +++ b/src/LegacyArithmetic/Double/Proofs/ShiftLeft.v @@ -3,8 +3,8 @@ Require Import Crypto.LegacyArithmetic.Interface. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. Require Import Crypto.LegacyArithmetic.Double.Proofs.ShiftLeftRightTactic. -Require Import Crypto.Util.ZUtil. -(*Require Import Crypto.Util.Tactics.*) +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Definitions. Local Open Scope Z_scope. diff --git a/src/LegacyArithmetic/Double/Proofs/ShiftLeftRightTactic.v b/src/LegacyArithmetic/Double/Proofs/ShiftLeftRightTactic.v index 41234bf6e..98cf3cf9c 100644 --- a/src/LegacyArithmetic/Double/Proofs/ShiftLeftRightTactic.v +++ b/src/LegacyArithmetic/Double/Proofs/ShiftLeftRightTactic.v @@ -1,8 +1,17 @@ Require Import Coq.ZArith.ZArith. Require Import Crypto.LegacyArithmetic.Interface. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Testbit. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.Ztestbit. Require Import Crypto.Util.Tactics.UniquePose. Require Import Crypto.Util.Tactics.BreakMatch. +Require Export Crypto.Util.ZUtil.ZSimplify.Autogenerated. +Require Export Crypto.Util.ZUtil.ZSimplify.Core. +Require Export Crypto.Util.ZUtil.ZSimplify.Simple. +Require Export Crypto.Util.ZUtil.LandLorShiftBounds. Local Open Scope Z_scope. diff --git a/src/LegacyArithmetic/Double/Proofs/ShiftRight.v b/src/LegacyArithmetic/Double/Proofs/ShiftRight.v index 16e7c5d6a..245e03480 100644 --- a/src/LegacyArithmetic/Double/Proofs/ShiftRight.v +++ b/src/LegacyArithmetic/Double/Proofs/ShiftRight.v @@ -3,8 +3,6 @@ Require Import Crypto.LegacyArithmetic.Interface. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. Require Import Crypto.LegacyArithmetic.Double.Proofs.ShiftLeftRightTactic. -Require Import Crypto.Util.ZUtil. -(*Require Import Crypto.Util.Tactics.*) Local Open Scope Z_scope. diff --git a/src/LegacyArithmetic/Double/Proofs/ShiftRightDoubleWordImmediate.v b/src/LegacyArithmetic/Double/Proofs/ShiftRightDoubleWordImmediate.v index 00a6d03cd..7210d80d5 100644 --- a/src/LegacyArithmetic/Double/Proofs/ShiftRightDoubleWordImmediate.v +++ b/src/LegacyArithmetic/Double/Proofs/ShiftRightDoubleWordImmediate.v @@ -3,8 +3,8 @@ Require Import Crypto.LegacyArithmetic.Interface. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. Require Import Crypto.LegacyArithmetic.Double.Proofs.ShiftLeftRightTactic. -Require Import Crypto.Util.ZUtil. -(*Require Import Crypto.Util.Tactics.*) +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Definitions. Local Open Scope Z_scope. diff --git a/src/LegacyArithmetic/Double/Proofs/SpreadLeftImmediate.v b/src/LegacyArithmetic/Double/Proofs/SpreadLeftImmediate.v index c50d43616..0cbc237d2 100644 --- a/src/LegacyArithmetic/Double/Proofs/SpreadLeftImmediate.v +++ b/src/LegacyArithmetic/Double/Proofs/SpreadLeftImmediate.v @@ -3,7 +3,11 @@ Require Import Crypto.LegacyArithmetic.Interface. Require Import Crypto.LegacyArithmetic.InterfaceProofs. Require Import Crypto.LegacyArithmetic.Double.Core. Require Import Crypto.LegacyArithmetic.Double.Proofs.Decode. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Modulo. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.SpecializeBy. Require Import Crypto.Util.Notations. diff --git a/src/LegacyArithmetic/Interface.v b/src/LegacyArithmetic/Interface.v index 4a671eb4c..9a652bbd4 100644 --- a/src/LegacyArithmetic/Interface.v +++ b/src/LegacyArithmetic/Interface.v @@ -1,6 +1,7 @@ (*** Interface for bounded arithmetic *) Require Import Coq.ZArith.ZArith. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. + Require Import Crypto.Util.Tuple. Require Import Crypto.Util.AutoRewrite. Require Import Crypto.Util.Notations. diff --git a/src/LegacyArithmetic/InterfaceProofs.v b/src/LegacyArithmetic/InterfaceProofs.v index 33917e00d..a2c8d9de5 100644 --- a/src/LegacyArithmetic/InterfaceProofs.v +++ b/src/LegacyArithmetic/InterfaceProofs.v @@ -1,7 +1,13 @@ (** * Alternate forms for Interface for bounded arithmetic *) Require Import Coq.ZArith.ZArith Coq.micromega.Psatz. Require Import Crypto.LegacyArithmetic.Interface. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Hints. +Require Import Crypto.Util.ZUtil.Hints.ZArith. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.EquivModulo. Require Import Crypto.Util.Tuple. Require Import Crypto.Util.AutoRewrite. Require Import Crypto.Util.Notations. diff --git a/src/LegacyArithmetic/MontgomeryReduction.v b/src/LegacyArithmetic/MontgomeryReduction.v index c3538dd01..786e08d28 100644 --- a/src/LegacyArithmetic/MontgomeryReduction.v +++ b/src/LegacyArithmetic/MontgomeryReduction.v @@ -5,7 +5,7 @@ Require Import Coq.ZArith.ZArith Coq.Lists.List Coq.Classes.Morphisms Coq.microm Require Import Crypto.Arithmetic.MontgomeryReduction.Definition. Require Import Crypto.Arithmetic.MontgomeryReduction.Proofs. Require Import Crypto.LegacyArithmetic.ZBounded. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.EquivModulo. Require Import Crypto.Util.Tactics.Test. Require Import Crypto.Util.Tactics.Not. Require Import Crypto.Util.LetIn. diff --git a/src/LegacyArithmetic/Pow2Base.v b/src/LegacyArithmetic/Pow2Base.v index 62f1f742d..c5c69e684 100644 --- a/src/LegacyArithmetic/Pow2Base.v +++ b/src/LegacyArithmetic/Pow2Base.v @@ -1,6 +1,5 @@ Require Import Coq.ZArith.Zpower Coq.ZArith.ZArith. Require Import Crypto.Util.ListUtil. -Require Import Crypto.Util.ZUtil. Require Import Coq.Lists.List. Local Open Scope Z_scope. diff --git a/src/LegacyArithmetic/Pow2BaseProofs.v b/src/LegacyArithmetic/Pow2BaseProofs.v index b6df85f5c..495636be7 100644 --- a/src/LegacyArithmetic/Pow2BaseProofs.v +++ b/src/LegacyArithmetic/Pow2BaseProofs.v @@ -2,7 +2,13 @@ Require Import Coq.ZArith.Zpower Coq.ZArith.ZArith Coq.micromega.Psatz. Require Import Coq.Numbers.Natural.Peano.NPeano. Require Import Coq.Lists.List. Require Import Coq.funind.Recdef. -Require Import Crypto.Util.ListUtil Crypto.Util.ZUtil Crypto.Util.NatUtil. +Require Import Crypto.Util.ListUtil Crypto.Util.NatUtil. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Testbit. +Require Import Crypto.Util.ZUtil.Pow2Mod. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Shift. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. Require Import Crypto.LegacyArithmetic.VerdiTactics. Require Import Crypto.Util.Tactics.SpecializeBy. Require Import Crypto.Util.Tactics.BreakMatch. @@ -282,7 +288,7 @@ Section Pow2BaseProofs. | |- _ => rewrite BaseSystemProofs.set_higher | |- _ => rewrite nth_default_base | |- _ => rewrite IHi - | |- _ => rewrite <-Z.lor_shiftl by (rewrite ?Z.pow2_mod_spec; try apply Z.mod_pos_bound; zero_bounds) + | |- _ => rewrite <-Z.lor_shiftl by (rewrite ?Z.pow2_mod_spec; try apply Z.mod_pos_bound; Z.zero_bounds) | |- context[min ?x ?y] => (rewrite Nat.min_l by omega || rewrite Nat.min_r by omega) | |- context[2 ^ ?a * _] => rewrite (Z.mul_comm (2 ^ a)); rewrite <-Z.shiftl_mul_pow2 | |- _ => solve [auto] @@ -452,7 +458,7 @@ Section UniformBase. intros; apply Z.eq_le_incl. f_equal; auto. + apply nth_default_preserves_properties_length_dep; - try solve [apply nth_default_preserves_properties; split; zero_bounds; rewrite limb_widths_uniform; auto || omega]. + try solve [apply nth_default_preserves_properties; split; Z.zero_bounds; rewrite limb_widths_uniform; auto || omega]. intros; apply nth_default_preserves_properties_length_dep; try solve [intros; omega]. let x := fresh "x" in intro x; intros; replace x with width; try symmetry; auto. diff --git a/src/LegacyArithmetic/ZBounded.v b/src/LegacyArithmetic/ZBounded.v index bccbf7428..2eec4122b 100644 --- a/src/LegacyArithmetic/ZBounded.v +++ b/src/LegacyArithmetic/ZBounded.v @@ -2,7 +2,7 @@ (** This file specifies a ℤ-like type of bounded integers, with operations for Montgomery Reduction and Barrett Reduction. *) Require Import Coq.ZArith.ZArith. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Notations. diff --git a/src/LegacyArithmetic/ZBoundedZ.v b/src/LegacyArithmetic/ZBoundedZ.v index fef654f47..2943598c1 100644 --- a/src/LegacyArithmetic/ZBoundedZ.v +++ b/src/LegacyArithmetic/ZBoundedZ.v @@ -1,7 +1,9 @@ (*** ℤ can be a bounded ℤ-Like type *) Require Import Coq.ZArith.ZArith Coq.micromega.Psatz. Require Import Crypto.LegacyArithmetic.ZBounded. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Pow2Mod. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.LetIn. Require Import Crypto.Util.Notations. diff --git a/src/Specific/Framework/ArithmeticSynthesis/Karatsuba.v b/src/Specific/Framework/ArithmeticSynthesis/Karatsuba.v index 7274d2c35..58f5279ab 100644 --- a/src/Specific/Framework/ArithmeticSynthesis/Karatsuba.v +++ b/src/Specific/Framework/ArithmeticSynthesis/Karatsuba.v @@ -6,7 +6,7 @@ Require Import Crypto.Arithmetic.Core. Import B. Require Import Crypto.Arithmetic.PrimeFieldTheorems. Require Crypto.Specific.Framework.CurveParameters. Require Import Crypto.Util.Decidable. -Require Import Crypto.Util.LetIn Crypto.Util.ZUtil. +Require Import Crypto.Util.LetIn. Require Import Crypto.Arithmetic.Karatsuba. Require Import Crypto.Util.Tactics.BreakMatch. Require Crypto.Util.Tuple. diff --git a/src/Specific/NISTP256/FancyMachine256/Core.v b/src/Specific/NISTP256/FancyMachine256/Core.v index fec353c6f..881fa2e1e 100644 --- a/src/Specific/NISTP256/FancyMachine256/Core.v +++ b/src/Specific/NISTP256/FancyMachine256/Core.v @@ -20,7 +20,6 @@ Require Import Crypto.Compilers.Linearize. Require Import Crypto.Compilers.Inline. Require Import Crypto.Compilers.CommonSubexpressionElimination. Require Export Crypto.Compilers.Reify. -Require Export Crypto.Util.ZUtil. Require Export Crypto.Util.Option. Require Export Crypto.Util.Notations. Require Import Crypto.Util.ListUtil. diff --git a/src/Specific/NISTP256/FancyMachine256/Montgomery.v b/src/Specific/NISTP256/FancyMachine256/Montgomery.v index b8510614e..4caecca6b 100644 --- a/src/Specific/NISTP256/FancyMachine256/Montgomery.v +++ b/src/Specific/NISTP256/FancyMachine256/Montgomery.v @@ -1,6 +1,7 @@ Require Import Crypto.Specific.NISTP256.FancyMachine256.Core. Require Import Crypto.LegacyArithmetic.MontgomeryReduction. Require Import Crypto.Arithmetic.MontgomeryReduction.Proofs. +Require Import Crypto.Util.ZUtil.EquivModulo. Section expression. Context (ops : fancy_machine.instructions (2 * 128)) (props : fancy_machine.arithmetic ops) (modulus : Z) (m' : Z) (Hm : modulus <> 0) (H : 0 <= modulus < 2^256) (Hm' : 0 <= m' < 2^256). diff --git a/src/Util/FixedWordSizesEquality.v b/src/Util/FixedWordSizesEquality.v index bd71f5b80..797c7ee3b 100644 --- a/src/Util/FixedWordSizesEquality.v +++ b/src/Util/FixedWordSizesEquality.v @@ -4,7 +4,8 @@ Require Import Coq.Arith.Arith. Require Import bbv.WordScope. Require Import Crypto.Util.FixedWordSizes. Require Import Crypto.Util.WordUtil. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Log2. +Require Import Crypto.Util.ZUtil.Z2Nat. Require Import Crypto.Util.Tactics.BreakMatch. Definition wordT_beq_hetero {logsz1 logsz2} : wordT logsz1 -> wordT logsz2 -> bool diff --git a/src/Util/NUtil.v b/src/Util/NUtil.v index 8c354690b..1d78e3276 100644 --- a/src/Util/NUtil.v +++ b/src/Util/NUtil.v @@ -1,42 +1,14 @@ Require Import Coq.NArith.NArith. Require Import Coq.Numbers.Natural.Peano.NPeano. Require Import Crypto.Util.NatUtil Crypto.Util.Decidable. +Require Export Crypto.Util.NUtil.WithoutReferenceToZ. Require bbv.WordScope. Require Import bbv.NatLib. Require Crypto.Util.WordUtil. +Require Import Crypto.Util.ZUtil.Z2Nat. +Require Import Crypto.Util.ZUtil.Shift. Module N. - Lemma size_le a b : (a <= b -> N.size a <= N.size b)%N. - Proof. - destruct (dec (a=0)%N), (dec (b=0)%N); subst; auto using N.le_0_l. - { destruct a; auto. } - { rewrite !N.size_log2 by assumption. - rewrite <-N.succ_le_mono. - apply N.log2_le_mono. } - Qed. - - Lemma le_to_nat a b : (a <= b)%N <-> (N.to_nat a <= N.to_nat b)%nat. - Proof. - rewrite <-N.lt_succ_r. - rewrite <-Nat.lt_succ_r. - rewrite <-Nnat.N2Nat.inj_succ. - rewrite <-NatUtil.Nat2N_inj_lt. - rewrite !Nnat.N2Nat.id. - reflexivity. - Qed. - - Lemma size_nat_equiv : forall n, N.size_nat n = N.to_nat (N.size n). - Proof. - destruct n as [|p]; auto; simpl; induction p as [p IHp|p IHp|]; simpl; auto; rewrite IHp, Pnat.Pos2Nat.inj_succ; reflexivity. - Qed. - - Lemma size_nat_le a b : (a <= b)%N -> (N.size_nat a <= N.size_nat b)%nat. - Proof. - rewrite !size_nat_equiv. - rewrite <-le_to_nat. - apply size_le. - Qed. - Lemma shiftr_size : forall n bound, N.size_nat n <= bound -> N.shiftr_nat n bound = 0%N. Proof. @@ -45,7 +17,7 @@ Module N. rewrite Nshiftr_nat_equiv. destruct (N.eq_dec n 0); subst; [apply N.shiftr_0_l|]. apply N.shiftr_eq_0. - rewrite size_nat_equiv in *. + rewrite N.size_nat_equiv in *. rewrite N.size_log2 in * by auto. apply N.le_succ_l. rewrite <- N.compare_le_iff. @@ -114,7 +86,7 @@ Module N. apply WordUtil.bound_check_nat_N. apply Znat.Nat2Z.inj_lt. rewrite Znat.Z2Nat.id by omega. - rewrite ZUtil.Z.pow_Zpow. + rewrite Z.pow_Zpow. replace (Z.of_nat 2) with 2%Z by reflexivity. omega. Qed. @@ -134,13 +106,13 @@ Module N. rewrite WordUtil.wordToNat_combine. rewrite !Word.wordToNat_natToWord_idempotent by (rewrite Nnat.N2Nat.id; auto using ZToN_NPow2_lt). f_equal. - rewrite ZUtil.Z.lor_shiftl by auto. + rewrite Z.lor_shiftl by auto. rewrite !Z_N_nat. rewrite Znat.Z2Nat.inj_add by (try apply Z.shiftl_nonneg; omega). f_equal. rewrite Z.shiftl_mul_pow2 by auto. rewrite Znat.Z2Nat.inj_mul by omega. - rewrite <-ZUtil.Z.pow_Z2N_Zpow by omega. + rewrite <-Z.pow_Z2N_Zpow by omega. rewrite Nat.mul_comm. f_equal. Qed. diff --git a/src/Util/NUtil/WithoutReferenceToZ.v b/src/Util/NUtil/WithoutReferenceToZ.v new file mode 100644 index 000000000..1955b53d2 --- /dev/null +++ b/src/Util/NUtil/WithoutReferenceToZ.v @@ -0,0 +1,54 @@ +(** NUtil that doesn't depend on ZUtil stuff *) +(** Should probably come up with a better organization of this stuff *) +Require Import Coq.NArith.NArith. +Require Import Coq.Numbers.Natural.Peano.NPeano. +Require Import Crypto.Util.NatUtil Crypto.Util.Decidable. + +Module N. + Lemma size_le a b : (a <= b -> N.size a <= N.size b)%N. + Proof. + destruct (dec (a=0)%N), (dec (b=0)%N); subst; auto using N.le_0_l. + { destruct a; auto. } + { rewrite !N.size_log2 by assumption. + rewrite <-N.succ_le_mono. + apply N.log2_le_mono. } + Qed. + + Lemma N_le_1_l : forall p, (1 <= N.pos p)%N. + Proof. + destruct p; cbv; congruence. + Qed. + + Lemma Pos_land_upper_bound_l : forall a b, (Pos.land a b <= N.pos a)%N. + Proof. + induction a as [a IHa|a IHa|]; destruct b as [b|b|]; try solve [cbv; congruence]; + simpl; specialize (IHa b); case_eq (Pos.land a b); intro p; simpl; + try (apply N_le_1_l || apply N.le_0_l); intro land_eq; + rewrite land_eq in *; unfold N.le, N.compare in *; + rewrite ?Pos.compare_xI_xI, ?Pos.compare_xO_xI, ?Pos.compare_xO_xO; + try assumption. + destruct (p ?=a)%positive; cbv; congruence. + Qed. + + Lemma le_to_nat a b : (a <= b)%N <-> (N.to_nat a <= N.to_nat b)%nat. + Proof. + rewrite <-N.lt_succ_r. + rewrite <-Nat.lt_succ_r. + rewrite <-Nnat.N2Nat.inj_succ. + rewrite <-NatUtil.Nat2N_inj_lt. + rewrite !Nnat.N2Nat.id. + reflexivity. + Qed. + + Lemma size_nat_equiv : forall n, N.size_nat n = N.to_nat (N.size n). + Proof. + destruct n as [|p]; auto; simpl; induction p as [p IHp|p IHp|]; simpl; auto; rewrite IHp, Pnat.Pos2Nat.inj_succ; reflexivity. + Qed. + + Lemma size_nat_le a b : (a <= b)%N -> (N.size_nat a <= N.size_nat b)%nat. + Proof. + rewrite !size_nat_equiv. + rewrite <-le_to_nat. + apply size_le. + Qed. +End N. diff --git a/src/Util/NumTheoryUtil.v b/src/Util/NumTheoryUtil.v index a59a6bbaa..3db3e3dca 100644 --- a/src/Util/NumTheoryUtil.v +++ b/src/Util/NumTheoryUtil.v @@ -1,6 +1,10 @@ Require Import Coq.ZArith.Zpower Coq.ZArith.Znumtheory Coq.ZArith.ZArith Coq.ZArith.Zdiv. Require Import Coq.omega.Omega Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Arith. -Require Import Crypto.Util.NatUtil Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Divide. +Require Import Crypto.Util.ZUtil.Modulo. +Require Import Crypto.Util.ZUtil.Odd. +Require Import Crypto.Util.NatUtil. +Require Import Crypto.Util.ZUtil.Tactics.PrimeBound. Require Export Crypto.Util.FixCoqMistakes. Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z. @@ -48,9 +52,9 @@ Hypothesis prime_p : prime p. Hypothesis neq_p_2 : p <> 2. (* Euler's Criterion is also provable with p = 2, but we do not need it and are lazy.*) Hypothesis x_id : x * 2 + 1 = p. -Lemma lt_1_p : 1 < p. Proof using prime_p. prime_bound. Qed. -Lemma x_pos: 0 < x. Proof using prime_p x_id. prime_bound. Qed. -Lemma x_nonneg: 0 <= x. Proof using prime_p x_id. prime_bound. Qed. +Lemma lt_1_p : 1 < p. Proof using prime_p. Z.prime_bound. Qed. +Lemma x_pos: 0 < x. Proof using prime_p x_id. Z.prime_bound. Qed. +Lemma x_nonneg: 0 <= x. Proof using prime_p x_id. Z.prime_bound. Qed. Lemma x_id_inv : x = (p - 1) / 2. Proof using x_id. @@ -85,12 +89,12 @@ Lemma fermat_little: forall a (a_nonzero : a mod p <> 0), Proof using prime_p. intros a a_nonzero. assert (rel_prime a p). { - apply rel_prime_mod_rev; try prime_bound. - assert (0 < p) as p_pos by prime_bound. + apply rel_prime_mod_rev; try Z.prime_bound. + assert (0 < p) as p_pos by Z.prime_bound. apply rel_prime_le_prime; auto; pose proof (Z.mod_pos_bound a p p_pos). omega. } - rewrite (Coqprime.PrimalityTest.Zp.Zpower_mod_is_gpow _ _ _ lt_1_p) by (auto || prime_bound). + rewrite (Coqprime.PrimalityTest.Zp.Zpower_mod_is_gpow _ _ _ lt_1_p) by (auto || Z.prime_bound). rewrite <- mod_p_order. apply Coqprime.PrimalityTest.EGroup.fermat_gen; try apply Z.eq_dec. apply Coqprime.PrimalityTest.Zp.in_mod_ZPGroup; auto. @@ -126,14 +130,14 @@ Proof using Type*. assert (b mod p <> 0) as b_nonzero. { intuition. rewrite <- Z.pow_2_r in a_square. - rewrite Z.mod_exp_0 in a_square by prime_bound. + rewrite Z.mod_exp_0 in a_square by Z.prime_bound. rewrite <- a_square in a_nonzero. auto. } pose proof (squared_fermat_little b b_nonzero). - rewrite Z.mod_pow in * by prime_bound. + rewrite Z.mod_pow in * by Z.prime_bound. rewrite <- a_square. - rewrite Z.mod_mod; prime_bound. + rewrite Z.mod_mod; Z.prime_bound. Qed. Lemma exists_primitive_root_power : @@ -174,10 +178,10 @@ Proof using Type*. intros a a_range pow_a_x. destruct (exists_primitive_root_power) as [y [in_ZPGroup_y [y_order gpow_y]]]; auto. destruct (gpow_y a a_range) as [j [j_range pow_y_j]]; clear gpow_y. - rewrite Z.mod_pow in pow_a_x by prime_bound. + rewrite Z.mod_pow in pow_a_x by Z.prime_bound. replace a with (a mod p) in pow_y_j by (apply Z.mod_small; omega). rewrite <- pow_y_j in pow_a_x. - rewrite <- Z.mod_pow in pow_a_x by prime_bound. + rewrite <- Z.mod_pow in pow_a_x by Z.prime_bound. rewrite <- Z.pow_mul_r in pow_a_x by omega. assert (p - 1 | j * x) as divide_mul_j_x. { rewrite <- Coqprime.PrimalityTest.Zp.phi_is_order in y_order. @@ -196,7 +200,7 @@ Proof using Type*. rewrite Z.mul_comm. rewrite x_id_inv in divide_mul_j_x; auto. apply (Z.divide_mul_div _ j 2) in divide_mul_j_x; - try (apply prime_pred_divide2 || prime_bound); auto. + try (apply prime_pred_divide2 || Z.prime_bound); auto. rewrite <- Zdivide_Zdiv_eq by (auto || omega). rewrite Zplus_diag_eq_mult_2. replace (a mod p) with a in pow_y_j by (symmetry; apply Z.mod_small; omega). @@ -296,8 +300,8 @@ Proof. apply (euler_criterion (p / 2) p prime_p). + auto. + apply div2_p_1mod4; auto. - + prime_bound. - + apply minus1_even_pow; [apply divide2_1mod4 | | apply Z_div_pos]; prime_bound. + + Z.prime_bound. + + apply minus1_even_pow; [apply divide2_1mod4 | | apply Z_div_pos]; Z.prime_bound. Qed. diff --git a/src/Util/QUtil.v b/src/Util/QUtil.v index dbdf4fba0..2c2dfd8c8 100644 --- a/src/Util/QUtil.v +++ b/src/Util/QUtil.v @@ -1,6 +1,8 @@ Require Import Coq.ZArith.ZArith Coq.QArith.QArith QArith.Qround. +Require Import Coq.micromega.Lia. Require Import Crypto.Util.Decidable. -Require Import ZUtil Lia. +Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.Morphisms. Local Open Scope Z_scope. diff --git a/src/Util/Tuple.v b/src/Util/Tuple.v index cf7732a4c..eda97f556 100644 --- a/src/Util/Tuple.v +++ b/src/Util/Tuple.v @@ -618,6 +618,27 @@ Proof. auto using fieldwiseb'_fieldwise'. Qed. +Lemma map_ext {A B} (f g : A -> B) n (t : tuple A n) : + (forall x : A, f x = g x) -> + map f t = map g t. +Proof. + destruct n; [reflexivity|]; cbn in *. + induction n; cbn in *; intro H; auto; [ ]. + rewrite IHn by assumption. + rewrite H; reflexivity. +Qed. + +Lemma map_ext_In {A B} (f g : A -> B) n (t : tuple A n) : + (forall x, In x (to_list n t) -> f x = g x) -> + map f t = map g t. +Proof. + destruct n; [reflexivity|]; cbn in *. + induction n; cbn in *; intro H; auto; [ ]. + destruct t. + rewrite IHn by auto using in_cons. + rewrite H; auto using in_eq. +Qed. + Fixpoint from_list_default' {T} (d y:T) (n:nat) (xs:list T) : tuple' T n := match n return tuple' T n with diff --git a/src/Util/WordUtil.v b/src/Util/WordUtil.v index 2faafe6c6..932f48fee 100644 --- a/src/Util/WordUtil.v +++ b/src/Util/WordUtil.v @@ -9,12 +9,15 @@ Require Import Coq.Bool.Bool. Require Import Crypto.Util.Bool. Require Import Crypto.Util.NatUtil. -Require Import Crypto.Util.ZUtil. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.DestructHead. Require Import Crypto.Util.Tactics.RewriteHyp. Require Import Crypto.Util.Sigma. +Require Import Crypto.Util.ZUtil.LandLorShiftBounds. +Require Import Crypto.Util.ZUtil.N2Z. +Require Import Crypto.Util.ZUtil.Definitions. + Require Import bbv.WordScope. Require Import bbv.Nomega. diff --git a/src/Util/ZUtil.v b/src/Util/ZUtil.v index 270bd0c90..765142c39 100644 --- a/src/Util/ZUtil.v +++ b/src/Util/ZUtil.v @@ -1,1520 +1,69 @@ -Require Import Coq.ZArith.Zpower Coq.ZArith.Znumtheory Coq.ZArith.ZArith Coq.ZArith.Zdiv. -Require Import Coq.Classes.RelationClasses Coq.Classes.Morphisms. -Require Import Coq.Structures.Equalities. -Require Import Coq.omega.Omega Coq.micromega.Psatz Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Arith. -Require Import Crypto.Util.NatUtil. -Require Import Crypto.Util.Tactics.SpecializeBy. -Require Import Crypto.Util.Tactics.BreakMatch. -Require Import Crypto.Util.Tactics.Contains. -Require Import Crypto.Util.Tactics.Not. -Require Import Crypto.Util.Bool. -Require Import Crypto.Util.Notations. -Require Import Coq.Lists.List. -Require Export Crypto.Util.FixCoqMistakes. -Require Export Crypto.Util.ZUtil.Definitions. -Require Export Crypto.Util.ZUtil.Div. -Require Export Crypto.Util.ZUtil.Le. -Require Export Crypto.Util.ZUtil.EquivModulo. -Require Export Crypto.Util.ZUtil.Hints. -Require Export Crypto.Util.ZUtil.Land. -Require Export Crypto.Util.ZUtil.Modulo. -Require Export Crypto.Util.ZUtil.Modulo.PullPush. -Require Export Crypto.Util.ZUtil.Morphisms. -Require Export Crypto.Util.ZUtil.Notations. -Require Export Crypto.Util.ZUtil.Pow2Mod. -Require Export Crypto.Util.ZUtil.Quot. -Require Export Crypto.Util.ZUtil.Sgn. -Require Export Crypto.Util.ZUtil.Tactics. -Require Export Crypto.Util.ZUtil.Testbit. -Require Export Crypto.Util.ZUtil.ZSimplify. -Import Nat. -Local Open Scope Z. - -Module Z. - Lemma mul_comm3 x y z : x * (y * z) = y * (x * z). - Proof. lia. Qed. - - Lemma pos_pow_nat_pos : forall x n, - Z.pos x ^ Z.of_nat n > 0. - Proof. - do 2 (try intros x n; induction n as [|n]; subst; simpl in *; auto with zarith). - rewrite <- Pos.add_1_r, Zpower_pos_is_exp. - apply Zmult_gt_0_compat; auto; reflexivity. - Qed. - - (** TODO: Should we get rid of this duplicate? *) - Notation gt0_neq0 := Z.positive_is_nonzero (only parsing). - - Lemma pow_Z2N_Zpow : forall a n, 0 <= a -> - ((Z.to_nat a) ^ n = Z.to_nat (a ^ Z.of_nat n)%Z)%nat. - Proof. - intros a n H; induction n as [|n IHn]; try reflexivity. - rewrite Nat2Z.inj_succ. - rewrite pow_succ_r by apply le_0_n. - rewrite Z.pow_succ_r by apply Zle_0_nat. - rewrite IHn. - rewrite Z2Nat.inj_mul; auto using Z.pow_nonneg. - Qed. - - Lemma pow_Zpow : forall a n : nat, Z.of_nat (a ^ n) = Z.of_nat a ^ Z.of_nat n. - Proof with auto using Zle_0_nat, Z.pow_nonneg. - intros; apply Z2Nat.inj... - rewrite <- pow_Z2N_Zpow, !Nat2Z.id... - Qed. - Hint Rewrite pow_Zpow : push_Zof_nat. - Hint Rewrite <- pow_Zpow : pull_Zof_nat. - - Lemma Zpow_sub_1_nat_pow a v - : (Z.pos a^Z.of_nat v - 1 = Z.of_nat (Z.to_nat (Z.pos a)^v - 1))%Z. - Proof. - rewrite <- (Z2Nat.id (Z.pos a)) at 1 by lia. - change 2%Z with (Z.of_nat 2); change 1%Z with (Z.of_nat 1); - autorewrite with pull_Zof_nat. - rewrite Nat2Z.inj_sub - by (change 1%nat with (Z.to_nat (Z.pos a)^0)%nat; apply Nat.pow_le_mono_r; simpl; lia). - reflexivity. - Qed. - Hint Rewrite Zpow_sub_1_nat_pow : pull_Zof_nat. - Hint Rewrite <- Zpow_sub_1_nat_pow : push_Zof_nat. - - Lemma divide_mul_div: forall a b c (a_nonzero : a <> 0) (c_nonzero : c <> 0), - (a | b * (a / c)) -> (c | a) -> (c | b). - Proof. - intros ? ? ? ? ? divide_a divide_c_a; do 2 Z.divide_exists_mul. - rewrite divide_c_a in divide_a. - rewrite Z.div_mul' in divide_a by auto. - replace (b * k) with (k * b) in divide_a by ring. - replace (c * k * k0) with (k * (k0 * c)) in divide_a by ring. - rewrite Z.mul_cancel_l in divide_a by (intuition auto with nia; rewrite H in divide_c_a; ring_simplify in divide_a; intuition). - eapply Zdivide_intro; eauto. - Qed. - - Lemma divide2_even_iff : forall n, (2 | n) <-> Z.even n = true. - Proof. - intros n; split. { - intro divide2_n. - Z.divide_exists_mul; [ | pose proof (Z.mod_pos_bound n 2); omega]. - rewrite divide2_n. - apply Z.even_mul. - } { - intro n_even. - pose proof (Zmod_even n) as H. - rewrite n_even in H. - apply Zmod_divide; omega || auto. - } - Qed. - - Lemma prime_odd_or_2 : forall p (prime_p : prime p), p = 2 \/ Z.odd p = true. - Proof. - intros p prime_p. - apply Decidable.imp_not_l; try apply Z.eq_decidable. - intros p_neq2. - pose proof (Zmod_odd p) as mod_odd. - destruct (Sumbool.sumbool_of_bool (Z.odd p)) as [? | p_not_odd]; auto. - rewrite p_not_odd in mod_odd. - apply Zmod_divides in mod_odd; try omega. - destruct mod_odd as [c c_id]. - rewrite Z.mul_comm in c_id. - apply Zdivide_intro in c_id. - apply prime_divisors in c_id; auto. - destruct c_id; [omega | destruct H; [omega | destruct H; auto] ]. - pose proof (prime_ge_2 p prime_p); omega. - Qed. - - Lemma shiftr_add_shiftl_high : forall n m a b, 0 <= n <= m -> 0 <= a < 2 ^ n -> - Z.shiftr (a + (Z.shiftl b n)) m = Z.shiftr b (m - n). - Proof. - intros n m a b H H0. - rewrite !Z.shiftr_div_pow2, Z.shiftl_mul_pow2 by omega. - replace (2 ^ m) with (2 ^ n * 2 ^ (m - n)) by - (rewrite <-Z.pow_add_r by omega; f_equal; ring). - rewrite <-Z.div_div, Z.div_add, (Z.div_small a) ; try solve - [assumption || apply Z.pow_nonzero || apply Z.pow_pos_nonneg; omega]. - f_equal; ring. - Qed. - Hint Rewrite Z.shiftr_add_shiftl_high using zutil_arith : pull_Zshift. - Hint Rewrite <- Z.shiftr_add_shiftl_high using zutil_arith : push_Zshift. - - Lemma shiftr_add_shiftl_low : forall n m a b, 0 <= m <= n -> 0 <= a < 2 ^ n -> - Z.shiftr (a + (Z.shiftl b n)) m = Z.shiftr a m + Z.shiftr b (m - n). - Proof. - intros n m a b H H0. - rewrite !Z.shiftr_div_pow2, Z.shiftl_mul_pow2, Z.shiftr_mul_pow2 by omega. - replace (2 ^ n) with (2 ^ (n - m) * 2 ^ m) by - (rewrite <-Z.pow_add_r by omega; f_equal; ring). - rewrite Z.mul_assoc, Z.div_add by (apply Z.pow_nonzero; omega). - repeat f_equal; ring. - Qed. - Hint Rewrite Z.shiftr_add_shiftl_low using zutil_arith : pull_Zshift. - Hint Rewrite <- Z.shiftr_add_shiftl_low using zutil_arith : push_Zshift. - - Lemma testbit_add_shiftl_high : forall i, (0 <= i) -> forall a b n, (0 <= n <= i) -> - 0 <= a < 2 ^ n -> - Z.testbit (a + Z.shiftl b n) i = Z.testbit b (i - n). - Proof. - intros i ?. - apply natlike_ind with (x := i); [ intros a b n | intros x H0 H1 a b n | ]; intros; try assumption; - (destruct (Z.eq_dec 0 n); [ subst; rewrite Z.pow_0_r in *; - replace a with 0 by omega; f_equal; ring | ]); try omega. - rewrite <-Z.add_1_r at 1. rewrite <-Z.shiftr_spec by assumption. - replace (Z.succ x - n) with (x - (n - 1)) by ring. - rewrite shiftr_add_shiftl_low, <-Z.shiftl_opp_r with (a := b) by omega. - rewrite <-H1 with (a := Z.shiftr a 1); try omega; [ repeat f_equal; ring | ]. - rewrite Z.shiftr_div_pow2 by omega. - split; apply Z.div_pos || apply Z.div_lt_upper_bound; - try solve [rewrite ?Z.pow_1_r; omega]. - rewrite <-Z.pow_add_r by omega. - replace (1 + (n - 1)) with n by ring; omega. - Qed. - Hint Rewrite testbit_add_shiftl_high using zutil_arith : Ztestbit. - - Lemma nonneg_pow_pos a b : 0 < a -> 0 < a^b -> 0 <= b. - Proof. - destruct (Z_lt_le_dec b 0); intros; auto. - erewrite Z.pow_neg_r in * by eassumption. - omega. - Qed. - Hint Resolve nonneg_pow_pos (fun n => nonneg_pow_pos 2 n Z.lt_0_2) : zarith. - Lemma nonneg_pow_pos_helper a b dummy : 0 < a -> 0 <= dummy < a^b -> 0 <= b. - Proof. eauto with zarith omega. Qed. - Hint Resolve nonneg_pow_pos_helper (fun n dummy => nonneg_pow_pos_helper 2 n dummy Z.lt_0_2) : zarith. - - Lemma testbit_add_shiftl_full i (Hi : 0 <= i) a b n (Ha : 0 <= a < 2^n) - : Z.testbit (a + b << n) i - = if (i <? n) then Z.testbit a i else Z.testbit b (i - n). - Proof. - assert (0 < 2^n) by omega. - assert (0 <= n) by eauto 2 with zarith. - pose proof (Zlt_cases i n); break_match; autorewrite with Ztestbit; reflexivity. - Qed. - Hint Rewrite testbit_add_shiftl_full using zutil_arith : Ztestbit. - - Lemma land_add_land : forall n m a b, (m <= n)%nat -> - Z.land ((Z.land a (Z.ones (Z.of_nat n))) + (Z.shiftl b (Z.of_nat n))) (Z.ones (Z.of_nat m)) = Z.land a (Z.ones (Z.of_nat m)). - Proof. - intros n m a b H. - rewrite !Z.land_ones by apply Nat2Z.is_nonneg. - rewrite Z.shiftl_mul_pow2 by apply Nat2Z.is_nonneg. - replace (b * 2 ^ Z.of_nat n) with - ((b * 2 ^ Z.of_nat (n - m)) * 2 ^ Z.of_nat m) by - (rewrite (le_plus_minus m n) at 2; try assumption; - rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg; ring). - rewrite Z.mod_add by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat m)); omega). - symmetry. apply Znumtheory.Zmod_div_mod; try (apply Z.pow_pos_nonneg; omega). - rewrite (le_plus_minus m n) by assumption. - rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg. - apply Z.divide_factor_l. - Qed. - - Lemma div_pow2succ : forall n x, (0 <= x) -> - n / 2 ^ Z.succ x = Z.div2 (n / 2 ^ x). - Proof. - intros. - rewrite Z.pow_succ_r, Z.mul_comm by auto. - rewrite <- Z.div_div by (try apply Z.pow_nonzero; omega). - rewrite Zdiv2_div. - reflexivity. - Qed. - - Lemma shiftr_succ : forall n x, - Z.shiftr n (Z.succ x) = Z.shiftr (Z.shiftr n x) 1. - Proof. - intros. - rewrite Z.shiftr_shiftr by omega. - reflexivity. - Qed. - Hint Rewrite Z.shiftr_succ using zutil_arith : push_Zshift. - Hint Rewrite <- Z.shiftr_succ using zutil_arith : pull_Zshift. - - Lemma pow2_lt_or_divides : forall a b, 0 <= b -> - 2 ^ a < 2 ^ b \/ (2 ^ a) mod 2 ^ b = 0. - Proof. - intros a b H. - destruct (Z_lt_dec a b); [left|right]. - { apply Z.pow_lt_mono_r; auto; omega. } - { replace a with (a - b + b) by ring. - rewrite Z.pow_add_r by omega. - apply Z.mod_mul, Z.pow_nonzero; omega. } - Qed. - - Lemma odd_mod : forall a b, (b <> 0)%Z -> - Z.odd (a mod b) = if Z.odd b then xorb (Z.odd a) (Z.odd (a / b)) else Z.odd a. - Proof. - intros a b H. - rewrite Zmod_eq_full by assumption. - rewrite <-Z.add_opp_r, Z.odd_add, Z.odd_opp, Z.odd_mul. - case_eq (Z.odd b); intros; rewrite ?Bool.andb_true_r, ?Bool.andb_false_r; auto using Bool.xorb_false_r. - Qed. - - Lemma mod_same_pow : forall a b c, 0 <= c <= b -> a ^ b mod a ^ c = 0. - Proof. - intros a b c H. - replace b with (b - c + c) by ring. - rewrite Z.pow_add_r by omega. - apply Z_mod_mult. - Qed. - Hint Rewrite mod_same_pow using zutil_arith : zsimplify. - - Lemma ones_succ : forall x, (0 <= x) -> - Z.ones (Z.succ x) = 2 ^ x + Z.ones x. - Proof. - unfold Z.ones; intros. - rewrite !Z.shiftl_1_l. - rewrite Z.add_pred_r. - apply Z.succ_inj. - rewrite !Z.succ_pred. - rewrite Z.pow_succ_r; omega. - Qed. - - Lemma div_floor : forall a b c, 0 < b -> a < b * (Z.succ c) -> a / b <= c. - Proof. - intros. - apply Z.lt_succ_r. - apply Z.div_lt_upper_bound; try omega. - Qed. - - Lemma shiftr_1_r_le : forall a b, a <= b -> - Z.shiftr a 1 <= Z.shiftr b 1. - Proof. - intros. - rewrite !Z.shiftr_div_pow2, Z.pow_1_r by omega. - apply Z.div_le_mono; omega. - Qed. - Hint Resolve shiftr_1_r_le : zarith. - - Lemma shiftr_le : forall a b i : Z, 0 <= i -> a <= b -> a >> i <= b >> i. - Proof. - intros a b i ?; revert a b. apply natlike_ind with (x := i); intros; auto. - rewrite !shiftr_succ, shiftr_1_r_le; eauto. reflexivity. - Qed. - Hint Resolve shiftr_le : zarith. - - Lemma ones_pred : forall i, 0 < i -> Z.ones (Z.pred i) = Z.shiftr (Z.ones i) 1. - Proof. - induction i as [|p|p]; [ | | pose proof (Pos2Z.neg_is_neg p) ]; try omega. - intros. - unfold Z.ones. - rewrite !Z.shiftl_1_l, Z.shiftr_div_pow2, <-!Z.sub_1_r, Z.pow_1_r, <-!Z.add_opp_r by omega. - replace (2 ^ (Z.pos p)) with (2 ^ (Z.pos p - 1)* 2). - rewrite Z.div_add_l by omega. - reflexivity. - change 2 with (2 ^ 1) at 2. - rewrite <-Z.pow_add_r by (pose proof (Pos2Z.is_pos p); omega). - f_equal. omega. - Qed. - Hint Rewrite <- ones_pred using zutil_arith : push_Zshift. - - Lemma shiftr_ones' : forall a n, 0 <= a < 2 ^ n -> forall i, (0 <= i) -> - Z.shiftr a i <= Z.ones (n - i) \/ n <= i. - Proof. - intros a n H. - apply natlike_ind. - + unfold Z.ones. - rewrite Z.shiftr_0_r, Z.shiftl_1_l, Z.sub_0_r. - omega. - + intros x H0 H1. - destruct (Z_lt_le_dec x n); try omega. - intuition auto with zarith lia. - left. - rewrite shiftr_succ. - replace (n - Z.succ x) with (Z.pred (n - x)) by omega. - rewrite Z.ones_pred by omega. - apply Z.shiftr_1_r_le. - assumption. - Qed. - - Lemma shiftr_ones : forall a n i, 0 <= a < 2 ^ n -> (0 <= i) -> (i <= n) -> - Z.shiftr a i <= Z.ones (n - i) . - Proof. - intros a n i G G0 G1. - destruct (Z_le_lt_eq_dec i n G1). - + destruct (Z.shiftr_ones' a n G i G0); omega. - + subst; rewrite Z.sub_diag. - destruct (Z.eq_dec a 0). - - subst; rewrite Z.shiftr_0_l; reflexivity. - - rewrite Z.shiftr_eq_0; try omega; try reflexivity. - apply Z.log2_lt_pow2; omega. - Qed. - Hint Resolve shiftr_ones : zarith. - - Lemma shiftr_upper_bound : forall a n, 0 <= n -> 0 <= a <= 2 ^ n -> Z.shiftr a n <= 1. - Proof. - intros a ? ? [a_nonneg a_upper_bound]. - apply Z_le_lt_eq_dec in a_upper_bound. - destruct a_upper_bound. - + destruct (Z.eq_dec 0 a). - - subst; rewrite Z.shiftr_0_l; omega. - - rewrite Z.shiftr_eq_0; auto; try omega. - apply Z.log2_lt_pow2; auto; omega. - + subst. - rewrite Z.shiftr_div_pow2 by assumption. - rewrite Z.div_same; try omega. - assert (0 < 2 ^ n) by (apply Z.pow_pos_nonneg; omega). - omega. - Qed. - Hint Resolve shiftr_upper_bound : zarith. - - Lemma lor_shiftl : forall a b n, 0 <= n -> 0 <= a < 2 ^ n -> - Z.lor a (Z.shiftl b n) = a + (Z.shiftl b n). - Proof. - intros a b n H H0. - apply Z.bits_inj'; intros t ?. - rewrite Z.lor_spec, Z.shiftl_spec by assumption. - destruct (Z_lt_dec t n). - + rewrite Z.testbit_add_shiftl_low by omega. - rewrite Z.testbit_neg_r with (n := t - n) by omega. - apply Bool.orb_false_r. - + rewrite testbit_add_shiftl_high by omega. - replace (Z.testbit a t) with false; [ apply Bool.orb_false_l | ]. - symmetry. - apply Z.testbit_false; try omega. - rewrite Z.div_small; try reflexivity. - split; try eapply Z.lt_le_trans with (m := 2 ^ n); try omega. - apply Z.pow_le_mono_r; omega. - Qed. - Hint Rewrite <- Z.lor_shiftl using zutil_arith : convert_to_Ztestbit. - - Lemma lor_shiftl' : forall a b n, 0 <= n -> 0 <= a < 2 ^ n -> - Z.lor (Z.shiftl b n) a = (Z.shiftl b n) + a. - Proof. - intros; rewrite Z.lor_comm, Z.add_comm; apply lor_shiftl; assumption. - Qed. - Hint Rewrite <- Z.lor_shiftl' using zutil_arith : convert_to_Ztestbit. - - Lemma shiftl_spec_full a n m - : Z.testbit (a << n) m = if Z_lt_dec m n - then false - else if Z_le_dec 0 m - then Z.testbit a (m - n) - else false. - Proof. - repeat break_match; auto using Z.shiftl_spec_low, Z.shiftl_spec, Z.testbit_neg_r with omega. - Qed. - Hint Rewrite shiftl_spec_full : Ztestbit_full. - - Lemma shiftr_spec_full a n m - : Z.testbit (a >> n) m = if Z_lt_dec m (-n) - then false - else if Z_le_dec 0 m - then Z.testbit a (m + n) - else false. - Proof. - rewrite <- Z.shiftl_opp_r, shiftl_spec_full, Z.sub_opp_r; reflexivity. - Qed. - Hint Rewrite shiftr_spec_full : Ztestbit_full. - - Lemma lnot_sub1 x : Z.lnot (x-1) = (-x). - Proof. - replace (-x) with (- (1) - (x - 1)) by omega. - rewrite <-(Z.add_lnot_diag (x-1)); omega. - Qed. - - Lemma lnot_opp x : Z.lnot (- x) = x-1. - Proof. - rewrite <-Z.lnot_involutive, lnot_sub1; reflexivity. - Qed. - - Lemma testbit_sub_pow2 n i x (i_range:0 <= i < n) (x_range:0 < x < 2 ^ n) : - Z.testbit (2 ^ n - x) i = negb (Z.testbit (x - 1) i). - Proof. - rewrite <-Z.lnot_spec, lnot_sub1 by omega. - rewrite <-(Z.mod_pow2_bits_low (-x) _ _ (proj2 i_range)). - f_equal. - rewrite Z.mod_opp_l_nz; autorewrite with zsimplify; omega. - Qed. - - Lemma ones_nonneg : forall i, (0 <= i) -> 0 <= Z.ones i. - Proof. - apply natlike_ind. - + unfold Z.ones. simpl; omega. - + intros. - rewrite Z.ones_succ by assumption. - Z.zero_bounds. - Qed. - Hint Resolve ones_nonneg : zarith. - - Lemma ones_pos_pos : forall i, (0 < i) -> 0 < Z.ones i. - Proof. - intros. - unfold Z.ones. - rewrite Z.shiftl_1_l. - apply Z.lt_succ_lt_pred. - apply Z.pow_gt_1; omega. - Qed. - Hint Resolve ones_pos_pos : zarith. - - Lemma pow2_mod_id_iff : forall a n, 0 <= n -> - (Z.pow2_mod a n = a <-> 0 <= a < 2 ^ n). - Proof. - intros a n H. - rewrite Z.pow2_mod_spec by assumption. - assert (0 < 2 ^ n) by Z.zero_bounds. - rewrite Z.mod_small_iff by omega. - split; intros; intuition omega. - Qed. - - Lemma testbit_false_bound : forall a x, 0 <= x -> - (forall n, ~ (n < x) -> Z.testbit a n = false) -> - a < 2 ^ x. - Proof. - intros a x H H0. - assert (H1 : a = Z.pow2_mod a x). { - apply Z.bits_inj'; intros. - rewrite Z.testbit_pow2_mod by omega; break_match; auto. - } - rewrite H1. - rewrite Z.pow2_mod_spec; try apply Z.mod_pos_bound; Z.zero_bounds. - Qed. - - Lemma lor_range : forall x y n, 0 <= x < 2 ^ n -> 0 <= y < 2 ^ n -> - 0 <= Z.lor x y < 2 ^ n. - Proof. - intros x y n H H0; assert (0 <= n) by auto with zarith omega. - repeat match goal with - | |- _ => progress intros - | |- _ => rewrite Z.lor_spec - | |- _ => rewrite Z.testbit_eqb by auto with zarith omega - | |- _ => rewrite !Z.div_small by (split; try omega; eapply Z.lt_le_trans; - [ intuition eassumption | apply Z.pow_le_mono_r; omega]) - | |- _ => split - | |- _ => apply testbit_false_bound - | |- _ => solve [auto with zarith] - | |- _ => solve [apply Z.lor_nonneg; intuition auto] - end. - Qed. - Hint Resolve lor_range : zarith. - - Lemma lor_shiftl_bounds : forall x y n m, - (0 <= n)%Z -> (0 <= m)%Z -> - (0 <= x < 2 ^ m)%Z -> - (0 <= y < 2 ^ n)%Z -> - (0 <= Z.lor y (Z.shiftl x n) < 2 ^ (n + m))%Z. - Proof. - intros x y n m H H0 H1 H2. - apply Z.lor_range. - { split; try omega. - apply Z.lt_le_trans with (m := (2 ^ n)%Z); try omega. - apply Z.pow_le_mono_r; omega. } - { rewrite Z.shiftl_mul_pow2 by omega. - rewrite Z.pow_add_r by omega. - split; Z.zero_bounds. - rewrite Z.mul_comm. - apply Z.mul_lt_mono_pos_l; omega. } - Qed. - - Lemma N_le_1_l : forall p, (1 <= N.pos p)%N. - Proof. - destruct p; cbv; congruence. - Qed. - - Lemma Pos_land_upper_bound_l : forall a b, (Pos.land a b <= N.pos a)%N. - Proof. - induction a as [a IHa|a IHa|]; destruct b as [b|b|]; try solve [cbv; congruence]; - simpl; specialize (IHa b); case_eq (Pos.land a b); intro p; simpl; - try (apply N_le_1_l || apply N.le_0_l); intro land_eq; - rewrite land_eq in *; unfold N.le, N.compare in *; - rewrite ?Pos.compare_xI_xI, ?Pos.compare_xO_xI, ?Pos.compare_xO_xO; - try assumption. - destruct (p ?=a)%positive; cbv; congruence. - Qed. - - Lemma land_upper_bound_l : forall a b, (0 <= a) -> (0 <= b) -> - Z.land a b <= a. - Proof. - intros a b H H0. - destruct a, b; try solve [exfalso; auto]; try solve [cbv; congruence]. - cbv [Z.land]. - rewrite <-N2Z.inj_pos, <-N2Z.inj_le. - auto using Pos_land_upper_bound_l. - Qed. - - Lemma land_upper_bound_r : forall a b, (0 <= a) -> (0 <= b) -> - Z.land a b <= b. - Proof. - intros. - rewrite Z.land_comm. - auto using Z.land_upper_bound_l. - Qed. - - Lemma le_fold_right_max : forall low l x, (forall y, In y l -> low <= y) -> - In x l -> x <= fold_right Z.max low l. - Proof. - induction l as [|a l IHl]; intros ? lower_bound In_list; [cbv [In] in *; intuition | ]. - simpl. - destruct (in_inv In_list); subst. - + apply Z.le_max_l. - + etransitivity. - - apply IHl; auto; intuition auto with datatypes. - - apply Z.le_max_r. - Qed. - - Lemma le_fold_right_max_initial : forall low l, low <= fold_right Z.max low l. - Proof. - induction l as [|a l IHl]; intros; try reflexivity. - etransitivity; [ apply IHl | apply Z.le_max_r ]. - Qed. - - Lemma add_compare_mono_r: forall n m p, (n + p ?= m + p) = (n ?= m). - Proof. - intros n m p. - rewrite <-!(Z.add_comm p). - apply Z.add_compare_mono_l. - Qed. - - Lemma compare_add_shiftl : forall x1 y1 x2 y2 n, 0 <= n -> - Z.pow2_mod x1 n = x1 -> Z.pow2_mod x2 n = x2 -> - x1 + (y1 << n) ?= x2 + (y2 << n) = - if Z.eq_dec y1 y2 - then x1 ?= x2 - else y1 ?= y2. - Proof. - repeat match goal with - | |- _ => progress intros - | |- _ => progress subst y1 - | |- _ => rewrite Z.shiftl_mul_pow2 by omega - | |- _ => rewrite add_compare_mono_r - | |- _ => rewrite <-Z.mul_sub_distr_r - | |- _ => break_innermost_match_step - | H : Z.pow2_mod _ _ = _ |- _ => rewrite pow2_mod_id_iff in H by omega - | H : ?a <> ?b |- _ = (?a ?= ?b) => - case_eq (a ?= b); rewrite ?Z.compare_eq_iff, ?Z.compare_gt_iff, ?Z.compare_lt_iff - | |- _ + (_ * _) > _ + (_ * _) => cbv [Z.gt] - | |- _ + (_ * ?x) < _ + (_ * ?x) => - apply Z.lt_sub_lt_add; apply Z.lt_le_trans with (m := 1 * x); [omega|] - | |- _ => apply Z.mul_le_mono_nonneg_r; omega - | |- _ => reflexivity - | |- _ => congruence - end. - Qed. - - Lemma ones_le x y : x <= y -> Z.ones x <= Z.ones y. - Proof. - rewrite !Z.ones_equiv; auto with zarith. - Qed. - Hint Resolve ones_le : zarith. - - Lemma mul_div_le x y z - (Hx : 0 <= x) (Hy : 0 <= y) (Hz : 0 < z) - (Hyz : y <= z) - : x * y / z <= x. - Proof. - transitivity (x * z / z); [ | rewrite Z.div_mul by lia; lia ]. - apply Z_div_le; nia. - Qed. - - Hint Resolve mul_div_le : zarith. - - Lemma div_mul_diff_exact a b c - (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) - : c * a / b = c * (a / b) + (c * (a mod b)) / b. - Proof. - rewrite (Z_div_mod_eq a b) at 1 by lia. - rewrite Z.mul_add_distr_l. - replace (c * (b * (a / b))) with ((c * (a / b)) * b) by lia. - rewrite Z.div_add_l by lia. - lia. - Qed. - - Lemma div_mul_diff_exact' a b c - (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) - : c * (a / b) = c * a / b - (c * (a mod b)) / b. - Proof. - rewrite div_mul_diff_exact by assumption; lia. - Qed. - - Lemma div_mul_diff_exact'' a b c - (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) - : a * c / b = (a / b) * c + (c * (a mod b)) / b. - Proof. - rewrite (Z.mul_comm a c), div_mul_diff_exact by lia; lia. - Qed. - - Lemma div_mul_diff_exact''' a b c - (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) - : (a / b) * c = a * c / b - (c * (a mod b)) / b. - Proof. - rewrite (Z.mul_comm a c), div_mul_diff_exact by lia; lia. - Qed. - - Lemma div_mul_diff a b c - (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) - : c * a / b - c * (a / b) <= c. - Proof. - rewrite div_mul_diff_exact by assumption. - ring_simplify; auto with zarith. - Qed. - - Lemma div_mul_le_le a b c - : 0 <= a -> 0 < b -> 0 <= c -> c * (a / b) <= c * a / b <= c * (a / b) + c. - Proof. - pose proof (Z.div_mul_diff a b c); split; try apply Z.div_mul_le; lia. - Qed. - - Lemma div_mul_le_le_offset a b c - : 0 <= a -> 0 < b -> 0 <= c -> c * a / b - c <= c * (a / b). - Proof. - pose proof (Z.div_mul_le_le a b c); lia. - Qed. - - Hint Resolve Zmult_le_compat_r Zmult_le_compat_l Z_div_le Z.div_mul_le_le_offset Z.add_le_mono Z.sub_le_mono : zarith. - - Lemma log2_nonneg' n a : n <= 0 -> n <= Z.log2 a. - Proof. - intros; transitivity 0; auto with zarith. - Qed. - - Hint Resolve log2_nonneg' : zarith. - - Lemma le_lt_to_log2 x y z : 0 <= z -> 0 < y -> 2^x <= y < 2^z -> x <= Z.log2 y < z. - Proof. - destruct (Z_le_gt_dec 0 x); auto with concl_log2 lia. - Qed. - - Lemma div_x_y_x x y : 0 < x -> 0 < y -> x / y / x = 1 / y. - Proof. - intros; rewrite Z.div_div, (Z.mul_comm y x), <- Z.div_div, Z.div_same by lia. - reflexivity. - Qed. - - Hint Rewrite div_x_y_x using zutil_arith : zsimplify. - - Lemma mod_opp_l_z_iff a b (H : b <> 0) : a mod b = 0 <-> (-a) mod b = 0. - Proof. - split; intro H'; apply Z.mod_opp_l_z in H'; rewrite ?Z.opp_involutive in H'; assumption. - Qed. - - Lemma opp_eq_0_iff a : -a = 0 <-> a = 0. - Proof. omega. Qed. - - Hint Rewrite <- mod_opp_l_z_iff using zutil_arith : zsimplify. - Hint Rewrite opp_eq_0_iff : zsimplify. - - Lemma sub_pos_bound a b X : 0 <= a < X -> 0 <= b < X -> -X < a - b < X. - Proof. lia. Qed. - - Lemma shiftl_opp_l a n - : Z.shiftl (-a) n = - Z.shiftl a n - (if Z_zerop (a mod 2 ^ (- n)) then 0 else 1). - Proof. - destruct (Z_dec 0 n) as [ [?|?] | ? ]; - subst; - rewrite ?Z.pow_neg_r by omega; - autorewrite with zsimplify_const; - [ | | simpl; omega ]. - { rewrite !Z.shiftl_mul_pow2 by omega. - nia. } - { rewrite !Z.shiftl_div_pow2 by omega. - rewrite Z.div_opp_l_complete by auto with zarith. - reflexivity. } - Qed. - Hint Rewrite shiftl_opp_l : push_Zshift. - Hint Rewrite <- shiftl_opp_l : pull_Zshift. - - Lemma shiftr_opp_l a n - : Z.shiftr (-a) n = - Z.shiftr a n - (if Z_zerop (a mod 2 ^ n) then 0 else 1). - Proof. - unfold Z.shiftr; rewrite shiftl_opp_l at 1; rewrite Z.opp_involutive. - reflexivity. - Qed. - Hint Rewrite shiftr_opp_l : push_Zshift. - Hint Rewrite <- shiftr_opp_l : pull_Zshift. - - Lemma sub_pos_bound_div a b X : 0 <= a < X -> 0 <= b < X -> -1 <= (a - b) / X <= 0. - Proof. - intros H0 H1; pose proof (Z.sub_pos_bound a b X H0 H1). - assert (Hn : -X <= a - b) by lia. - assert (Hp : a - b <= X - 1) by lia. - split; etransitivity; [ | apply Z_div_le, Hn; lia | apply Z_div_le, Hp; lia | ]; - instantiate; autorewrite with zsimplify; try reflexivity. - Qed. - - Hint Resolve (fun a b X H0 H1 => proj1 (Z.sub_pos_bound_div a b X H0 H1)) - (fun a b X H0 H1 => proj1 (Z.sub_pos_bound_div a b X H0 H1)) : zarith. - - Lemma sub_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (a - b) / X = if a <? b then -1 else 0. - Proof. - intros H0 H1; pose proof (Z.sub_pos_bound_div a b X H0 H1). - destruct (a <? b) eqn:?; Z.ltb_to_lt. - { cut ((a - b) / X <> 0); [ lia | ]. - autorewrite with zstrip_div; auto with zarith lia. } - { autorewrite with zstrip_div; auto with zarith lia. } - Qed. - - Lemma add_opp_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (-b + a) / X = if a <? b then -1 else 0. - Proof. - rewrite !(Z.add_comm (-_)), !Z.add_opp_r. - apply Z.sub_pos_bound_div_eq. - Qed. - - Hint Rewrite Z.sub_pos_bound_div_eq Z.add_opp_pos_bound_div_eq using zutil_arith : zstrip_div. - - Lemma div_small_sym a b : 0 <= a < b -> 0 = a / b. - Proof. intros; symmetry; apply Z.div_small; assumption. Qed. - - Lemma mod_small_sym a b : 0 <= a < b -> a = a mod b. - Proof. intros; symmetry; apply Z.mod_small; assumption. Qed. - - Hint Resolve div_small_sym mod_small_sym : zarith. - - Lemma mod_eq_le_to_eq a b : 0 < a <= b -> a mod b = 0 -> a = b. - Proof. - intros H H'. - assert (a = b * (a / b)) by auto with zarith lia. - assert (a / b = 1) by nia. - nia. - Qed. - Hint Resolve mod_eq_le_to_eq : zarith. - - Lemma mod_eq_le_div_1 a b : 0 < a <= b -> a mod b = 0 -> a / b = 1. - Proof. auto with zarith. Qed. - Hint Resolve mod_eq_le_div_1 : zarith. - Hint Rewrite mod_eq_le_div_1 using zutil_arith : zsimplify. - - Lemma mod_neq_0_le_to_neq a b : a mod b <> 0 -> a <> b. - Proof. repeat intro; subst; autorewrite with zsimplify in *; lia. Qed. - Hint Resolve mod_neq_0_le_to_neq : zarith. - - Lemma div_small_neg x y : 0 < -x <= y -> x / y = -1. - Proof. - intro H; rewrite <- (Z.opp_involutive x). - rewrite Z.div_opp_l_complete by lia. - generalize dependent (-x); clear x; intros x H. - pose proof (mod_neq_0_le_to_neq x y). - autorewrite with zsimplify; edestruct Z_zerop; autorewrite with zsimplify in *; lia. - Qed. - Hint Rewrite div_small_neg using zutil_arith : zsimplify. - - Lemma div_sub_small x y z : 0 <= x < z -> 0 <= y <= z -> (x - y) / z = if x <? y then -1 else 0. - Proof. - pose proof (Zlt_cases x y). - (destruct (x <? y) eqn:?); - intros; autorewrite with zsimplify; try lia. - Qed. - Hint Rewrite div_sub_small using zutil_arith : zsimplify. - - Lemma le_lt_trans n m p : n <= m -> m < p -> n < p. - Proof. lia. Qed. - - Lemma mul_div_lt_by_le x y z b : 0 <= y < z -> 0 <= x < b -> x * y / z < b. - Proof. - intros [? ?] [? ?]; eapply Z.le_lt_trans; [ | eassumption ]. - auto with zarith. - Qed. - Hint Resolve mul_div_lt_by_le : zarith. - - Definition pow_sub_r' - := fun a b c y H0 H1 => @Logic.eq_trans _ _ _ y (@Z.pow_sub_r a b c H0 H1). - Definition pow_sub_r'_sym - := fun a b c y p H0 H1 => Logic.eq_sym (@Logic.eq_trans _ y _ _ (Logic.eq_sym p) (@Z.pow_sub_r a b c H0 H1)). - Hint Resolve pow_sub_r' pow_sub_r'_sym Z.eq_le_incl : zarith. - Hint Resolve (fun b => f_equal (fun e => b ^ e)) (fun e => f_equal (fun b => b ^ e)) : zarith. - Definition mul_div_le' - := fun x y z w p H0 H1 H2 H3 => @Z.le_trans _ _ w (@Z.mul_div_le x y z H0 H1 H2 H3) p. - Hint Resolve mul_div_le' : zarith. - Lemma mul_div_le'' x y z w : y <= w -> 0 <= x -> 0 <= y -> 0 < z -> x <= z -> x * y / z <= w. - Proof. - rewrite (Z.mul_comm x y); intros; apply mul_div_le'; assumption. - Qed. - Hint Resolve mul_div_le'' : zarith. - - Lemma two_p_two_eq_four : 2^(2) = 4. - Proof. reflexivity. Qed. - Hint Rewrite <- two_p_two_eq_four : push_Zpow. - - Lemma base_pow_neg b n : n < 0 -> b^n = 0. - Proof. - destruct n; intro H; try reflexivity; compute in H; congruence. - Qed. - Hint Rewrite base_pow_neg using zutil_arith : zsimplify. - - Lemma div_mod' a b : b <> 0 -> a = (a / b) * b + a mod b. - Proof. intro; etransitivity; [ apply (Z.div_mod a b); assumption | lia ]. Qed. - Hint Rewrite <- div_mod' using zutil_arith : zsimplify. - - Lemma div_mod'' a b : b <> 0 -> a = a mod b + b * (a / b). - Proof. intro; etransitivity; [ apply (Z.div_mod a b); assumption | lia ]. Qed. - Hint Rewrite <- div_mod'' using zutil_arith : zsimplify. - - Lemma div_mod''' a b : b <> 0 -> a = a mod b + (a / b) * b. - Proof. intro; etransitivity; [ apply (Z.div_mod a b); assumption | lia ]. Qed. - Hint Rewrite <- div_mod''' using zutil_arith : zsimplify. - - Definition opp_distr_if (b : bool) x y : -(if b then x else y) = if b then -x else -y. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite opp_distr_if : push_Zopp. - Hint Rewrite <- opp_distr_if : pull_Zopp. - - Lemma mul_r_distr_if (b : bool) x y z : z * (if b then x else y) = if b then z * x else z * y. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite mul_r_distr_if : push_Zmul. - Hint Rewrite <- mul_r_distr_if : pull_Zmul. - - Lemma mul_l_distr_if (b : bool) x y z : (if b then x else y) * z = if b then x * z else y * z. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite mul_l_distr_if : push_Zmul. - Hint Rewrite <- mul_l_distr_if : pull_Zmul. - - Lemma add_r_distr_if (b : bool) x y z : z + (if b then x else y) = if b then z + x else z + y. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite add_r_distr_if : push_Zadd. - Hint Rewrite <- add_r_distr_if : pull_Zadd. - - Lemma add_l_distr_if (b : bool) x y z : (if b then x else y) + z = if b then x + z else y + z. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite add_l_distr_if : push_Zadd. - Hint Rewrite <- add_l_distr_if : pull_Zadd. - - Lemma sub_r_distr_if (b : bool) x y z : z - (if b then x else y) = if b then z - x else z - y. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite sub_r_distr_if : push_Zsub. - Hint Rewrite <- sub_r_distr_if : pull_Zsub. - - Lemma sub_l_distr_if (b : bool) x y z : (if b then x else y) - z = if b then x - z else y - z. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite sub_l_distr_if : push_Zsub. - Hint Rewrite <- sub_l_distr_if : pull_Zsub. - - Lemma div_r_distr_if (b : bool) x y z : z / (if b then x else y) = if b then z / x else z / y. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite div_r_distr_if : push_Zdiv. - Hint Rewrite <- div_r_distr_if : pull_Zdiv. - - Lemma div_l_distr_if (b : bool) x y z : (if b then x else y) / z = if b then x / z else y / z. - Proof. destruct b; reflexivity. Qed. - Hint Rewrite div_l_distr_if : push_Zdiv. - Hint Rewrite <- div_l_distr_if : pull_Zdiv. - - Lemma sub_mod_mod_0 x d : (x - x mod d) mod d = 0. - Proof. - destruct (Z_zerop d); subst; autorewrite with push_Zmod zsimplify; reflexivity. - Qed. - Hint Resolve sub_mod_mod_0 : zarith. - Hint Rewrite sub_mod_mod_0 : zsimplify. - - Lemma div_between n a b : 0 <= n -> b <> 0 -> n * b <= a < (1 + n) * b -> a / b = n. - Proof. intros; Z.div_mod_to_quot_rem_in_goal; nia. Qed. - Hint Rewrite div_between using zutil_arith : zsimplify. - - Lemma mod_small_n n a b : 0 <= n -> b <> 0 -> n * b <= a < (1 + n) * b -> a mod b = a - n * b. - Proof. intros; erewrite Zmod_eq_full, div_between by eassumption. reflexivity. Qed. - Hint Rewrite mod_small_n using zutil_arith : zsimplify. - - Lemma div_between_1 a b : b <> 0 -> b <= a < 2 * b -> a / b = 1. - Proof. intros; rewrite (div_between 1) by lia; reflexivity. Qed. - Hint Rewrite div_between_1 using zutil_arith : zsimplify. - - Lemma mod_small_1 a b : b <> 0 -> b <= a < 2 * b -> a mod b = a - b. - Proof. intros; rewrite (mod_small_n 1) by lia; lia. Qed. - Hint Rewrite mod_small_1 using zutil_arith : zsimplify. - - Lemma div_between_if n a b : 0 <= n -> b <> 0 -> n * b <= a < (2 + n) * b -> (a / b = if (1 + n) * b <=? a then 1 + n else n)%Z. - Proof. - intros. - break_match; Z.ltb_to_lt; - apply div_between; lia. - Qed. - - Lemma mod_small_n_if n a b : 0 <= n -> b <> 0 -> n * b <= a < (2 + n) * b -> a mod b = a - (if (1 + n) * b <=? a then (1 + n) else n) * b. - Proof. intros; erewrite Zmod_eq_full, div_between_if by eassumption; autorewrite with zsimplify_const. reflexivity. Qed. - - Lemma div_between_0_if a b : b <> 0 -> 0 <= a < 2 * b -> a / b = if b <=? a then 1 else 0. - Proof. intros; rewrite (div_between_if 0) by lia; autorewrite with zsimplify_const; reflexivity. Qed. - - Lemma mod_small_0_if a b : b <> 0 -> 0 <= a < 2 * b -> a mod b = a - if b <=? a then b else 0. - Proof. intros; rewrite (mod_small_n_if 0) by lia; autorewrite with zsimplify_const. break_match; lia. Qed. - - Lemma mul_mod_distr_r_full a b c : (a * c) mod (b * c) = (a mod b * c). - Proof. - destruct (Z_zerop b); [ | destruct (Z_zerop c) ]; subst; - autorewrite with zsimplify; auto using Z.mul_mod_distr_r. - Qed. - - Lemma mul_mod_distr_l_full a b c : (c * a) mod (c * b) = c * (a mod b). - Proof. - destruct (Z_zerop b); [ | destruct (Z_zerop c) ]; subst; - autorewrite with zsimplify; auto using Z.mul_mod_distr_l. - Qed. - - Lemma lt_mul_2_mod_sub : forall a b, b <> 0 -> b <= a < 2 * b -> a mod b = a - b. - Proof. - intros a b H H0. - replace (a mod b) with ((1 * b + (a - b)) mod b) by (f_equal; ring). - rewrite Z.mod_add_l by auto. - apply Z.mod_small. - omega. - Qed. - - - Lemma leb_add_same x y : (x <=? y + x) = (0 <=? y). - Proof. destruct (x <=? y + x) eqn:?, (0 <=? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. - Hint Rewrite leb_add_same : zsimplify. - - Lemma ltb_add_same x y : (x <? y + x) = (0 <? y). - Proof. destruct (x <? y + x) eqn:?, (0 <? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. - Hint Rewrite ltb_add_same : zsimplify. - - Lemma geb_add_same x y : (x >=? y + x) = (0 >=? y). - Proof. destruct (x >=? y + x) eqn:?, (0 >=? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. - Hint Rewrite geb_add_same : zsimplify. - - Lemma gtb_add_same x y : (x >? y + x) = (0 >? y). - Proof. destruct (x >? y + x) eqn:?, (0 >? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. - Hint Rewrite gtb_add_same : zsimplify. - - Lemma shiftl_add x y z : 0 <= z -> (x + y) << z = (x << z) + (y << z). - Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. - Hint Rewrite shiftl_add using zutil_arith : push_Zshift. - Hint Rewrite <- shiftl_add using zutil_arith : pull_Zshift. - - Lemma shiftr_add x y z : z <= 0 -> (x + y) >> z = (x >> z) + (y >> z). - Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. - Hint Rewrite shiftr_add using zutil_arith : push_Zshift. - Hint Rewrite <- shiftr_add using zutil_arith : pull_Zshift. - - Lemma shiftl_sub x y z : 0 <= z -> (x - y) << z = (x << z) - (y << z). - Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. - Hint Rewrite shiftl_sub using zutil_arith : push_Zshift. - Hint Rewrite <- shiftl_sub using zutil_arith : pull_Zshift. - - Lemma shiftr_sub x y z : z <= 0 -> (x - y) >> z = (x >> z) - (y >> z). - Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. - Hint Rewrite shiftr_sub using zutil_arith : push_Zshift. - Hint Rewrite <- shiftr_sub using zutil_arith : pull_Zshift. - - Lemma shl_shr_lt x y n m (Hx : 0 <= x < 2^n) (Hy : 0 <= y < 2^n) (Hm : 0 <= m <= n) - : 0 <= (x >> (n - m)) + ((y << m) mod 2^n) < 2^n. - Proof. - cut (0 <= (x >> (n - m)) + ((y << m) mod 2^n) <= 2^n - 1); [ omega | ]. - assert (0 <= x <= 2^n - 1) by omega. - assert (0 <= y <= 2^n - 1) by omega. - assert (0 < 2 ^ (n - m)) by auto with zarith. - assert (0 <= y mod 2 ^ (n - m) < 2^(n-m)) by auto with zarith. - assert (0 <= y mod 2 ^ (n - m) <= 2 ^ (n - m) - 1) by omega. - assert (0 <= (y mod 2 ^ (n - m)) * 2^m <= (2^(n-m) - 1)*2^m) by auto with zarith. - assert (0 <= x / 2^(n-m) < 2^n / 2^(n-m)). - { split; Z.zero_bounds. - apply Z.div_lt_upper_bound; autorewrite with pull_Zpow zsimplify; nia. } - autorewrite with Zshift_to_pow. - split; Z.zero_bounds. - replace (2^n) with (2^(n-m) * 2^m) by (autorewrite with pull_Zpow; f_equal; omega). - rewrite Zmult_mod_distr_r. - autorewrite with pull_Zpow zsimplify push_Zmul in * |- . - nia. - Qed. - - Lemma add_shift_mod x y n m - (Hx : 0 <= x < 2^n) (Hy : 0 <= y) - (Hn : 0 <= n) (Hm : 0 < m) - : (x + y << n) mod (m * 2^n) = x + (y mod m) << n. - Proof. - pose proof (Z.mod_bound_pos y m). - specialize_by omega. - assert (0 < 2^n) by auto with zarith. - autorewrite with Zshift_to_pow. - rewrite Zplus_mod, !Zmult_mod_distr_r. - rewrite Zplus_mod, !Zmod_mod, <- Zplus_mod. - rewrite !(Zmod_eq (_ + _)) by nia. - etransitivity; [ | apply Z.add_0_r ]. - rewrite <- !Z.add_opp_r, <- !Z.add_assoc. - repeat apply f_equal. - ring_simplify. - cut (((x + y mod m * 2 ^ n) / (m * 2 ^ n)) = 0); [ nia | ]. - apply Z.div_small; split; nia. - Qed. - - Lemma add_mul_mod x y n m - (Hx : 0 <= x < 2^n) (Hy : 0 <= y) - (Hn : 0 <= n) (Hm : 0 < m) - : (x + y * 2^n) mod (m * 2^n) = x + (y mod m) * 2^n. - Proof. - generalize (add_shift_mod x y n m). - autorewrite with Zshift_to_pow; auto. - Qed. - - Lemma lt_pow_2_shiftr : forall a n, 0 <= a < 2 ^ n -> a >> n = 0. - Proof. - intros a n H. - destruct (Z_le_dec 0 n). - + rewrite Z.shiftr_div_pow2 by assumption. - auto using Z.div_small. - + assert (2 ^ n = 0) by (apply Z.pow_neg_r; omega). - omega. - Qed. - - Hint Rewrite Z.pow2_bits_eqb using zutil_arith : Ztestbit. - Lemma pow_2_shiftr : forall n, 0 <= n -> (2 ^ n) >> n = 1. - Proof. - intros; apply Z.bits_inj'; intros. - replace 1 with (2 ^ 0) by ring. - repeat match goal with - | |- _ => progress intros - | |- _ => progress rewrite ?Z.eqb_eq, ?Z.eqb_neq in * - | |- _ => progress autorewrite with Ztestbit - | |- context[Z.eqb ?a ?b] => case_eq (Z.eqb a b) - | |- _ => reflexivity || omega - end. - Qed. - - Lemma lt_mul_2_pow_2_shiftr : forall a n, 0 <= a < 2 * 2 ^ n -> - a >> n = if Z_lt_dec a (2 ^ n) then 0 else 1. - Proof. - intros a n H; break_match; [ apply lt_pow_2_shiftr; omega | ]. - destruct (Z_le_dec 0 n). - + replace (2 * 2 ^ n) with (2 ^ (n + 1)) in * - by (rewrite Z.pow_add_r; try omega; ring). - pose proof (Z.shiftr_ones a (n + 1) n H). - pose proof (Z.shiftr_le (2 ^ n) a n). - specialize_by omega. - replace (n + 1 - n) with 1 in * by ring. - replace (Z.ones 1) with 1 in * by reflexivity. - rewrite pow_2_shiftr in * by omega. - omega. - + assert (2 ^ n = 0) by (apply Z.pow_neg_r; omega). - omega. - Qed. - - Lemma shiftr_nonneg_le : forall a n, 0 <= a -> 0 <= n -> a >> n <= a. - Proof. - intros. - repeat match goal with - | [ H : _ <= _ |- _ ] - => rewrite Z.lt_eq_cases in H - | [ H : _ \/ _ |- _ ] => destruct H - | _ => progress subst - | _ => progress autorewrite with zsimplify Zshift_to_pow - | _ => solve [ auto with zarith omega ] - end. - Qed. - Hint Resolve shiftr_nonneg_le : zarith. - - Lemma log2_pred_pow2_full a : Z.log2 (Z.pred (2^a)) = Z.max 0 (Z.pred a). - Proof. - destruct (Z_dec 0 a) as [ [?|?] | ?]. - { rewrite Z.log2_pred_pow2 by assumption. - apply Z.max_case_strong; omega. } - { autorewrite with zsimplify; simpl. - apply Z.max_case_strong; omega. } - { subst; compute; reflexivity. } - Qed. - Hint Rewrite log2_pred_pow2_full : zsimplify. - - Lemma log2_up_le_full a : a <= 2^Z.log2_up a. - Proof. - destruct (Z_dec 1 a) as [ [ ? | ? ] | ? ]; - first [ apply Z.log2_up_spec; assumption - | rewrite Z.log2_up_eqn0 by omega; simpl; omega ]. - Qed. - - Lemma log2_up_le_pow2_full : forall a b : Z, (0 <= b)%Z -> (a <= 2 ^ b)%Z <-> (Z.log2_up a <= b)%Z. - Proof. - intros a b H. - destruct (Z_lt_le_dec 0 a); [ apply Z.log2_up_le_pow2; assumption | ]. - split; transitivity 0%Z; try omega; auto with zarith. - rewrite Z.log2_up_eqn0 by omega. - reflexivity. - Qed. - - Lemma ones_lt_pow2 x y : 0 <= x <= y -> Z.ones x < 2^y. - Proof. - rewrite Z.ones_equiv, Z.lt_pred_le. - auto with zarith. - Qed. - Hint Resolve ones_lt_pow2 : zarith. - - Lemma log2_ones_full x : Z.log2 (Z.ones x) = Z.max 0 (Z.pred x). - Proof. - rewrite Z.ones_equiv, log2_pred_pow2_full; reflexivity. - Qed. - Hint Rewrite log2_ones_full : zsimplify. - - Lemma log2_ones_lt x y : 0 < x <= y -> Z.log2 (Z.ones x) < y. - Proof. - rewrite log2_ones_full; apply Z.max_case_strong; omega. - Qed. - Hint Resolve log2_ones_lt : zarith. - - Lemma log2_ones_le x y : 0 <= x <= y -> Z.log2 (Z.ones x) <= y. - Proof. - rewrite log2_ones_full; apply Z.max_case_strong; omega. - Qed. - Hint Resolve log2_ones_le : zarith. - - Lemma log2_ones_lt_nonneg x y : 0 < y -> x <= y -> Z.log2 (Z.ones x) < y. - Proof. - rewrite log2_ones_full; apply Z.max_case_strong; omega. - Qed. - Hint Resolve log2_ones_lt_nonneg : zarith. - - Lemma log2_lt_pow2_alt a b : 0 < b -> (a < 2^b <-> Z.log2 a < b). - Proof. - destruct (Z_lt_le_dec 0 a); auto using Z.log2_lt_pow2; []. - rewrite Z.log2_nonpos by omega. - split; auto with zarith; []. - intro; eapply le_lt_trans; [ eassumption | ]. - auto with zarith. - Qed. - - Section ZInequalities. - Lemma land_le : forall x y, (0 <= x)%Z -> (Z.land x y <= x)%Z. - Proof. - intros x y H; apply Z.ldiff_le; [assumption|]. - rewrite Z.ldiff_land, Z.land_comm, Z.land_assoc. - rewrite <- Z.land_0_l with (a := y); f_equal. - rewrite Z.land_comm, Z.land_lnot_diag. - reflexivity. - Qed. - - Lemma lor_lower : forall x y, (0 <= x)%Z -> (0 <= y)%Z -> (x <= Z.lor x y)%Z. - Proof. - intros x y H H0; apply Z.ldiff_le; [apply Z.lor_nonneg; auto|]. - rewrite Z.ldiff_land. - apply Z.bits_inj_iff'; intros k Hpos; apply Z.le_ge in Hpos. - rewrite Z.testbit_0_l, Z.land_spec, Z.lnot_spec, Z.lor_spec; - [|apply Z.ge_le; assumption]. - induction (Z.testbit x k), (Z.testbit y k); cbv; reflexivity. - Qed. - - Lemma lor_le : forall x y z, - (0 <= x)%Z - -> (x <= y)%Z - -> (y <= z)%Z - -> (Z.lor x y <= (2 ^ Z.log2_up (z+1)) - 1)%Z. - Proof. - intros x y z H H0 H1; apply Z.ldiff_le. - - - apply Z.le_add_le_sub_r. - replace 1%Z with (2 ^ 0)%Z by (cbv; reflexivity). - rewrite Z.add_0_l. - apply Z.pow_le_mono_r; [cbv; reflexivity|]. - apply Z.log2_up_nonneg. - - - destruct (Z_lt_dec 0 z). - - + assert (forall a, a - 1 = Z.pred a)%Z as HP by (intro; omega); - rewrite HP, <- Z.ones_equiv; clear HP. - apply Z.ldiff_ones_r_low; [apply Z.lor_nonneg; split; omega|]. - rewrite Z.log2_up_eqn, Z.log2_lor; try omega. - apply Z.lt_succ_r. - apply Z.max_case_strong; intros; apply Z.log2_le_mono; omega. - - + replace z with 0%Z by omega. - replace y with 0%Z by omega. - replace x with 0%Z by omega. - cbv; reflexivity. - Qed. - - Lemma pow2_ge_0: forall a, (0 <= 2 ^ a)%Z. - Proof. - intros; apply Z.pow_nonneg; omega. - Qed. - - Lemma pow2_gt_0: forall a, (0 <= a)%Z -> (0 < 2 ^ a)%Z. - Proof. - intros; apply Z.pow_pos_nonneg; [|assumption]; omega. - Qed. - - Local Ltac solve_pow2 := - repeat match goal with - | [|- _ /\ _] => split - | [|- (0 < 2 ^ _)%Z] => apply pow2_gt_0 - | [|- (0 <= 2 ^ _)%Z] => apply pow2_ge_0 - | [|- (2 ^ _ <= 2 ^ _)%Z] => apply Z.pow_le_mono_r - | [|- (_ <= _)%Z] => omega - | [|- (_ < _)%Z] => omega - end. - - Lemma pow2_mod_range : forall a n m, - (0 <= n) -> - (n <= m) -> - (0 <= Z.pow2_mod a n < 2 ^ m). - Proof. - intros; unfold Z.pow2_mod. - rewrite Z.land_ones; [|assumption]. - split; [apply Z.mod_pos_bound, pow2_gt_0; assumption|]. - eapply Z.lt_le_trans; [apply Z.mod_pos_bound, pow2_gt_0; assumption|]. - apply Z.pow_le_mono; [|assumption]. - split; simpl; omega. - Qed. - - Lemma shiftr_range : forall a n m, - (0 <= n)%Z -> - (0 <= m)%Z -> - (0 <= a < 2 ^ (n + m))%Z -> - (0 <= Z.shiftr a n < 2 ^ m)%Z. - Proof. - intros a n m H0 H1 H2; destruct H2. - split; [apply Z.shiftr_nonneg; assumption|]. - rewrite Z.shiftr_div_pow2; [|assumption]. - apply Z.div_lt_upper_bound; [apply pow2_gt_0; assumption|]. - eapply Z.lt_le_trans; [eassumption|apply Z.eq_le_incl]. - apply Z.pow_add_r; omega. - Qed. - - - Lemma shiftr_le_mono: forall a b c d, - (0 <= a)%Z - -> (0 <= d)%Z - -> (a <= c)%Z - -> (d <= b)%Z - -> (Z.shiftr a b <= Z.shiftr c d)%Z. - Proof. - intros. - repeat rewrite Z.shiftr_div_pow2; [|omega|omega]. - etransitivity; [apply Z.div_le_compat_l | apply Z.div_le_mono]; solve_pow2. - Qed. - - Lemma shiftl_le_mono: forall a b c d, - (0 <= a)%Z - -> (0 <= b)%Z - -> (a <= c)%Z - -> (b <= d)%Z - -> (Z.shiftl a b <= Z.shiftl c d)%Z. - Proof. - intros. - repeat rewrite Z.shiftl_mul_pow2; [|omega|omega]. - etransitivity; [apply Z.mul_le_mono_nonneg_l|apply Z.mul_le_mono_nonneg_r]; solve_pow2. - Qed. - End ZInequalities. - - Lemma max_log2_up x y : Z.max (Z.log2_up x) (Z.log2_up y) = Z.log2_up (Z.max x y). - Proof. apply Z.max_monotone; intros ??; apply Z.log2_up_le_mono. Qed. - Hint Rewrite max_log2_up : push_Zmax. - Hint Rewrite <- max_log2_up : pull_Zmax. - - Lemma lor_bounds x y : 0 <= x -> 0 <= y - -> Z.max x y <= Z.lor x y <= 2^Z.log2_up (Z.max x y + 1) - 1. - Proof. - apply Z.max_case_strong; intros; split; - try solve [ eauto using lor_lower, Z.le_trans, lor_le with omega - | rewrite Z.lor_comm; eauto using lor_lower, Z.le_trans, lor_le with omega ]. - Qed. - Lemma lor_bounds_lower x y : 0 <= x -> 0 <= y - -> Z.max x y <= Z.lor x y. - Proof. intros; apply lor_bounds; assumption. Qed. - Lemma lor_bounds_upper x y : Z.lor x y <= 2^Z.log2_up (Z.max x y + 1) - 1. - Proof. - pose proof (proj2 (Z.lor_neg x y)). - destruct (Z_lt_le_dec x 0), (Z_lt_le_dec y 0); - try solve [ intros; apply lor_bounds; assumption ]; - transitivity (2^0-1); - try apply Z.sub_le_mono_r, Z.pow_le_mono_r, Z.log2_up_nonneg; - simpl; omega. - Qed. - Lemma lor_bounds_gen_lower x y l : 0 <= x -> 0 <= y -> l <= Z.max x y - -> l <= Z.lor x y. - Proof. - intros; etransitivity; - solve [ apply lor_bounds; auto - | eauto ]. - Qed. - Lemma lor_bounds_gen_upper x y u : x <= u -> y <= u - -> Z.lor x y <= 2^Z.log2_up (u + 1) - 1. - Proof. - intros; etransitivity; [ apply lor_bounds_upper | ]. - apply Z.sub_le_mono_r, Z.pow_le_mono_r, Z.log2_up_le_mono, Z.max_case_strong; - omega. - Qed. - Lemma lor_bounds_gen x y l u : 0 <= x -> 0 <= y -> l <= Z.max x y -> x <= u -> y <= u - -> l <= Z.lor x y <= 2^Z.log2_up (u + 1) - 1. - Proof. auto using lor_bounds_gen_lower, lor_bounds_gen_upper. Qed. - - Lemma log2_up_le_full_max a : Z.max a 1 <= 2^Z.log2_up a. - Proof. - apply Z.max_case_strong; auto using Z.log2_up_le_full. - intros; rewrite Z.log2_up_eqn0 by assumption; reflexivity. - Qed. - Lemma log2_up_le_1 a : Z.log2_up a <= 1 <-> a <= 2. - Proof. - pose proof (Z.log2_nonneg (Z.pred a)). - destruct (Z_dec a 2) as [ [ ? | ? ] | ? ]. - { rewrite (proj2 (Z.log2_up_null a)) by omega; split; omega. } - { rewrite Z.log2_up_eqn by omega. - split; try omega; intro. - assert (Z.log2 (Z.pred a) = 0) by omega. - assert (Z.pred a <= 1) by (apply Z.log2_null; omega). - omega. } - { subst; cbv -[Z.le]; split; omega. } - Qed. - Lemma log2_up_1_le a : 1 <= Z.log2_up a <-> 2 <= a. - Proof. - pose proof (Z.log2_nonneg (Z.pred a)). - destruct (Z_dec a 2) as [ [ ? | ? ] | ? ]. - { rewrite (proj2 (Z.log2_up_null a)) by omega; split; omega. } - { rewrite Z.log2_up_eqn by omega; omega. } - { subst; cbv -[Z.le]; split; omega. } - Qed. - - Lemma shiftl_le_Proper2 y - : Proper (Z.le ==> Z.le) (fun x => Z.shiftl x y). - Proof. - unfold Basics.flip in *. - pose proof (Zle_cases 0 y) as Hx. - intros x x' H. - pose proof (Zle_cases 0 x) as Hy. - pose proof (Zle_cases 0 x') as Hy'. - destruct (0 <=? y), (0 <=? x), (0 <=? x'); - autorewrite with Zshift_to_pow; - Z.replace_all_neg_with_pos; - autorewrite with pull_Zopp; - rewrite ?Z.div_opp_l_complete; - repeat destruct (Z_zerop _); - autorewrite with zsimplify_const pull_Zopp; - auto with zarith; - repeat match goal with - | [ |- context[-?x - ?y] ] - => replace (-x - y) with (-(x + y)) by omega - | _ => rewrite <- Z.opp_le_mono - | _ => rewrite <- Z.add_le_mono_r - | _ => solve [ auto with zarith ] - | [ |- ?x <= ?y + 1 ] - => cut (x <= y); [ omega | solve [ auto with zarith ] ] - | [ |- -_ <= _ ] - => solve [ transitivity (-0); auto with zarith ] - end. - { repeat match goal with H : context[_ mod _] |- _ => revert H end; - Z.div_mod_to_quot_rem_in_goal; nia. } - Qed. - - Lemma shiftl_le_Proper1 x - (R := fun b : bool => if b then Z.le else Basics.flip Z.le) - : Proper (R (0 <=? x) ==> Z.le) (Z.shiftl x). - Proof. - unfold Basics.flip in *. - pose proof (Zle_cases 0 x) as Hx. - intros y y' H. - pose proof (Zle_cases 0 y) as Hy. - pose proof (Zle_cases 0 y') as Hy'. - destruct (0 <=? x), (0 <=? y), (0 <=? y'); subst R; cbv beta iota in *; - autorewrite with Zshift_to_pow; - Z.replace_all_neg_with_pos; - autorewrite with pull_Zopp; - rewrite ?Z.div_opp_l_complete; - repeat destruct (Z_zerop _); - autorewrite with zsimplify_const pull_Zopp; - auto with zarith; - repeat match goal with - | [ |- context[-?x - ?y] ] - => replace (-x - y) with (-(x + y)) by omega - | _ => rewrite <- Z.opp_le_mono - | _ => rewrite <- Z.add_le_mono_r - | _ => solve [ auto with zarith ] - | [ |- ?x <= ?y + 1 ] - => cut (x <= y); [ omega | solve [ auto with zarith ] ] - | [ |- context[2^?x] ] - => lazymatch goal with - | [ H : 1 < 2^x |- _ ] => fail - | [ H : 0 < 2^x |- _ ] => fail - | [ H : 0 <= 2^x |- _ ] => fail - | _ => first [ assert (1 < 2^x) by auto with zarith - | assert (0 < 2^x) by auto with zarith - | assert (0 <= 2^x) by auto with zarith ] - end - | [ H : ?x <= ?y |- _ ] - => is_var x; is_var y; - lazymatch goal with - | [ H : 2^x <= 2^y |- _ ] => fail - | [ H : 2^x < 2^y |- _ ] => fail - | _ => assert (2^x <= 2^y) by auto with zarith - end - | [ H : ?x <= ?y, H' : ?f ?x = ?k, H'' : ?f ?y <> ?k |- _ ] - => let Hn := fresh in - assert (Hn : x <> y) by congruence; - assert (x < y) by omega; clear H Hn - | [ H : ?x <= ?y, H' : ?f ?x <> ?k, H'' : ?f ?y = ?k |- _ ] - => let Hn := fresh in - assert (Hn : x <> y) by congruence; - assert (x < y) by omega; clear H Hn - | _ => solve [ repeat match goal with H : context[_ mod _] |- _ => revert H end; - Z.div_mod_to_quot_rem_in_goal; subst; - lazymatch goal with - | [ |- _ <= (?a * ?q + ?r) * ?q' ] - => transitivity (q * (a * q') + r * q'); - [ assert (0 < a * q') by nia; nia - | nia ] - end ] - end. - { replace y' with (y + (y' - y)) by omega. - rewrite Z.pow_add_r, <- Zdiv_Zdiv by auto with zarith. - assert (y < y') by (assert (y <> y') by congruence; omega). - assert (1 < 2^(y'-y)) by auto with zarith. - assert (0 < x / 2^y) - by (repeat match goal with H : context[_ mod _] |- _ => revert H end; - Z.div_mod_to_quot_rem_in_goal; nia). - assert (2^y <= x) - by (repeat match goal with H : context[_ / _] |- _ => revert H end; - Z.div_mod_to_quot_rem_in_goal; nia). - match goal with - | [ |- ?x + 1 <= ?y ] => cut (x < y); [ omega | ] - end. - auto with zarith. } - Qed. - - Lemma shiftr_le_Proper2 y - : Proper (Z.le ==> Z.le) (fun x => Z.shiftr x y). - Proof. apply shiftl_le_Proper2. Qed. - - Lemma shiftr_le_Proper1 x - (R := fun b : bool => if b then Z.le else Basics.flip Z.le) - : Proper (R (x <? 0) ==> Z.le) (Z.shiftr x). - Proof. - subst R; intros y y' H'; unfold Z.shiftr; apply shiftl_le_Proper1. - unfold Basics.flip in *. - pose proof (Zle_cases 0 x). - pose proof (Zlt_cases x 0). - destruct (0 <=? x), (x <? 0); try omega. - Qed. -End Z. - -Module N2Z. - Lemma inj_land n m : Z.of_N (N.land n m) = Z.land (Z.of_N n) (Z.of_N m). - Proof. destruct n, m; reflexivity. Qed. - Hint Rewrite inj_land : push_Zof_N. - Hint Rewrite <- inj_land : pull_Zof_N. - - Lemma inj_lor n m : Z.of_N (N.lor n m) = Z.lor (Z.of_N n) (Z.of_N m). - Proof. destruct n, m; reflexivity. Qed. - Hint Rewrite inj_lor : push_Zof_N. - Hint Rewrite <- inj_lor : pull_Zof_N. - - Lemma inj_shiftl: forall x y, Z.of_N (N.shiftl x y) = Z.shiftl (Z.of_N x) (Z.of_N y). - Proof. - intros x y. - apply Z.bits_inj_iff'; intros k Hpos. - rewrite Z2N.inj_testbit; [|assumption]. - rewrite Z.shiftl_spec; [|assumption]. - - assert ((Z.to_N k) >= y \/ (Z.to_N k) < y)%N as g by ( - unfold N.ge, N.lt; induction (N.compare (Z.to_N k) y); [left|auto|left]; - intro H; inversion H). - - destruct g as [g|g]; - [ rewrite N.shiftl_spec_high; [|apply N2Z.inj_le; rewrite Z2N.id|apply N.ge_le] - | rewrite N.shiftl_spec_low]; try assumption. - - - rewrite <- N2Z.inj_testbit; f_equal. - rewrite N2Z.inj_sub, Z2N.id; [reflexivity|assumption|apply N.ge_le; assumption]. - - - apply N2Z.inj_lt in g. - rewrite Z2N.id in g; [symmetry|assumption]. - apply Z.testbit_neg_r; omega. - Qed. - Hint Rewrite inj_shiftl : push_Zof_N. - Hint Rewrite <- inj_shiftl : pull_Zof_N. - - Lemma inj_shiftr: forall x y, Z.of_N (N.shiftr x y) = Z.shiftr (Z.of_N x) (Z.of_N y). - Proof. - intros. - apply Z.bits_inj_iff'; intros k Hpos. - rewrite Z2N.inj_testbit; [|assumption]. - rewrite Z.shiftr_spec, N.shiftr_spec; [|apply N2Z.inj_le; rewrite Z2N.id|]; try assumption. - rewrite <- N2Z.inj_testbit; f_equal. - rewrite N2Z.inj_add; f_equal. - apply Z2N.id; assumption. - Qed. - Hint Rewrite inj_shiftr : push_Zof_N. - Hint Rewrite <- inj_shiftr : pull_Zof_N. -End N2Z. - -Module Export BoundsTactics. - Ltac prime_bound := Z.prime_bound. - Ltac zero_bounds := Z.zero_bounds. -End BoundsTactics. +Require Coq.ZArith.Zpower Coq.ZArith.Znumtheory Coq.ZArith.ZArith Coq.ZArith.Zdiv. +Require Coq.omega.Omega Coq.micromega.Psatz Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Arith. +Require Crypto.Util.ZUtil.AddGetCarry. +Require Crypto.Util.ZUtil.AddModulo. +Require Crypto.Util.ZUtil.CC. +Require Crypto.Util.ZUtil.CPS. +Require Crypto.Util.ZUtil.Definitions. +Require Crypto.Util.ZUtil.DistrIf. +Require Crypto.Util.ZUtil.Div. +Require Crypto.Util.ZUtil.Div.Bootstrap. +Require Crypto.Util.ZUtil.Divide. +Require Crypto.Util.ZUtil.EquivModulo. +Require Crypto.Util.ZUtil.Ge. +Require Crypto.Util.ZUtil.Hints. +Require Crypto.Util.ZUtil.Hints.Core. +Require Crypto.Util.ZUtil.Hints.PullPush. +Require Crypto.Util.ZUtil.Hints.ZArith. +Require Crypto.Util.ZUtil.Hints.Ztestbit. +Require Crypto.Util.ZUtil.Land. +Require Crypto.Util.ZUtil.LandLorBounds. +Require Crypto.Util.ZUtil.LandLorShiftBounds. +Require Crypto.Util.ZUtil.Le. +Require Crypto.Util.ZUtil.Lnot. +Require Crypto.Util.ZUtil.Log2. +Require Crypto.Util.ZUtil.ModInv. +Require Crypto.Util.ZUtil.Modulo. +Require Crypto.Util.ZUtil.Modulo.Bootstrap. +Require Crypto.Util.ZUtil.Modulo.PullPush. +Require Crypto.Util.ZUtil.Morphisms. +Require Crypto.Util.ZUtil.Mul. +Require Crypto.Util.ZUtil.MulSplit. +Require Crypto.Util.ZUtil.N2Z. +Require Crypto.Util.ZUtil.Notations. +Require Crypto.Util.ZUtil.Odd. +Require Crypto.Util.ZUtil.Ones. +Require Crypto.Util.ZUtil.Opp. +Require Crypto.Util.ZUtil.Peano. +Require Crypto.Util.ZUtil.Pow. +Require Crypto.Util.ZUtil.Pow2. +Require Crypto.Util.ZUtil.Pow2Mod. +Require Crypto.Util.ZUtil.Quot. +Require Crypto.Util.ZUtil.Rshi. +Require Crypto.Util.ZUtil.Sgn. +Require Crypto.Util.ZUtil.Shift. +Require Crypto.Util.ZUtil.Sorting. +Require Crypto.Util.ZUtil.Stabilization. +Require Crypto.Util.ZUtil.Tactics. +Require Crypto.Util.ZUtil.Tactics.CompareToSgn. +Require Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Crypto.Util.ZUtil.Tactics.DivideExistsMul. +Require Crypto.Util.ZUtil.Tactics.LinearSubstitute. +Require Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Crypto.Util.ZUtil.Tactics.PeelLe. +Require Crypto.Util.ZUtil.Tactics.PrimeBound. +Require Crypto.Util.ZUtil.Tactics.PullPush. +Require Crypto.Util.ZUtil.Tactics.PullPush.Modulo. +Require Crypto.Util.ZUtil.Tactics.ReplaceNegWithPos. +Require Crypto.Util.ZUtil.Tactics.RewriteModSmall. +Require Crypto.Util.ZUtil.Tactics.SimplifyFractionsLe. +Require Crypto.Util.ZUtil.Tactics.SplitMinMax. +Require Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Crypto.Util.ZUtil.Tactics.Ztestbit. +Require Crypto.Util.ZUtil.Testbit. +Require Crypto.Util.ZUtil.Z2Nat. +Require Crypto.Util.ZUtil.ZSimplify. +Require Crypto.Util.ZUtil.ZSimplify.Autogenerated. +Require Crypto.Util.ZUtil.ZSimplify.Core. +Require Crypto.Util.ZUtil.ZSimplify.Simple. +Require Crypto.Util.ZUtil.Zselect. diff --git a/src/Util/ZUtil/Definitions.v b/src/Util/ZUtil/Definitions.v index af2d8239e..4ef6b5403 100644 --- a/src/Util/ZUtil/Definitions.v +++ b/src/Util/ZUtil/Definitions.v @@ -84,4 +84,9 @@ Module Z. := if s =? 2^Z.log2 s then mul_split_at_bitwidth (Z.log2 s) x y else ((x * y) mod s, (x * y) / s). + + Definition round_lor_land_bound (x : Z) : Z + := if (0 <=? x)%Z + then 2^(Z.log2_up (x+1))-1 + else -2^(Z.log2_up (-x)). End Z. diff --git a/src/Util/ZUtil/DistrIf.v b/src/Util/ZUtil/DistrIf.v new file mode 100644 index 000000000..0d20fc1f4 --- /dev/null +++ b/src/Util/ZUtil/DistrIf.v @@ -0,0 +1,51 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints.Core. +Local Open Scope Z_scope. + +Module Z. + Definition opp_distr_if (b : bool) x y : -(if b then x else y) = if b then -x else -y. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite opp_distr_if : push_Zopp. + Hint Rewrite <- opp_distr_if : pull_Zopp. + + Lemma mul_r_distr_if (b : bool) x y z : z * (if b then x else y) = if b then z * x else z * y. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite mul_r_distr_if : push_Zmul. + Hint Rewrite <- mul_r_distr_if : pull_Zmul. + + Lemma mul_l_distr_if (b : bool) x y z : (if b then x else y) * z = if b then x * z else y * z. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite mul_l_distr_if : push_Zmul. + Hint Rewrite <- mul_l_distr_if : pull_Zmul. + + Lemma add_r_distr_if (b : bool) x y z : z + (if b then x else y) = if b then z + x else z + y. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite add_r_distr_if : push_Zadd. + Hint Rewrite <- add_r_distr_if : pull_Zadd. + + Lemma add_l_distr_if (b : bool) x y z : (if b then x else y) + z = if b then x + z else y + z. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite add_l_distr_if : push_Zadd. + Hint Rewrite <- add_l_distr_if : pull_Zadd. + + Lemma sub_r_distr_if (b : bool) x y z : z - (if b then x else y) = if b then z - x else z - y. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite sub_r_distr_if : push_Zsub. + Hint Rewrite <- sub_r_distr_if : pull_Zsub. + + Lemma sub_l_distr_if (b : bool) x y z : (if b then x else y) - z = if b then x - z else y - z. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite sub_l_distr_if : push_Zsub. + Hint Rewrite <- sub_l_distr_if : pull_Zsub. + + Lemma div_r_distr_if (b : bool) x y z : z / (if b then x else y) = if b then z / x else z / y. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite div_r_distr_if : push_Zdiv. + Hint Rewrite <- div_r_distr_if : pull_Zdiv. + + Lemma div_l_distr_if (b : bool) x y z : (if b then x else y) / z = if b then x / z else y / z. + Proof. destruct b; reflexivity. Qed. + Hint Rewrite div_l_distr_if : push_Zdiv. + Hint Rewrite <- div_l_distr_if : pull_Zdiv. +End Z. diff --git a/src/Util/ZUtil/Div.v b/src/Util/ZUtil/Div.v index 5ae17ad1a..7012f83c0 100644 --- a/src/Util/ZUtil/Div.v +++ b/src/Util/ZUtil/Div.v @@ -2,11 +2,14 @@ Require Import Coq.ZArith.ZArith Coq.micromega.Lia. Require Import Coq.ZArith.Znumtheory. Require Import Crypto.Util.ZUtil.Tactics.CompareToSgn. Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.ZUtil.Le. Require Import Crypto.Util.ZUtil.Hints.Core. Require Import Crypto.Util.ZUtil.Hints.ZArith. Require Import Crypto.Util.ZUtil.Hints.PullPush. +Require Import Crypto.Util.ZUtil.Hints. Require Import Crypto.Util.ZUtil.ZSimplify.Core. +Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z_scope. Module Z. @@ -262,4 +265,165 @@ Module Z. Lemma div_opp_r a b : a / (-b) = ((-a) / b). Proof. Z.div_mod_to_quot_rem; nia. Qed. Hint Resolve div_opp_r : zarith. + + Lemma div_floor : forall a b c, 0 < b -> a < b * (Z.succ c) -> a / b <= c. + Proof. + intros. + apply Z.lt_succ_r. + apply Z.div_lt_upper_bound; try omega. + Qed. + + Lemma mul_div_le x y z + (Hx : 0 <= x) (Hy : 0 <= y) (Hz : 0 < z) + (Hyz : y <= z) + : x * y / z <= x. + Proof. + transitivity (x * z / z); [ | rewrite Z.div_mul by lia; lia ]. + apply Z_div_le; nia. + Qed. + Hint Resolve mul_div_le : zarith. + + Lemma div_mul_diff_exact a b c + (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) + : c * a / b = c * (a / b) + (c * (a mod b)) / b. + Proof. + rewrite (Z_div_mod_eq a b) at 1 by lia. + rewrite Z.mul_add_distr_l. + replace (c * (b * (a / b))) with ((c * (a / b)) * b) by lia. + rewrite Z.div_add_l by lia. + lia. + Qed. + + Lemma div_mul_diff_exact' a b c + (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) + : c * (a / b) = c * a / b - (c * (a mod b)) / b. + Proof. + rewrite div_mul_diff_exact by assumption; lia. + Qed. + + Lemma div_mul_diff_exact'' a b c + (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) + : a * c / b = (a / b) * c + (c * (a mod b)) / b. + Proof. + rewrite (Z.mul_comm a c), div_mul_diff_exact by lia; lia. + Qed. + + Lemma div_mul_diff_exact''' a b c + (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) + : (a / b) * c = a * c / b - (c * (a mod b)) / b. + Proof. + rewrite (Z.mul_comm a c), div_mul_diff_exact by lia; lia. + Qed. + + Lemma div_mul_diff a b c + (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) + : c * a / b - c * (a / b) <= c. + Proof. + rewrite div_mul_diff_exact by assumption. + ring_simplify; auto with zarith. + Qed. + + Lemma div_mul_le_le a b c + : 0 <= a -> 0 < b -> 0 <= c -> c * (a / b) <= c * a / b <= c * (a / b) + c. + Proof. + pose proof (Z.div_mul_diff a b c); split; try apply Z.div_mul_le; lia. + Qed. + + Lemma div_mul_le_le_offset a b c + : 0 <= a -> 0 < b -> 0 <= c -> c * a / b - c <= c * (a / b). + Proof. + pose proof (Z.div_mul_le_le a b c); lia. + Qed. + Hint Resolve div_mul_le_le_offset : zarith. + + Lemma div_x_y_x x y : 0 < x -> 0 < y -> x / y / x = 1 / y. + Proof. + intros; rewrite Z.div_div, (Z.mul_comm y x), <- Z.div_div, Z.div_same by lia. + reflexivity. + Qed. + Hint Rewrite div_x_y_x using zutil_arith : zsimplify. + + Lemma sub_pos_bound_div a b X : 0 <= a < X -> 0 <= b < X -> -1 <= (a - b) / X <= 0. + Proof. + intros H0 H1; pose proof (Z.sub_pos_bound a b X H0 H1). + assert (Hn : -X <= a - b) by lia. + assert (Hp : a - b <= X - 1) by lia. + split; etransitivity; [ | apply Z_div_le, Hn; lia | apply Z_div_le, Hp; lia | ]; + instantiate; autorewrite with zsimplify; try reflexivity. + Qed. + + Hint Resolve (fun a b X H0 H1 => proj1 (Z.sub_pos_bound_div a b X H0 H1)) + (fun a b X H0 H1 => proj1 (Z.sub_pos_bound_div a b X H0 H1)) : zarith. + + Lemma sub_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (a - b) / X = if a <? b then -1 else 0. + Proof. + intros H0 H1; pose proof (Z.sub_pos_bound_div a b X H0 H1). + destruct (a <? b) eqn:?; Z.ltb_to_lt. + { cut ((a - b) / X <> 0); [ lia | ]. + autorewrite with zstrip_div; auto with zarith lia. } + { autorewrite with zstrip_div; auto with zarith lia. } + Qed. + + Lemma add_opp_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (-b + a) / X = if a <? b then -1 else 0. + Proof. + rewrite !(Z.add_comm (-_)), !Z.add_opp_r. + apply Z.sub_pos_bound_div_eq. + Qed. + + Hint Rewrite Z.sub_pos_bound_div_eq Z.add_opp_pos_bound_div_eq using zutil_arith : zstrip_div. + + Lemma div_small_sym a b : 0 <= a < b -> 0 = a / b. + Proof. intros; symmetry; apply Z.div_small; assumption. Qed. + Hint Resolve div_small_sym : zarith. + + Lemma mod_eq_le_div_1 a b : 0 < a <= b -> a mod b = 0 -> a / b = 1. + Proof. intros; Z.div_mod_to_quot_rem; nia. Qed. + Hint Resolve mod_eq_le_div_1 : zarith. + Hint Rewrite mod_eq_le_div_1 using zutil_arith : zsimplify. + + Lemma div_small_neg x y : 0 < -x <= y -> x / y = -1. + Proof. intros; Z.div_mod_to_quot_rem; nia. Qed. + Hint Rewrite div_small_neg using zutil_arith : zsimplify. + + Lemma div_sub_small x y z : 0 <= x < z -> 0 <= y <= z -> (x - y) / z = if x <? y then -1 else 0. + Proof. + pose proof (Zlt_cases x y). + (destruct (x <? y) eqn:?); + intros; autorewrite with zsimplify; try lia. + Qed. + Hint Rewrite div_sub_small using zutil_arith : zsimplify. + + Lemma mul_div_lt_by_le x y z b : 0 <= y < z -> 0 <= x < b -> x * y / z < b. + Proof. + intros [? ?] [? ?]; eapply Z.le_lt_trans; [ | eassumption ]. + auto with zarith. + Qed. + Hint Resolve mul_div_lt_by_le : zarith. + + Definition mul_div_le' + := fun x y z w p H0 H1 H2 H3 => @Z.le_trans _ _ w (@Z.mul_div_le x y z H0 H1 H2 H3) p. + Hint Resolve mul_div_le' : zarith. + Lemma mul_div_le'' x y z w : y <= w -> 0 <= x -> 0 <= y -> 0 < z -> x <= z -> x * y / z <= w. + Proof. + rewrite (Z.mul_comm x y); intros; apply mul_div_le'; assumption. + Qed. + Hint Resolve mul_div_le'' : zarith. + + Lemma div_between n a b : 0 <= n -> b <> 0 -> n * b <= a < (1 + n) * b -> a / b = n. + Proof. intros; Z.div_mod_to_quot_rem_in_goal; nia. Qed. + Hint Rewrite div_between using zutil_arith : zsimplify. + + Lemma div_between_1 a b : b <> 0 -> b <= a < 2 * b -> a / b = 1. + Proof. intros; rewrite (div_between 1) by lia; reflexivity. Qed. + Hint Rewrite div_between_1 using zutil_arith : zsimplify. + + Lemma div_between_if n a b : 0 <= n -> b <> 0 -> n * b <= a < (2 + n) * b -> (a / b = if (1 + n) * b <=? a then 1 + n else n)%Z. + Proof. + intros. + break_match; Z.ltb_to_lt; + apply div_between; lia. + Qed. + + Lemma div_between_0_if a b : b <> 0 -> 0 <= a < 2 * b -> a / b = if b <=? a then 1 else 0. + Proof. intros; rewrite (div_between_if 0) by lia; autorewrite with zsimplify_const; reflexivity. Qed. End Z. diff --git a/src/Util/ZUtil/Divide.v b/src/Util/ZUtil/Divide.v new file mode 100644 index 000000000..8609db5ad --- /dev/null +++ b/src/Util/ZUtil/Divide.v @@ -0,0 +1,36 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.ZArith.Znumtheory. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Tactics.DivideExistsMul. +Local Open Scope Z_scope. + +Module Z. + Lemma divide_mul_div: forall a b c (a_nonzero : a <> 0) (c_nonzero : c <> 0), + (a | b * (a / c)) -> (c | a) -> (c | b). + Proof. + intros ? ? ? ? ? divide_a divide_c_a; do 2 Z.divide_exists_mul. + rewrite divide_c_a in divide_a. + rewrite Z.div_mul' in divide_a by auto. + replace (b * k) with (k * b) in divide_a by ring. + replace (c * k * k0) with (k * (k0 * c)) in divide_a by ring. + rewrite Z.mul_cancel_l in divide_a by (intuition auto with nia; rewrite H in divide_c_a; ring_simplify in divide_a; intuition). + eapply Zdivide_intro; eauto. + Qed. + + Lemma divide2_even_iff : forall n, (2 | n) <-> Z.even n = true. + Proof. + intros n; split. { + intro divide2_n. + Z.divide_exists_mul; [ | pose proof (Z.mod_pos_bound n 2); omega]. + rewrite divide2_n. + apply Z.even_mul. + } { + intro n_even. + pose proof (Zmod_even n) as H. + rewrite n_even in H. + apply Zmod_divide; omega || auto. + } + Qed. +End Z. diff --git a/src/Util/ZUtil/Hints/ZArith.v b/src/Util/ZUtil/Hints/ZArith.v index 17e56f9cf..2aa70dc97 100644 --- a/src/Util/ZUtil/Hints/ZArith.v +++ b/src/Util/ZUtil/Hints/ZArith.v @@ -6,3 +6,5 @@ Hint Resolve (fun a b H => proj1 (Z.mod_pos_bound a b H)) (fun a b H => proj2 (Z Hint Resolve (fun n m => proj1 (Z.opp_le_mono n m)) : zarith. Hint Resolve (fun n m => proj1 (Z.pred_le_mono n m)) : zarith. Hint Resolve (fun a b => proj2 (Z.lor_nonneg a b)) : zarith. + +Hint Resolve Zmult_le_compat_r Zmult_le_compat_l Z_div_le Z.add_le_mono Z.sub_le_mono : zarith. diff --git a/src/Util/ZUtil/Land.v b/src/Util/ZUtil/Land.v index f46d541e9..7f27f942d 100644 --- a/src/Util/ZUtil/Land.v +++ b/src/Util/ZUtil/Land.v @@ -1,6 +1,8 @@ Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. Require Import Crypto.Util.ZUtil.Notations. -Local Open Scope Z_scope. +Require Import Crypto.Util.ZUtil.Definitions. +Local Open Scope bool_scope. Local Open Scope Z_scope. Module Z. Lemma land_same_r : forall a b, (a &' b) &' b = a &' b. @@ -10,4 +12,15 @@ Module Z. case_eq (Z.testbit b n); intros; rewrite ?Bool.andb_true_r, ?Bool.andb_false_r; reflexivity. Qed. + + Lemma land_m1'_l a : Z.land (-1) a = a. + Proof. apply Z.land_m1_l. Qed. + Hint Rewrite Z.land_m1_l land_m1'_l : zsimplify_const zsimplify zsimplify_fast. + + Lemma land_m1'_r a : Z.land a (-1) = a. + Proof. apply Z.land_m1_r. Qed. + Hint Rewrite Z.land_m1_r land_m1'_r : zsimplify_const zsimplify zsimplify_fast. + + Lemma sub_1_lt_le x y : (x - 1 < y) <-> (x <= y). + Proof. lia. Qed. End Z. diff --git a/src/Util/ZUtil/LandLorBounds.v b/src/Util/ZUtil/LandLorBounds.v new file mode 100644 index 000000000..1b10ecf97 --- /dev/null +++ b/src/Util/ZUtil/LandLorBounds.v @@ -0,0 +1,132 @@ +Require Import Coq.micromega.Lia. +Require Import Coq.ZArith.ZArith. +Require Import Coq.Classes.Morphisms. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Pow2. +Require Import Crypto.Util.ZUtil.Tactics.PeelLe. +Require Import Crypto.Util.ZUtil.Modulo.PullPush. +Require Import Crypto.Util.ZUtil.Ones. +Require Import Crypto.Util.ZUtil.Lnot. +Require Import Crypto.Util.ZUtil.Land. +Require Import Crypto.Util.Tactics.UniquePose. +Require Import Crypto.Util.Tactics.DestructHead. +Require Import Crypto.Util.Tactics.BreakMatch. +Local Open Scope Z_scope. + +Module Z. + Local Ltac saturate := + repeat first [ progress cbv [Z.round_lor_land_bound Proper respectful Basics.flip] in * + | progress cbn in * + | progress intros + | match goal with + | [ |- context[Z.log2_up ?x] ] + => unique pose proof (Z.log2_up_nonneg x) + | [ |- context[2^?x] ] + => unique assert (0 <= 2^x) by (apply Z.pow_nonneg; lia) + | [ H : 0 <= ?x |- context[2^?x] ] + => unique assert (0 < 2^x) by (apply Z.pow_pos_nonneg; lia) + | [ H : Pos.le ?x ?y |- context[Z.pos ?x] ] + => unique assert (Z.pos x <= Z.pos y) by lia + | [ H : Pos.le ?x ?y |- context[Z.pos (?x+1)] ] + => unique assert (Z.pos (x+1) <= Z.pos (y+1)) by lia + | [ H : Z.le ?x ?y |- context[2^Z.log2_up ?x] ] + => unique assert (2^Z.log2_up x <= 2^Z.log2_up y) by (Z.peel_le; lia) + end ]. + Local Ltac do_rewrites_step := + match goal with + | [ |- ?R ?x ?x ] => reflexivity + | [ |- context[Z.land (-2^_) (-2^_)] ] + => rewrite <- !Z.lnot_ones_equiv, <- !Z.lnot_lor, !Z.lor_ones_ones, !Z.lnot_ones_equiv + | [ |- context[Z.lor (-2^_) (-2^_)] ] + => rewrite <- !Z.lnot_ones_equiv, <- !Z.lnot_land, !Z.land_ones_ones, !Z.lnot_ones_equiv + | [ |- context[Z.land (2^_-1) (2^_-1)] ] + => rewrite !Z.sub_1_r, <- !Z.ones_equiv, !Z.land_ones_ones, !Z.ones_equiv, <- !Z.sub_1_r + | [ |- context[Z.lor (2^_-1) (2^_-1)] ] + => rewrite !Z.sub_1_r, <- !Z.ones_equiv, !Z.lor_ones_ones, !Z.ones_equiv, <- !Z.sub_1_r + | [ |- context[Z.land (2^?x-1) (-2^?y)] ] + => rewrite (@Z.land_comm (2^x-1) (-2^y)) + | [ |- context[Z.lor (2^?x-1) (-2^?y)] ] + => rewrite (@Z.lor_comm (2^x-1) (-2^y)) + | [ |- context[Z.land (-2^_) (2^_-1)] ] + => rewrite !Z.sub_1_r, <- !Z.ones_equiv, !Z.land_ones, ?Z.ones_equiv, <- ?Z.sub_1_r by lia + | [ |- context[Z.lor (-2^?x) (2^?y-1)] ] + => rewrite <- !Z.lnot_ones_equiv, <- (Z.lnot_involutive (2^y-1)), <- !Z.lnot_land, ?Z.lnot_ones_equiv, (Z.lnot_sub1 (2^y)), !Z.ones_equiv, ?Z.lnot_equiv, <- !Z.sub_1_r + | [ |- context[-?x mod ?y] ] + => rewrite (@Z.opp_mod_mod_push x y) by Z.NoZMod + | [ H : ?x <= ?x |- _ ] => clear H + | [ H : ?x < ?y, H' : ?y <= ?z |- _ ] => unique assert (x < z) by lia + | [ H : ?x < ?y, H' : ?a <= ?x |- _ ] => unique assert (a < y) by lia + | [ H : 2^?x < 2^?y |- context[2^?x mod 2^?y] ] + => repeat first [ rewrite (Z.mod_small (2^x) (2^y)) by lia + | rewrite !(@Z_mod_nz_opp_full (2^x) (2^y)) ] + | [ H : ?x < ?y, H' : context[?x mod ?y] |- _ ] => rewrite (Z.mod_small x y) in H' by lia + | [ |- context[2^?x mod 2^?y] ] + => let H := fresh in + destruct (@Z.pow2_lt_or_divides x y ltac:(lia)) as [H|H]; + [ repeat first [ rewrite (Z.mod_small (2^x) (2^y)) by lia + | rewrite !(@Z_mod_nz_opp_full (2^x) (2^y)) ] + | rewrite H ] + | _ => progress autorewrite with zsimplify_const + end. + Local Ltac do_rewrites := repeat do_rewrites_step. + Local Ltac fin_t := + repeat first [ progress destruct_head'_and + | match goal with + | [ H : orb _ _ = _ |- _ ] + => progress rewrite ?Bool.orb_true_iff, ?Bool.orb_false_iff, ?Z.ltb_lt, ?Z.ltb_ge in * + end + | break_innermost_match_step + | progress destruct_head'_or + | lia + | progress Z.peel_le ]. + Local Ltac t := + saturate; do_rewrites; fin_t. + + Local Instance land_round_Proper_pos_r x + : Proper (Pos.le ==> Z.le) + (fun y => + Z.land (Z.round_lor_land_bound x) (Z.round_lor_land_bound (Z.pos y))). + Proof. destruct x; t. Qed. + + Local Instance land_round_Proper_pos_l y + : Proper (Pos.le ==> Z.le) + (fun x => + Z.land (Z.round_lor_land_bound (Z.pos x)) (Z.round_lor_land_bound y)). + Proof. destruct y; t. Qed. + + Local Instance lor_round_Proper_pos_r x + : Proper (Pos.le ==> Z.le) + (fun y => + Z.lor (Z.round_lor_land_bound x) (Z.round_lor_land_bound (Z.pos y))). + Proof. destruct x; t. Qed. + + Local Instance lor_round_Proper_pos_l y + : Proper (Pos.le ==> Z.le) + (fun x => + Z.lor (Z.round_lor_land_bound (Z.pos x)) (Z.round_lor_land_bound y)). + Proof. destruct y; t. Qed. + + Local Instance land_round_Proper_neg_r x + : Proper (Basics.flip Pos.le ==> Z.le) + (fun y => + Z.land (Z.round_lor_land_bound x) (Z.round_lor_land_bound (Z.neg y))). + Proof. destruct x; t. Qed. + + Local Instance land_round_Proper_neg_l y + : Proper (Basics.flip Pos.le ==> Z.le) + (fun x => + Z.land (Z.round_lor_land_bound (Z.neg x)) (Z.round_lor_land_bound y)). + Proof. destruct y; t. Qed. + + Local Instance lor_round_Proper_neg_r x + : Proper (Basics.flip Pos.le ==> Z.le) + (fun y => + Z.lor (Z.round_lor_land_bound x) (Z.round_lor_land_bound (Z.neg y))). + Proof. destruct x; t. Qed. + + Local Instance lor_round_Proper_neg_l y + : Proper (Basics.flip Pos.le ==> Z.le) + (fun x => + Z.lor (Z.round_lor_land_bound (Z.neg x)) (Z.round_lor_land_bound y)). + Proof. destruct y; t. Qed. +End Z. diff --git a/src/Util/ZUtil/LandLorShiftBounds.v b/src/Util/ZUtil/LandLorShiftBounds.v new file mode 100644 index 000000000..e978ab6b0 --- /dev/null +++ b/src/Util/ZUtil/LandLorShiftBounds.v @@ -0,0 +1,340 @@ +Require Import Coq.Classes.Morphisms. +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Hints.ZArith. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Pow. +Require Import Crypto.Util.ZUtil.Pow2. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Testbit. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.ReplaceNegWithPos. +Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. +Require Import Crypto.Util.NUtil.WithoutReferenceToZ. +Local Open Scope Z_scope. + +Module Z. + Lemma lor_range : forall x y n, 0 <= x < 2 ^ n -> 0 <= y < 2 ^ n -> + 0 <= Z.lor x y < 2 ^ n. + Proof. + intros x y n H H0; assert (0 <= n) by auto with zarith omega. + repeat match goal with + | |- _ => progress intros + | |- _ => rewrite Z.lor_spec + | |- _ => rewrite Z.testbit_eqb by auto with zarith omega + | |- _ => rewrite !Z.div_small by (split; try omega; eapply Z.lt_le_trans; + [ intuition eassumption | apply Z.pow_le_mono_r; omega]) + | |- _ => split + | |- _ => apply Z.testbit_false_bound + | |- _ => solve [auto with zarith] + | |- _ => solve [apply Z.lor_nonneg; intuition auto] + end. + Qed. + Hint Resolve lor_range : zarith. + + Lemma lor_shiftl_bounds : forall x y n m, + (0 <= n)%Z -> (0 <= m)%Z -> + (0 <= x < 2 ^ m)%Z -> + (0 <= y < 2 ^ n)%Z -> + (0 <= Z.lor y (Z.shiftl x n) < 2 ^ (n + m))%Z. + Proof. + intros x y n m H H0 H1 H2. + apply Z.lor_range. + { split; try omega. + apply Z.lt_le_trans with (m := (2 ^ n)%Z); try omega. + apply Z.pow_le_mono_r; omega. } + { rewrite Z.shiftl_mul_pow2 by omega. + rewrite Z.pow_add_r by omega. + split; Z.zero_bounds. + rewrite Z.mul_comm. + apply Z.mul_lt_mono_pos_l; omega. } + Qed. + + Lemma land_upper_bound_l : forall a b, (0 <= a) -> (0 <= b) -> + Z.land a b <= a. + Proof. + intros a b H H0. + destruct a, b; try solve [exfalso; auto]; try solve [cbv; congruence]. + cbv [Z.land]. + rewrite <-N2Z.inj_pos, <-N2Z.inj_le. + auto using N.Pos_land_upper_bound_l. + Qed. + + Lemma land_upper_bound_r : forall a b, (0 <= a) -> (0 <= b) -> + Z.land a b <= b. + Proof. + intros. + rewrite Z.land_comm. + auto using Z.land_upper_bound_l. + Qed. + + Section ZInequalities. + Lemma land_le : forall x y, (0 <= x)%Z -> (Z.land x y <= x)%Z. + Proof. + intros x y H; apply Z.ldiff_le; [assumption|]. + rewrite Z.ldiff_land, Z.land_comm, Z.land_assoc. + rewrite <- Z.land_0_l with (a := y); f_equal. + rewrite Z.land_comm, Z.land_lnot_diag. + reflexivity. + Qed. + + Lemma lor_lower : forall x y, (0 <= x)%Z -> (0 <= y)%Z -> (x <= Z.lor x y)%Z. + Proof. + intros x y H H0; apply Z.ldiff_le; [apply Z.lor_nonneg; auto|]. + rewrite Z.ldiff_land. + apply Z.bits_inj_iff'; intros k Hpos; apply Z.le_ge in Hpos. + rewrite Z.testbit_0_l, Z.land_spec, Z.lnot_spec, Z.lor_spec; + [|apply Z.ge_le; assumption]. + induction (Z.testbit x k), (Z.testbit y k); cbv; reflexivity. + Qed. + + Lemma lor_le : forall x y z, + (0 <= x)%Z + -> (x <= y)%Z + -> (y <= z)%Z + -> (Z.lor x y <= (2 ^ Z.log2_up (z+1)) - 1)%Z. + Proof. + intros x y z H H0 H1; apply Z.ldiff_le. + + - apply Z.le_add_le_sub_r. + replace 1%Z with (2 ^ 0)%Z by (cbv; reflexivity). + rewrite Z.add_0_l. + apply Z.pow_le_mono_r; [cbv; reflexivity|]. + apply Z.log2_up_nonneg. + + - destruct (Z_lt_dec 0 z). + + + assert (forall a, a - 1 = Z.pred a)%Z as HP by (intro; omega); + rewrite HP, <- Z.ones_equiv; clear HP. + apply Z.ldiff_ones_r_low; [apply Z.lor_nonneg; split; omega|]. + rewrite Z.log2_up_eqn, Z.log2_lor; try omega. + apply Z.lt_succ_r. + apply Z.max_case_strong; intros; apply Z.log2_le_mono; omega. + + + replace z with 0%Z by omega. + replace y with 0%Z by omega. + replace x with 0%Z by omega. + cbv; reflexivity. + Qed. + + Local Ltac solve_pow2 := + repeat match goal with + | [|- _ /\ _] => split + | [|- (0 < 2 ^ _)%Z] => apply Z.pow2_gt_0 + | [|- (0 <= 2 ^ _)%Z] => apply Z.pow2_ge_0 + | [|- (2 ^ _ <= 2 ^ _)%Z] => apply Z.pow_le_mono_r + | [|- (_ <= _)%Z] => omega + | [|- (_ < _)%Z] => omega + end. + + Lemma pow2_mod_range : forall a n m, + (0 <= n) -> + (n <= m) -> + (0 <= Z.pow2_mod a n < 2 ^ m). + Proof. + intros; unfold Z.pow2_mod. + rewrite Z.land_ones; [|assumption]. + split; [apply Z.mod_pos_bound, Z.pow2_gt_0; assumption|]. + eapply Z.lt_le_trans; [apply Z.mod_pos_bound, Z.pow2_gt_0; assumption|]. + apply Z.pow_le_mono; [|assumption]. + split; simpl; omega. + Qed. + + Lemma shiftr_range : forall a n m, + (0 <= n)%Z -> + (0 <= m)%Z -> + (0 <= a < 2 ^ (n + m))%Z -> + (0 <= Z.shiftr a n < 2 ^ m)%Z. + Proof. + intros a n m H0 H1 H2; destruct H2. + split; [apply Z.shiftr_nonneg; assumption|]. + rewrite Z.shiftr_div_pow2; [|assumption]. + apply Z.div_lt_upper_bound; [apply Z.pow2_gt_0; assumption|]. + eapply Z.lt_le_trans; [eassumption|apply Z.eq_le_incl]. + apply Z.pow_add_r; omega. + Qed. + + + Lemma shiftr_le_mono: forall a b c d, + (0 <= a)%Z + -> (0 <= d)%Z + -> (a <= c)%Z + -> (d <= b)%Z + -> (Z.shiftr a b <= Z.shiftr c d)%Z. + Proof. + intros. + repeat rewrite Z.shiftr_div_pow2; [|omega|omega]. + etransitivity; [apply Z.div_le_compat_l | apply Z.div_le_mono]; solve_pow2. + Qed. + + Lemma shiftl_le_mono: forall a b c d, + (0 <= a)%Z + -> (0 <= b)%Z + -> (a <= c)%Z + -> (b <= d)%Z + -> (Z.shiftl a b <= Z.shiftl c d)%Z. + Proof. + intros. + repeat rewrite Z.shiftl_mul_pow2; [|omega|omega]. + etransitivity; [apply Z.mul_le_mono_nonneg_l|apply Z.mul_le_mono_nonneg_r]; solve_pow2. + Qed. + End ZInequalities. + + Lemma lor_bounds x y : 0 <= x -> 0 <= y + -> Z.max x y <= Z.lor x y <= 2^Z.log2_up (Z.max x y + 1) - 1. + Proof. + apply Z.max_case_strong; intros; split; + try solve [ eauto using lor_lower, Z.le_trans, lor_le with omega + | rewrite Z.lor_comm; eauto using lor_lower, Z.le_trans, lor_le with omega ]. + Qed. + Lemma lor_bounds_lower x y : 0 <= x -> 0 <= y + -> Z.max x y <= Z.lor x y. + Proof. intros; apply lor_bounds; assumption. Qed. + Lemma lor_bounds_upper x y : Z.lor x y <= 2^Z.log2_up (Z.max x y + 1) - 1. + Proof. + pose proof (proj2 (Z.lor_neg x y)). + destruct (Z_lt_le_dec x 0), (Z_lt_le_dec y 0); + try solve [ intros; apply lor_bounds; assumption ]; + transitivity (2^0-1); + try apply Z.sub_le_mono_r, Z.pow_le_mono_r, Z.log2_up_nonneg; + simpl; omega. + Qed. + Lemma lor_bounds_gen_lower x y l : 0 <= x -> 0 <= y -> l <= Z.max x y + -> l <= Z.lor x y. + Proof. + intros; etransitivity; + solve [ apply lor_bounds; auto + | eauto ]. + Qed. + Lemma lor_bounds_gen_upper x y u : x <= u -> y <= u + -> Z.lor x y <= 2^Z.log2_up (u + 1) - 1. + Proof. + intros; etransitivity; [ apply lor_bounds_upper | ]. + apply Z.sub_le_mono_r, Z.pow_le_mono_r, Z.log2_up_le_mono, Z.max_case_strong; + omega. + Qed. + Lemma lor_bounds_gen x y l u : 0 <= x -> 0 <= y -> l <= Z.max x y -> x <= u -> y <= u + -> l <= Z.lor x y <= 2^Z.log2_up (u + 1) - 1. + Proof. auto using lor_bounds_gen_lower, lor_bounds_gen_upper. Qed. + + Lemma shiftl_le_Proper2 y + : Proper (Z.le ==> Z.le) (fun x => Z.shiftl x y). + Proof. + unfold Basics.flip in *. + pose proof (Zle_cases 0 y) as Hx. + intros x x' H. + pose proof (Zle_cases 0 x) as Hy. + pose proof (Zle_cases 0 x') as Hy'. + destruct (0 <=? y), (0 <=? x), (0 <=? x'); + autorewrite with Zshift_to_pow; + Z.replace_all_neg_with_pos; + autorewrite with pull_Zopp; + rewrite ?Z.div_opp_l_complete; + repeat destruct (Z_zerop _); + autorewrite with zsimplify_const pull_Zopp; + auto with zarith; + repeat match goal with + | [ |- context[-?x - ?y] ] + => replace (-x - y) with (-(x + y)) by omega + | _ => rewrite <- Z.opp_le_mono + | _ => rewrite <- Z.add_le_mono_r + | _ => solve [ auto with zarith ] + | [ |- ?x <= ?y + 1 ] + => cut (x <= y); [ omega | solve [ auto with zarith ] ] + | [ |- -_ <= _ ] + => solve [ transitivity (-0); auto with zarith ] + end. + { repeat match goal with H : context[_ mod _] |- _ => revert H end; + Z.div_mod_to_quot_rem_in_goal; nia. } + Qed. + + Lemma shiftl_le_Proper1 x + (R := fun b : bool => if b then Z.le else Basics.flip Z.le) + : Proper (R (0 <=? x) ==> Z.le) (Z.shiftl x). + Proof. + unfold Basics.flip in *. + pose proof (Zle_cases 0 x) as Hx. + intros y y' H. + pose proof (Zle_cases 0 y) as Hy. + pose proof (Zle_cases 0 y') as Hy'. + destruct (0 <=? x), (0 <=? y), (0 <=? y'); subst R; cbv beta iota in *; + autorewrite with Zshift_to_pow; + Z.replace_all_neg_with_pos; + autorewrite with pull_Zopp; + rewrite ?Z.div_opp_l_complete; + repeat destruct (Z_zerop _); + autorewrite with zsimplify_const pull_Zopp; + auto with zarith; + repeat match goal with + | [ |- context[-?x - ?y] ] + => replace (-x - y) with (-(x + y)) by omega + | _ => rewrite <- Z.opp_le_mono + | _ => rewrite <- Z.add_le_mono_r + | _ => solve [ auto with zarith ] + | [ |- ?x <= ?y + 1 ] + => cut (x <= y); [ omega | solve [ auto with zarith ] ] + | [ |- context[2^?x] ] + => lazymatch goal with + | [ H : 1 < 2^x |- _ ] => fail + | [ H : 0 < 2^x |- _ ] => fail + | [ H : 0 <= 2^x |- _ ] => fail + | _ => first [ assert (1 < 2^x) by auto with zarith + | assert (0 < 2^x) by auto with zarith + | assert (0 <= 2^x) by auto with zarith ] + end + | [ H : ?x <= ?y |- _ ] + => is_var x; is_var y; + lazymatch goal with + | [ H : 2^x <= 2^y |- _ ] => fail + | [ H : 2^x < 2^y |- _ ] => fail + | _ => assert (2^x <= 2^y) by auto with zarith + end + | [ H : ?x <= ?y, H' : ?f ?x = ?k, H'' : ?f ?y <> ?k |- _ ] + => let Hn := fresh in + assert (Hn : x <> y) by congruence; + assert (x < y) by omega; clear H Hn + | [ H : ?x <= ?y, H' : ?f ?x <> ?k, H'' : ?f ?y = ?k |- _ ] + => let Hn := fresh in + assert (Hn : x <> y) by congruence; + assert (x < y) by omega; clear H Hn + | _ => solve [ repeat match goal with H : context[_ mod _] |- _ => revert H end; + Z.div_mod_to_quot_rem_in_goal; subst; + lazymatch goal with + | [ |- _ <= (?a * ?q + ?r) * ?q' ] + => transitivity (q * (a * q') + r * q'); + [ assert (0 < a * q') by nia; nia + | nia ] + end ] + end. + { replace y' with (y + (y' - y)) by omega. + rewrite Z.pow_add_r, <- Zdiv_Zdiv by auto with zarith. + assert (y < y') by (assert (y <> y') by congruence; omega). + assert (1 < 2^(y'-y)) by auto with zarith. + assert (0 < x / 2^y) + by (repeat match goal with H : context[_ mod _] |- _ => revert H end; + Z.div_mod_to_quot_rem_in_goal; nia). + assert (2^y <= x) + by (repeat match goal with H : context[_ / _] |- _ => revert H end; + Z.div_mod_to_quot_rem_in_goal; nia). + match goal with + | [ |- ?x + 1 <= ?y ] => cut (x < y); [ omega | ] + end. + auto with zarith. } + Qed. + + Lemma shiftr_le_Proper2 y + : Proper (Z.le ==> Z.le) (fun x => Z.shiftr x y). + Proof. apply shiftl_le_Proper2. Qed. + + Lemma shiftr_le_Proper1 x + (R := fun b : bool => if b then Z.le else Basics.flip Z.le) + : Proper (R (x <? 0) ==> Z.le) (Z.shiftr x). + Proof. + subst R; intros y y' H'; unfold Z.shiftr; apply shiftl_le_Proper1. + unfold Basics.flip in *. + pose proof (Zle_cases 0 x). + pose proof (Zlt_cases x 0). + destruct (0 <=? x), (x <? 0); try omega. + Qed. +End Z. diff --git a/src/Util/ZUtil/Le.v b/src/Util/ZUtil/Le.v index ab7767de7..ca180c556 100644 --- a/src/Util/ZUtil/Le.v +++ b/src/Util/ZUtil/Le.v @@ -1,9 +1,58 @@ Require Import Coq.ZArith.ZArith. Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Local Open Scope Z_scope. Module Z. Lemma positive_is_nonzero : forall x, x > 0 -> x <> 0. Proof. intros; omega. Qed. Hint Resolve positive_is_nonzero : zarith. + + Lemma le_lt_trans n m p : n <= m -> m < p -> n < p. + Proof. lia. Qed. + + Lemma le_fold_right_max : forall low l x, (forall y, List.In y l -> low <= y) -> + List.In x l -> x <= List.fold_right Z.max low l. + Proof. + induction l as [|a l IHl]; intros ? lower_bound In_list; [cbv [List.In] in *; intuition | ]. + simpl. + destruct (List.in_inv In_list); subst. + + apply Z.le_max_l. + + etransitivity. + - apply IHl; auto; intuition auto with datatypes. + - apply Z.le_max_r. + Qed. + + Lemma le_fold_right_max_initial : forall low l, low <= List.fold_right Z.max low l. + Proof. + induction l as [|a l IHl]; intros; try reflexivity. + etransitivity; [ apply IHl | apply Z.le_max_r ]. + Qed. + + Lemma add_compare_mono_r: forall n m p, (n + p ?= m + p) = (n ?= m). + Proof. + intros n m p. + rewrite <-!(Z.add_comm p). + apply Z.add_compare_mono_l. + Qed. + + Lemma leb_add_same x y : (x <=? y + x) = (0 <=? y). + Proof. destruct (x <=? y + x) eqn:?, (0 <=? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. + Hint Rewrite leb_add_same : zsimplify. + + Lemma ltb_add_same x y : (x <? y + x) = (0 <? y). + Proof. destruct (x <? y + x) eqn:?, (0 <? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. + Hint Rewrite ltb_add_same : zsimplify. + + Lemma geb_add_same x y : (x >=? y + x) = (0 >=? y). + Proof. destruct (x >=? y + x) eqn:?, (0 >=? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. + Hint Rewrite geb_add_same : zsimplify. + + Lemma gtb_add_same x y : (x >? y + x) = (0 >? y). + Proof. destruct (x >? y + x) eqn:?, (0 >? y) eqn:?; Z.ltb_to_lt; try reflexivity; omega. Qed. + Hint Rewrite gtb_add_same : zsimplify. + + Lemma sub_pos_bound a b X : 0 <= a < X -> 0 <= b < X -> -X < a - b < X. + Proof. lia. Qed. End Z. diff --git a/src/Util/ZUtil/Lnot.v b/src/Util/ZUtil/Lnot.v new file mode 100644 index 000000000..c4c747c76 --- /dev/null +++ b/src/Util/ZUtil/Lnot.v @@ -0,0 +1,16 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Local Open Scope Z_scope. + +Module Z. + Lemma lnot_equiv n : Z.lnot n = Z.pred (-n). + Proof. reflexivity. Qed. + + Lemma lnot_sub1 n : Z.lnot (n-1) = -n. + Proof. rewrite lnot_equiv; lia. Qed. + + Lemma lnot_opp x : Z.lnot (- x) = x-1. + Proof. + rewrite <-Z.lnot_involutive, lnot_sub1; reflexivity. + Qed. +End Z. diff --git a/src/Util/ZUtil/Log2.v b/src/Util/ZUtil/Log2.v new file mode 100644 index 000000000..90c43b7fb --- /dev/null +++ b/src/Util/ZUtil/Log2.v @@ -0,0 +1,90 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Hints.ZArith. +Require Import Crypto.Util.ZUtil.Pow. +Require Import Crypto.Util.ZUtil.ZSimplify.Core. +Require Import Crypto.Util.ZUtil.ZSimplify.Simple. +Local Open Scope Z_scope. + +Module Z. + Lemma log2_nonneg' n a : n <= 0 -> n <= Z.log2 a. + Proof. + intros; transitivity 0; auto with zarith. + Qed. + Hint Resolve log2_nonneg' : zarith. + + Lemma le_lt_to_log2 x y z : 0 <= z -> 0 < y -> 2^x <= y < 2^z -> x <= Z.log2 y < z. + Proof. + destruct (Z_le_gt_dec 0 x); auto with concl_log2 lia. + Qed. + + Lemma log2_pred_pow2_full a : Z.log2 (Z.pred (2^a)) = Z.max 0 (Z.pred a). + Proof. + destruct (Z_dec 0 a) as [ [?|?] | ?]. + { rewrite Z.log2_pred_pow2 by assumption; lia. } + { autorewrite with zsimplify; simpl. + apply Z.max_case_strong; try omega. + + } + { subst; compute; reflexivity. } + Qed. + Hint Rewrite log2_pred_pow2_full : zsimplify. + + Lemma log2_up_le_full a : a <= 2^Z.log2_up a. + Proof. + destruct (Z_dec 1 a) as [ [ ? | ? ] | ? ]; + first [ apply Z.log2_up_spec; assumption + | rewrite Z.log2_up_eqn0 by omega; simpl; omega ]. + Qed. + + Lemma log2_up_le_pow2_full : forall a b : Z, (0 <= b)%Z -> (a <= 2 ^ b)%Z <-> (Z.log2_up a <= b)%Z. + Proof. + intros a b H. + destruct (Z_lt_le_dec 0 a); [ apply Z.log2_up_le_pow2; assumption | ]. + split; transitivity 0%Z; try omega; auto with zarith. + rewrite Z.log2_up_eqn0 by omega. + reflexivity. + Qed. + + Lemma log2_lt_pow2_alt a b : 0 < b -> (a < 2^b <-> Z.log2 a < b). + Proof. + destruct (Z_lt_le_dec 0 a); auto using Z.log2_lt_pow2; []. + rewrite Z.log2_nonpos by omega. + split; auto with zarith; []. + intro; eapply Z.le_lt_trans; [ eassumption | ]. + auto with zarith. + Qed. + + Lemma max_log2_up x y : Z.max (Z.log2_up x) (Z.log2_up y) = Z.log2_up (Z.max x y). + Proof. apply Z.max_monotone; intros ??; apply Z.log2_up_le_mono. Qed. + Hint Rewrite max_log2_up : push_Zmax. + Hint Rewrite <- max_log2_up : pull_Zmax. + + Lemma log2_up_le_full_max a : Z.max a 1 <= 2^Z.log2_up a. + Proof. + apply Z.max_case_strong; auto using Z.log2_up_le_full. + intros; rewrite Z.log2_up_eqn0 by assumption; reflexivity. + Qed. + Lemma log2_up_le_1 a : Z.log2_up a <= 1 <-> a <= 2. + Proof. + pose proof (Z.log2_nonneg (Z.pred a)). + destruct (Z_dec a 2) as [ [ ? | ? ] | ? ]. + { rewrite (proj2 (Z.log2_up_null a)) by omega; split; omega. } + { rewrite Z.log2_up_eqn by omega. + split; try omega; intro. + assert (Z.log2 (Z.pred a) = 0) by omega. + assert (Z.pred a <= 1) by (apply Z.log2_null; omega). + omega. } + { subst; cbv -[Z.le]; split; omega. } + Qed. + Lemma log2_up_1_le a : 1 <= Z.log2_up a <-> 2 <= a. + Proof. + pose proof (Z.log2_nonneg (Z.pred a)). + destruct (Z_dec a 2) as [ [ ? | ? ] | ? ]. + { rewrite (proj2 (Z.log2_up_null a)) by omega; split; omega. } + { rewrite Z.log2_up_eqn by omega; omega. } + { subst; cbv -[Z.le]; split; omega. } + Qed. +End Z. diff --git a/src/Util/ZUtil/Modulo.v b/src/Util/ZUtil/Modulo.v index 84917a454..567d106e3 100644 --- a/src/Util/ZUtil/Modulo.v +++ b/src/Util/ZUtil/Modulo.v @@ -4,6 +4,7 @@ Require Import Crypto.Util.ZUtil.ZSimplify.Core. Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.ZUtil.Tactics.ReplaceNegWithPos. +Require Import Crypto.Util.ZUtil.Tactics.PullPush.Modulo. Require Import Crypto.Util.ZUtil.Div. Require Import Crypto.Util.Tactics.BreakMatch. Require Import Crypto.Util.Tactics.DestructHead. @@ -287,4 +288,85 @@ Module Z. Lemma mod_opp_r a b : a mod (-b) = -((-a) mod b). Proof. pose proof (Z.div_opp_r a b); Z.div_mod_to_quot_rem; nia. Qed. Hint Resolve mod_opp_r : zarith. + + Lemma mod_same_pow : forall a b c, 0 <= c <= b -> a ^ b mod a ^ c = 0. + Proof. + intros a b c H. + replace b with (b - c + c) by ring. + rewrite Z.pow_add_r by omega. + apply Z_mod_mult. + Qed. + Hint Rewrite mod_same_pow using zutil_arith : zsimplify. + + Lemma mod_opp_l_z_iff a b (H : b <> 0) : a mod b = 0 <-> (-a) mod b = 0. + Proof. + split; intro H'; apply Z.mod_opp_l_z in H'; rewrite ?Z.opp_involutive in H'; assumption. + Qed. + Hint Rewrite <- mod_opp_l_z_iff using zutil_arith : zsimplify. + + Lemma mod_small_sym a b : 0 <= a < b -> a = a mod b. + Proof. intros; symmetry; apply Z.mod_small; assumption. Qed. + Hint Resolve mod_small_sym : zarith. + + Lemma mod_eq_le_to_eq a b : 0 < a <= b -> a mod b = 0 -> a = b. + Proof. pose proof (Z.mod_eq_le_div_1 a b); intros; Z.div_mod_to_quot_rem; nia. Qed. + Hint Resolve mod_eq_le_to_eq : zarith. + + Lemma mod_neq_0_le_to_neq a b : a mod b <> 0 -> a <> b. + Proof. repeat intro; subst; autorewrite with zsimplify in *; lia. Qed. + Hint Resolve mod_neq_0_le_to_neq : zarith. + + Lemma div_mod' a b : b <> 0 -> a = (a / b) * b + a mod b. + Proof. intro; etransitivity; [ apply (Z.div_mod a b); assumption | lia ]. Qed. + Hint Rewrite <- div_mod' using zutil_arith : zsimplify. + + Lemma div_mod'' a b : b <> 0 -> a = a mod b + b * (a / b). + Proof. intro; etransitivity; [ apply (Z.div_mod a b); assumption | lia ]. Qed. + Hint Rewrite <- div_mod'' using zutil_arith : zsimplify. + + Lemma div_mod''' a b : b <> 0 -> a = a mod b + (a / b) * b. + Proof. intro; etransitivity; [ apply (Z.div_mod a b); assumption | lia ]. Qed. + Hint Rewrite <- div_mod''' using zutil_arith : zsimplify. + + Lemma sub_mod_mod_0 x d : (x - x mod d) mod d = 0. + Proof. + destruct (Z_zerop d); subst; push_Zmod; autorewrite with zsimplify; reflexivity. + Qed. + Hint Resolve sub_mod_mod_0 : zarith. + Hint Rewrite sub_mod_mod_0 : zsimplify. + + Lemma mod_small_n n a b : 0 <= n -> b <> 0 -> n * b <= a < (1 + n) * b -> a mod b = a - n * b. + Proof. intros; erewrite Zmod_eq_full, Z.div_between by eassumption. reflexivity. Qed. + Hint Rewrite mod_small_n using zutil_arith : zsimplify. + + Lemma mod_small_1 a b : b <> 0 -> b <= a < 2 * b -> a mod b = a - b. + Proof. intros; rewrite (mod_small_n 1) by lia; lia. Qed. + Hint Rewrite mod_small_1 using zutil_arith : zsimplify. + + Lemma mod_small_n_if n a b : 0 <= n -> b <> 0 -> n * b <= a < (2 + n) * b -> a mod b = a - (if (1 + n) * b <=? a then (1 + n) else n) * b. + Proof. intros; erewrite Zmod_eq_full, Z.div_between_if by eassumption; autorewrite with zsimplify_const. reflexivity. Qed. + + Lemma mod_small_0_if a b : b <> 0 -> 0 <= a < 2 * b -> a mod b = a - if b <=? a then b else 0. + Proof. intros; rewrite (mod_small_n_if 0) by lia; autorewrite with zsimplify_const. break_match; lia. Qed. + + Lemma mul_mod_distr_r_full a b c : (a * c) mod (b * c) = (a mod b * c). + Proof. + destruct (Z_zerop b); [ | destruct (Z_zerop c) ]; subst; + autorewrite with zsimplify; auto using Z.mul_mod_distr_r. + Qed. + + Lemma mul_mod_distr_l_full a b c : (c * a) mod (c * b) = c * (a mod b). + Proof. + destruct (Z_zerop b); [ | destruct (Z_zerop c) ]; subst; + autorewrite with zsimplify; auto using Z.mul_mod_distr_l. + Qed. + + Lemma lt_mul_2_mod_sub : forall a b, b <> 0 -> b <= a < 2 * b -> a mod b = a - b. + Proof. + intros a b H H0. + replace (a mod b) with ((1 * b + (a - b)) mod b) by (f_equal; ring). + rewrite Z.mod_add_l by auto. + apply Z.mod_small. + omega. + Qed. End Z. diff --git a/src/Util/ZUtil/Morphisms.v b/src/Util/ZUtil/Morphisms.v index 91f3dff3c..15a9fcf1a 100644 --- a/src/Util/ZUtil/Morphisms.v +++ b/src/Util/ZUtil/Morphisms.v @@ -6,6 +6,7 @@ Require Import Coq.Classes.Morphisms. Require Import Coq.Classes.RelationPairs. Require Import Crypto.Util.ZUtil.Definitions. Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.LandLorBounds. Require Import Crypto.Util.ZUtil.Tactics.PeelLe. Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem. Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. @@ -279,4 +280,13 @@ Module Z. Lemma shiftl_Zneg_Zneg_le_Proper_r x : Proper (Basics.flip Pos.le ==> Z.le) (fun p => Z.shiftl (Zneg p) (Zneg x)). Proof. shift_Proper_t'. Qed. Hint Resolve shiftl_Zneg_Zneg_le_Proper_r : zarith. + + Hint Resolve Z.land_round_Proper_pos_r : zarith. + Hint Resolve Z.land_round_Proper_pos_l : zarith. + Hint Resolve Z.lor_round_Proper_pos_r : zarith. + Hint Resolve Z.lor_round_Proper_pos_l : zarith. + Hint Resolve Z.land_round_Proper_neg_r : zarith. + Hint Resolve Z.land_round_Proper_neg_l : zarith. + Hint Resolve Z.lor_round_Proper_neg_r : zarith. + Hint Resolve Z.lor_round_Proper_neg_l : zarith. End Z. diff --git a/src/Util/ZUtil/Mul.v b/src/Util/ZUtil/Mul.v new file mode 100644 index 000000000..6cf851e4e --- /dev/null +++ b/src/Util/ZUtil/Mul.v @@ -0,0 +1,8 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Local Open Scope Z_scope. + +Module Z. + Lemma mul_comm3 x y z : x * (y * z) = y * (x * z). + Proof. lia. Qed. +End Z. diff --git a/src/Util/ZUtil/N2Z.v b/src/Util/ZUtil/N2Z.v new file mode 100644 index 000000000..928f0b334 --- /dev/null +++ b/src/Util/ZUtil/N2Z.v @@ -0,0 +1,53 @@ +Require Import Coq.ZArith.ZArith. +Require Import Crypto.Util.ZUtil.Hints.Core. +Local Open Scope Z_scope. + +Module N2Z. + Lemma inj_land n m : Z.of_N (N.land n m) = Z.land (Z.of_N n) (Z.of_N m). + Proof. destruct n, m; reflexivity. Qed. + Hint Rewrite inj_land : push_Zof_N. + Hint Rewrite <- inj_land : pull_Zof_N. + + Lemma inj_lor n m : Z.of_N (N.lor n m) = Z.lor (Z.of_N n) (Z.of_N m). + Proof. destruct n, m; reflexivity. Qed. + Hint Rewrite inj_lor : push_Zof_N. + Hint Rewrite <- inj_lor : pull_Zof_N. + + Lemma inj_shiftl: forall x y, Z.of_N (N.shiftl x y) = Z.shiftl (Z.of_N x) (Z.of_N y). + Proof. + intros x y. + apply Z.bits_inj_iff'; intros k Hpos. + rewrite Z2N.inj_testbit; [|assumption]. + rewrite Z.shiftl_spec; [|assumption]. + + assert ((Z.to_N k) >= y \/ (Z.to_N k) < y)%N as g by ( + unfold N.ge, N.lt; induction (N.compare (Z.to_N k) y); [left|auto|left]; + intro H; inversion H). + + destruct g as [g|g]; + [ rewrite N.shiftl_spec_high; [|apply N2Z.inj_le; rewrite Z2N.id|apply N.ge_le] + | rewrite N.shiftl_spec_low]; try assumption. + + - rewrite <- N2Z.inj_testbit; f_equal. + rewrite N2Z.inj_sub, Z2N.id; [reflexivity|assumption|apply N.ge_le; assumption]. + + - apply N2Z.inj_lt in g. + rewrite Z2N.id in g; [symmetry|assumption]. + apply Z.testbit_neg_r; omega. + Qed. + Hint Rewrite inj_shiftl : push_Zof_N. + Hint Rewrite <- inj_shiftl : pull_Zof_N. + + Lemma inj_shiftr: forall x y, Z.of_N (N.shiftr x y) = Z.shiftr (Z.of_N x) (Z.of_N y). + Proof. + intros. + apply Z.bits_inj_iff'; intros k Hpos. + rewrite Z2N.inj_testbit; [|assumption]. + rewrite Z.shiftr_spec, N.shiftr_spec; [|apply N2Z.inj_le; rewrite Z2N.id|]; try assumption. + rewrite <- N2Z.inj_testbit; f_equal. + rewrite N2Z.inj_add; f_equal. + apply Z2N.id; assumption. + Qed. + Hint Rewrite inj_shiftr : push_Zof_N. + Hint Rewrite <- inj_shiftr : pull_Zof_N. +End N2Z. diff --git a/src/Util/ZUtil/Odd.v b/src/Util/ZUtil/Odd.v new file mode 100644 index 000000000..37b8bd443 --- /dev/null +++ b/src/Util/ZUtil/Odd.v @@ -0,0 +1,32 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.ZArith.Znumtheory. +Require Import Coq.micromega.Lia. +Local Open Scope Z_scope. + +Module Z. + Lemma prime_odd_or_2 : forall p (prime_p : prime p), p = 2 \/ Z.odd p = true. + Proof. + intros p prime_p. + apply Decidable.imp_not_l; try apply Z.eq_decidable. + intros p_neq2. + pose proof (Zmod_odd p) as mod_odd. + destruct (Sumbool.sumbool_of_bool (Z.odd p)) as [? | p_not_odd]; auto. + rewrite p_not_odd in mod_odd. + apply Zmod_divides in mod_odd; try omega. + destruct mod_odd as [c c_id]. + rewrite Z.mul_comm in c_id. + apply Zdivide_intro in c_id. + apply prime_divisors in c_id; auto. + destruct c_id; [omega | destruct H; [omega | destruct H; auto] ]. + pose proof (prime_ge_2 p prime_p); omega. + Qed. + + Lemma odd_mod : forall a b, (b <> 0)%Z -> + Z.odd (a mod b) = if Z.odd b then xorb (Z.odd a) (Z.odd (a / b)) else Z.odd a. + Proof. + intros a b H. + rewrite Zmod_eq_full by assumption. + rewrite <-Z.add_opp_r, Z.odd_add, Z.odd_opp, Z.odd_mul. + case_eq (Z.odd b); intros; rewrite ?Bool.andb_true_r, ?Bool.andb_false_r; auto using Bool.xorb_false_r. + Qed. +End Z. diff --git a/src/Util/ZUtil/Ones.v b/src/Util/ZUtil/Ones.v new file mode 100644 index 000000000..e856f23a0 --- /dev/null +++ b/src/Util/ZUtil/Ones.v @@ -0,0 +1,177 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Pow2. +Require Import Crypto.Util.ZUtil.Log2. +Require Import Crypto.Util.ZUtil.Lnot. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Hints.ZArith. +Require Import Crypto.Util.ZUtil.ZSimplify.Simple. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.Tactics.BreakMatch. +Require Import Crypto.Util.Tactics.DestructHead. +Require Import Crypto.Util.Tactics.UniquePose. +Local Open Scope bool_scope. Local Open Scope Z_scope. + +Module Z. + Lemma ones_le x y : x <= y -> Z.ones x <= Z.ones y. + Proof. + rewrite !Z.ones_equiv; auto with zarith. + Qed. + Hint Resolve ones_le : zarith. + + Lemma ones_lt_pow2 x y : 0 <= x <= y -> Z.ones x < 2^y. + Proof. + rewrite Z.ones_equiv, Z.lt_pred_le. + auto with zarith. + Qed. + Hint Resolve ones_lt_pow2 : zarith. + + Lemma log2_ones_full x : Z.log2 (Z.ones x) = Z.max 0 (Z.pred x). + Proof. + rewrite Z.ones_equiv, Z.log2_pred_pow2_full; reflexivity. + Qed. + Hint Rewrite log2_ones_full : zsimplify. + + Lemma log2_ones_lt x y : 0 < x <= y -> Z.log2 (Z.ones x) < y. + Proof. + rewrite log2_ones_full; apply Z.max_case_strong; omega. + Qed. + Hint Resolve log2_ones_lt : zarith. + + Lemma log2_ones_le x y : 0 <= x <= y -> Z.log2 (Z.ones x) <= y. + Proof. + rewrite log2_ones_full; apply Z.max_case_strong; omega. + Qed. + Hint Resolve log2_ones_le : zarith. + + Lemma log2_ones_lt_nonneg x y : 0 < y -> x <= y -> Z.log2 (Z.ones x) < y. + Proof. + rewrite log2_ones_full; apply Z.max_case_strong; omega. + Qed. + Hint Resolve log2_ones_lt_nonneg : zarith. + + Lemma ones_pred : forall i, 0 < i -> Z.ones (Z.pred i) = Z.shiftr (Z.ones i) 1. + Proof. + induction i as [|p|p]; [ | | pose proof (Pos2Z.neg_is_neg p) ]; try omega. + intros. + unfold Z.ones. + rewrite !Z.shiftl_1_l, Z.shiftr_div_pow2, <-!Z.sub_1_r, Z.pow_1_r, <-!Z.add_opp_r by omega. + replace (2 ^ (Z.pos p)) with (2 ^ (Z.pos p - 1)* 2). + rewrite Z.div_add_l by omega. + reflexivity. + change 2 with (2 ^ 1) at 2. + rewrite <-Z.pow_add_r by (pose proof (Pos2Z.is_pos p); omega). + f_equal. omega. + Qed. + Hint Rewrite <- ones_pred using zutil_arith : push_Zshift. + + Lemma ones_succ : forall x, (0 <= x) -> + Z.ones (Z.succ x) = 2 ^ x + Z.ones x. + Proof. + unfold Z.ones; intros. + rewrite !Z.shiftl_1_l. + rewrite Z.add_pred_r. + apply Z.succ_inj. + rewrite !Z.succ_pred. + rewrite Z.pow_succ_r; omega. + Qed. + + Lemma ones_nonneg : forall i, (0 <= i) -> 0 <= Z.ones i. + Proof. + apply natlike_ind. + + unfold Z.ones. simpl; omega. + + intros. + rewrite Z.ones_succ by assumption. + Z.zero_bounds. + Qed. + Hint Resolve ones_nonneg : zarith. + + Lemma ones_pos_pos : forall i, (0 < i) -> 0 < Z.ones i. + Proof. + intros. + unfold Z.ones. + rewrite Z.shiftl_1_l. + apply Z.lt_succ_lt_pred. + apply Z.pow_gt_1; omega. + Qed. + Hint Resolve ones_pos_pos : zarith. + + Lemma lnot_ones_equiv n : Z.lnot (Z.ones n) = -2^n. + Proof. rewrite Z.ones_equiv, Z.lnot_equiv, <- ?Z.sub_1_r; lia. Qed. + + Lemma land_ones_ones n m + : Z.land (Z.ones n) (Z.ones m) + = Z.ones (if ((n <? 0) || (m <? 0)) + then Z.max n m + else Z.min n m). + Proof. + repeat first [ reflexivity + | break_innermost_match_step + | progress rewrite ?Bool.orb_true_iff in * + | progress rewrite ?Bool.orb_false_iff in * + | progress rewrite ?Z.ltb_lt, ?Z.ltb_ge in * + | progress destruct_head'_and + | apply Z.min_case_strong + | apply Z.max_case_strong + | progress intros + | progress destruct_head'_or + | rewrite !Z.pow_r_Zneg + | rewrite !Z.land_m1_l + | rewrite !Z.land_m1_r + | progress change (Z.pred 0) with (-1) + | rewrite Z.mod_small by lia + | match goal with + | [ H : ?x < 0 |- _ ] => is_var x; destruct x; try lia + | [ H : ?x <= Z.neg _ |- _ ] => is_var x; destruct x; try lia + | [ |- context[Z.ones (Z.neg ?x)] ] => rewrite (Z.ones_equiv (Z.neg x)) + | [ H : ?n <= ?m |- Z.land (Z.ones ?m) (Z.ones ?n) = _ ] + => rewrite (Z.land_comm (Z.ones m) (Z.ones n)) + | [ H : ?n <= ?m |- Z.land (Z.ones ?n) (Z.ones ?m) = _ ] + => progress rewrite ?Z.land_ones, ?Z.ones_equiv, <- ?Z.sub_1_r by auto + | [ H : ?n <= ?m |- _ ] + => is_var n; is_var m; unique pose proof (Z.pow_le_mono_r 2 n m ltac:(lia) H) + | [ |- context[2^?x] ] => unique pose proof (Z.pow2_gt_0 x ltac:(lia)) + end ]. + Qed. + Hint Rewrite land_ones_ones : zsimplify. + + Lemma lor_ones_ones n m + : Z.lor (Z.ones n) (Z.ones m) + = Z.ones (if ((n <? 0) || (m <? 0)) + then Z.min n m + else Z.max n m). + Proof. + destruct (Z_zerop n), (Z_zerop m); subst; + repeat first [ reflexivity + | break_innermost_match_step + | progress rewrite ?Bool.orb_true_iff in * + | progress rewrite ?Bool.orb_false_iff in * + | progress rewrite ?Z.ltb_lt, ?Z.ltb_ge in * + | progress destruct_head'_and + | apply Z.min_case_strong + | apply Z.max_case_strong + | progress intros + | progress destruct_head'_or + | rewrite !Z.pow_r_Zneg + | rewrite !Z.lor_m1_l + | rewrite !Z.lor_m1_r + | progress change (Z.pred 0) with (-1) + | rewrite Z.mod_small by lia + | lia + | match goal with + | [ H : ?x < 0 |- _ ] => is_var x; destruct x; try lia + | [ H : ?x <= Z.neg _ |- _ ] => is_var x; destruct x; try lia + | [ |- context[Z.ones (Z.neg ?x)] ] => rewrite (Z.ones_equiv (Z.neg x)) + | [ H : ?n <= ?m |- Z.lor (Z.ones ?m) (Z.ones ?n) = _ ] + => rewrite (Z.lor_comm (Z.ones m) (Z.ones n)) + | [ H : ?n <= ?m |- Z.lor (Z.ones ?n) (Z.ones ?m) = _ ] + => progress rewrite ?Z.lor_ones_low; try apply Z.log2_ones_lt_nonneg; rewrite ?Z.ones_equiv, <- ?Z.sub_1_r + | [ H : ?n <= ?m |- _ ] + => is_var n; is_var m; unique pose proof (Z.pow_le_mono_r 2 n m ltac:(lia) H) + | [ |- context[2^?x] ] => unique pose proof (Z.pow2_gt_0 x ltac:(lia)) + end ]. + Qed. + Hint Rewrite lor_ones_ones : zsimplify. +End Z. diff --git a/src/Util/ZUtil/Opp.v b/src/Util/ZUtil/Opp.v new file mode 100644 index 000000000..3cc18241b --- /dev/null +++ b/src/Util/ZUtil/Opp.v @@ -0,0 +1,11 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.ZSimplify.Core. +Local Open Scope Z_scope. + +Module Z. + Lemma opp_eq_0_iff a : -a = 0 <-> a = 0. + Proof. omega. Qed. + Hint Rewrite opp_eq_0_iff : zsimplify. +End Z. diff --git a/src/Util/ZUtil/Pow.v b/src/Util/ZUtil/Pow.v new file mode 100644 index 000000000..06ce2187b --- /dev/null +++ b/src/Util/ZUtil/Pow.v @@ -0,0 +1,44 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints.Core. +Local Open Scope Z_scope. + +Module Z. + Lemma base_pow_neg b n : n < 0 -> b^n = 0. + Proof. + destruct n; intro H; try reflexivity; compute in H; congruence. + Qed. + Hint Rewrite base_pow_neg using zutil_arith : zsimplify. + + Lemma nonneg_pow_pos a b : 0 < a -> 0 < a^b -> 0 <= b. + Proof. + destruct (Z_lt_le_dec b 0); intros; auto. + erewrite Z.pow_neg_r in * by eassumption. + omega. + Qed. + Hint Resolve nonneg_pow_pos (fun n => nonneg_pow_pos 2 n Z.lt_0_2) : zarith. + Lemma nonneg_pow_pos_helper a b dummy : 0 < a -> 0 <= dummy < a^b -> 0 <= b. + Proof. eauto with zarith omega. Qed. + Hint Resolve nonneg_pow_pos_helper (fun n dummy => nonneg_pow_pos_helper 2 n dummy Z.lt_0_2) : zarith. + + Lemma div_pow2succ : forall n x, (0 <= x) -> + n / 2 ^ Z.succ x = Z.div2 (n / 2 ^ x). + Proof. + intros. + rewrite Z.pow_succ_r, Z.mul_comm by auto. + rewrite <- Z.div_div by (try apply Z.pow_nonzero; omega). + rewrite Zdiv2_div. + reflexivity. + Qed. + + Definition pow_sub_r' + := fun a b c y H0 H1 => @Logic.eq_trans _ _ _ y (@Z.pow_sub_r a b c H0 H1). + Definition pow_sub_r'_sym + := fun a b c y p H0 H1 => Logic.eq_sym (@Logic.eq_trans _ y _ _ (Logic.eq_sym p) (@Z.pow_sub_r a b c H0 H1)). + Hint Resolve pow_sub_r' pow_sub_r'_sym Z.eq_le_incl : zarith. + Hint Resolve (fun b => f_equal (fun e => b ^ e)) (fun e => f_equal (fun b => b ^ e)) : zarith. + + Lemma two_p_two_eq_four : 2^(2) = 4. + Proof. reflexivity. Qed. + Hint Rewrite <- two_p_two_eq_four : push_Zpow. +End Z. diff --git a/src/Util/ZUtil/Pow2.v b/src/Util/ZUtil/Pow2.v new file mode 100644 index 000000000..bc3b01225 --- /dev/null +++ b/src/Util/ZUtil/Pow2.v @@ -0,0 +1,26 @@ +Require Import Coq.micromega.Lia. +Require Import Coq.ZArith.ZArith. +Local Open Scope Z_scope. + +Module Z. + Lemma pow2_ge_0: forall a, (0 <= 2 ^ a)%Z. + Proof. + intros; apply Z.pow_nonneg; omega. + Qed. + + Lemma pow2_gt_0: forall a, (0 <= a)%Z -> (0 < 2 ^ a)%Z. + Proof. + intros; apply Z.pow_pos_nonneg; [|assumption]; omega. + Qed. + + Lemma pow2_lt_or_divides : forall a b, 0 <= b -> + 2 ^ a < 2 ^ b \/ (2 ^ a) mod 2 ^ b = 0. + Proof. + intros a b H. + destruct (Z_lt_dec a b); [left|right]. + { apply Z.pow_lt_mono_r; auto; omega. } + { replace a with (a - b + b) by ring. + rewrite Z.pow_add_r by omega. + apply Z.mod_mul, Z.pow_nonzero; omega. } + Qed. +End Z. diff --git a/src/Util/ZUtil/Pow2Mod.v b/src/Util/ZUtil/Pow2Mod.v index 237ca19dc..74c22394a 100644 --- a/src/Util/ZUtil/Pow2Mod.v +++ b/src/Util/ZUtil/Pow2Mod.v @@ -3,6 +3,7 @@ Require Import Crypto.Util.ZUtil.Definitions. Require Import Crypto.Util.ZUtil.Notations. Require Import Crypto.Util.ZUtil.Hints.Core. Require Import Crypto.Util.ZUtil.Hints.Ztestbit. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. Require Import Crypto.Util.ZUtil.Testbit. Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z_scope. @@ -51,4 +52,14 @@ Module Z. auto with zarith. Qed. Hint Resolve pow2_mod_pos_bound : zarith. + + Lemma pow2_mod_id_iff : forall a n, 0 <= n -> + (Z.pow2_mod a n = a <-> 0 <= a < 2 ^ n). + Proof. + intros a n H. + rewrite Z.pow2_mod_spec by assumption. + assert (0 < 2 ^ n) by Z.zero_bounds. + rewrite Z.mod_small_iff by omega. + split; intros; intuition omega. + Qed. End Z. diff --git a/src/Util/ZUtil/Shift.v b/src/Util/ZUtil/Shift.v new file mode 100644 index 000000000..b5fb79c13 --- /dev/null +++ b/src/Util/ZUtil/Shift.v @@ -0,0 +1,393 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Ones. +Require Import Crypto.Util.ZUtil.Definitions. +Require Import Crypto.Util.ZUtil.Testbit. +Require Import Crypto.Util.ZUtil.Pow2Mod. +Require Import Crypto.Util.ZUtil.Le. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.Tactics.BreakMatch. +Require Import Crypto.Util.Tactics.SpecializeBy. +Local Open Scope Z_scope. + +Module Z. + Lemma shiftr_add_shiftl_high : forall n m a b, 0 <= n <= m -> 0 <= a < 2 ^ n -> + Z.shiftr (a + (Z.shiftl b n)) m = Z.shiftr b (m - n). + Proof. + intros n m a b H H0. + rewrite !Z.shiftr_div_pow2, Z.shiftl_mul_pow2 by omega. + replace (2 ^ m) with (2 ^ n * 2 ^ (m - n)) by + (rewrite <-Z.pow_add_r by omega; f_equal; ring). + rewrite <-Z.div_div, Z.div_add, (Z.div_small a) ; try solve + [assumption || apply Z.pow_nonzero || apply Z.pow_pos_nonneg; omega]. + f_equal; ring. + Qed. + Hint Rewrite Z.shiftr_add_shiftl_high using zutil_arith : pull_Zshift. + Hint Rewrite <- Z.shiftr_add_shiftl_high using zutil_arith : push_Zshift. + + Lemma shiftr_add_shiftl_low : forall n m a b, 0 <= m <= n -> 0 <= a < 2 ^ n -> + Z.shiftr (a + (Z.shiftl b n)) m = Z.shiftr a m + Z.shiftr b (m - n). + Proof. + intros n m a b H H0. + rewrite !Z.shiftr_div_pow2, Z.shiftl_mul_pow2, Z.shiftr_mul_pow2 by omega. + replace (2 ^ n) with (2 ^ (n - m) * 2 ^ m) by + (rewrite <-Z.pow_add_r by omega; f_equal; ring). + rewrite Z.mul_assoc, Z.div_add by (apply Z.pow_nonzero; omega). + repeat f_equal; ring. + Qed. + Hint Rewrite Z.shiftr_add_shiftl_low using zutil_arith : pull_Zshift. + Hint Rewrite <- Z.shiftr_add_shiftl_low using zutil_arith : push_Zshift. + + Lemma testbit_add_shiftl_high : forall i, (0 <= i) -> forall a b n, (0 <= n <= i) -> + 0 <= a < 2 ^ n -> + Z.testbit (a + Z.shiftl b n) i = Z.testbit b (i - n). + Proof. + intros i ?. + apply natlike_ind with (x := i); [ intros a b n | intros x H0 H1 a b n | ]; intros; try assumption; + (destruct (Z.eq_dec 0 n); [ subst; rewrite Z.pow_0_r in *; + replace a with 0 by omega; f_equal; ring | ]); try omega. + rewrite <-Z.add_1_r at 1. rewrite <-Z.shiftr_spec by assumption. + replace (Z.succ x - n) with (x - (n - 1)) by ring. + rewrite shiftr_add_shiftl_low, <-Z.shiftl_opp_r with (a := b) by omega. + rewrite <-H1 with (a := Z.shiftr a 1); try omega; [ repeat f_equal; ring | ]. + rewrite Z.shiftr_div_pow2 by omega. + split; apply Z.div_pos || apply Z.div_lt_upper_bound; + try solve [rewrite ?Z.pow_1_r; omega]. + rewrite <-Z.pow_add_r by omega. + replace (1 + (n - 1)) with n by ring; omega. + Qed. + Hint Rewrite testbit_add_shiftl_high using zutil_arith : Ztestbit. + + Lemma shiftr_succ : forall n x, + Z.shiftr n (Z.succ x) = Z.shiftr (Z.shiftr n x) 1. + Proof. + intros. + rewrite Z.shiftr_shiftr by omega. + reflexivity. + Qed. + Hint Rewrite Z.shiftr_succ using zutil_arith : push_Zshift. + Hint Rewrite <- Z.shiftr_succ using zutil_arith : pull_Zshift. + + Lemma shiftr_1_r_le : forall a b, a <= b -> + Z.shiftr a 1 <= Z.shiftr b 1. + Proof. + intros. + rewrite !Z.shiftr_div_pow2, Z.pow_1_r by omega. + apply Z.div_le_mono; omega. + Qed. + Hint Resolve shiftr_1_r_le : zarith. + + Lemma shiftr_le : forall a b i : Z, 0 <= i -> a <= b -> a >> i <= b >> i. + Proof. + intros a b i ?; revert a b. apply natlike_ind with (x := i); intros; auto. + rewrite !shiftr_succ, shiftr_1_r_le; eauto. reflexivity. + Qed. + Hint Resolve shiftr_le : zarith. + + Lemma shiftr_ones' : forall a n, 0 <= a < 2 ^ n -> forall i, (0 <= i) -> + Z.shiftr a i <= Z.ones (n - i) \/ n <= i. + Proof. + intros a n H. + apply natlike_ind. + + unfold Z.ones. + rewrite Z.shiftr_0_r, Z.shiftl_1_l, Z.sub_0_r. + omega. + + intros x H0 H1. + destruct (Z_lt_le_dec x n); try omega. + intuition auto with zarith lia. + left. + rewrite shiftr_succ. + replace (n - Z.succ x) with (Z.pred (n - x)) by omega. + rewrite Z.ones_pred by omega. + apply Z.shiftr_1_r_le. + assumption. + Qed. + + Lemma shiftr_ones : forall a n i, 0 <= a < 2 ^ n -> (0 <= i) -> (i <= n) -> + Z.shiftr a i <= Z.ones (n - i) . + Proof. + intros a n i G G0 G1. + destruct (Z_le_lt_eq_dec i n G1). + + destruct (Z.shiftr_ones' a n G i G0); omega. + + subst; rewrite Z.sub_diag. + destruct (Z.eq_dec a 0). + - subst; rewrite Z.shiftr_0_l; reflexivity. + - rewrite Z.shiftr_eq_0; try omega; try reflexivity. + apply Z.log2_lt_pow2; omega. + Qed. + Hint Resolve shiftr_ones : zarith. + + Lemma shiftr_upper_bound : forall a n, 0 <= n -> 0 <= a <= 2 ^ n -> Z.shiftr a n <= 1. + Proof. + intros a ? ? [a_nonneg a_upper_bound]. + apply Z_le_lt_eq_dec in a_upper_bound. + destruct a_upper_bound. + + destruct (Z.eq_dec 0 a). + - subst; rewrite Z.shiftr_0_l; omega. + - rewrite Z.shiftr_eq_0; auto; try omega. + apply Z.log2_lt_pow2; auto; omega. + + subst. + rewrite Z.shiftr_div_pow2 by assumption. + rewrite Z.div_same; try omega. + assert (0 < 2 ^ n) by (apply Z.pow_pos_nonneg; omega). + omega. + Qed. + Hint Resolve shiftr_upper_bound : zarith. + + Lemma lor_shiftl : forall a b n, 0 <= n -> 0 <= a < 2 ^ n -> + Z.lor a (Z.shiftl b n) = a + (Z.shiftl b n). + Proof. + intros a b n H H0. + apply Z.bits_inj'; intros t ?. + rewrite Z.lor_spec, Z.shiftl_spec by assumption. + destruct (Z_lt_dec t n). + + rewrite Z.testbit_add_shiftl_low by omega. + rewrite Z.testbit_neg_r with (n := t - n) by omega. + apply Bool.orb_false_r. + + rewrite testbit_add_shiftl_high by omega. + replace (Z.testbit a t) with false; [ apply Bool.orb_false_l | ]. + symmetry. + apply Z.testbit_false; try omega. + rewrite Z.div_small; try reflexivity. + split; try eapply Z.lt_le_trans with (m := 2 ^ n); try omega. + apply Z.pow_le_mono_r; omega. + Qed. + Hint Rewrite <- Z.lor_shiftl using zutil_arith : convert_to_Ztestbit. + + Lemma lor_shiftl' : forall a b n, 0 <= n -> 0 <= a < 2 ^ n -> + Z.lor (Z.shiftl b n) a = (Z.shiftl b n) + a. + Proof. + intros; rewrite Z.lor_comm, Z.add_comm; apply lor_shiftl; assumption. + Qed. + Hint Rewrite <- Z.lor_shiftl' using zutil_arith : convert_to_Ztestbit. + + Lemma shiftl_spec_full a n m + : Z.testbit (a << n) m = if Z_lt_dec m n + then false + else if Z_le_dec 0 m + then Z.testbit a (m - n) + else false. + Proof. + repeat break_match; auto using Z.shiftl_spec_low, Z.shiftl_spec, Z.testbit_neg_r with omega. + Qed. + Hint Rewrite shiftl_spec_full : Ztestbit_full. + + Lemma shiftr_spec_full a n m + : Z.testbit (a >> n) m = if Z_lt_dec m (-n) + then false + else if Z_le_dec 0 m + then Z.testbit a (m + n) + else false. + Proof. + rewrite <- Z.shiftl_opp_r, shiftl_spec_full, Z.sub_opp_r; reflexivity. + Qed. + Hint Rewrite shiftr_spec_full : Ztestbit_full. + + Lemma testbit_add_shiftl_full i (Hi : 0 <= i) a b n (Ha : 0 <= a < 2^n) + : Z.testbit (a + b << n) i + = if (i <? n) then Z.testbit a i else Z.testbit b (i - n). + Proof. + assert (0 < 2^n) by omega. + assert (0 <= n) by eauto 2 with zarith. + pose proof (Zlt_cases i n); break_match; autorewrite with Ztestbit; reflexivity. + Qed. + Hint Rewrite testbit_add_shiftl_full using zutil_arith : Ztestbit. + + Lemma land_add_land : forall n m a b, (m <= n)%nat -> + Z.land ((Z.land a (Z.ones (Z.of_nat n))) + (Z.shiftl b (Z.of_nat n))) (Z.ones (Z.of_nat m)) = Z.land a (Z.ones (Z.of_nat m)). + Proof. + intros n m a b H. + rewrite !Z.land_ones by apply Nat2Z.is_nonneg. + rewrite Z.shiftl_mul_pow2 by apply Nat2Z.is_nonneg. + replace (b * 2 ^ Z.of_nat n) with + ((b * 2 ^ Z.of_nat (n - m)) * 2 ^ Z.of_nat m) by + (rewrite (le_plus_minus m n) at 2; try assumption; + rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg; ring). + rewrite Z.mod_add by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat m)); omega). + symmetry. apply Znumtheory.Zmod_div_mod; try (apply Z.pow_pos_nonneg; omega). + rewrite (le_plus_minus m n) by assumption. + rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg. + apply Z.divide_factor_l. + Qed. + + Lemma shiftl_add x y z : 0 <= z -> (x + y) << z = (x << z) + (y << z). + Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. + Hint Rewrite shiftl_add using zutil_arith : push_Zshift. + Hint Rewrite <- shiftl_add using zutil_arith : pull_Zshift. + + Lemma shiftr_add x y z : z <= 0 -> (x + y) >> z = (x >> z) + (y >> z). + Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. + Hint Rewrite shiftr_add using zutil_arith : push_Zshift. + Hint Rewrite <- shiftr_add using zutil_arith : pull_Zshift. + + Lemma shiftl_sub x y z : 0 <= z -> (x - y) << z = (x << z) - (y << z). + Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. + Hint Rewrite shiftl_sub using zutil_arith : push_Zshift. + Hint Rewrite <- shiftl_sub using zutil_arith : pull_Zshift. + + Lemma shiftr_sub x y z : z <= 0 -> (x - y) >> z = (x >> z) - (y >> z). + Proof. intros; autorewrite with Zshift_to_pow; lia. Qed. + Hint Rewrite shiftr_sub using zutil_arith : push_Zshift. + Hint Rewrite <- shiftr_sub using zutil_arith : pull_Zshift. + + Lemma compare_add_shiftl : forall x1 y1 x2 y2 n, 0 <= n -> + Z.pow2_mod x1 n = x1 -> Z.pow2_mod x2 n = x2 -> + x1 + (y1 << n) ?= x2 + (y2 << n) = + if Z.eq_dec y1 y2 + then x1 ?= x2 + else y1 ?= y2. + Proof. + repeat match goal with + | |- _ => progress intros + | |- _ => progress subst y1 + | |- _ => rewrite Z.shiftl_mul_pow2 by omega + | |- _ => rewrite Z.add_compare_mono_r + | |- _ => rewrite <-Z.mul_sub_distr_r + | |- _ => break_innermost_match_step + | H : Z.pow2_mod _ _ = _ |- _ => rewrite Z.pow2_mod_id_iff in H by omega + | H : ?a <> ?b |- _ = (?a ?= ?b) => + case_eq (a ?= b); rewrite ?Z.compare_eq_iff, ?Z.compare_gt_iff, ?Z.compare_lt_iff + | |- _ + (_ * _) > _ + (_ * _) => cbv [Z.gt] + | |- _ + (_ * ?x) < _ + (_ * ?x) => + apply Z.lt_sub_lt_add; apply Z.lt_le_trans with (m := 1 * x); [omega|] + | |- _ => apply Z.mul_le_mono_nonneg_r; omega + | |- _ => reflexivity + | |- _ => congruence + end. + Qed. + + Lemma shiftl_opp_l a n + : Z.shiftl (-a) n = - Z.shiftl a n - (if Z_zerop (a mod 2 ^ (- n)) then 0 else 1). + Proof. + destruct (Z_dec 0 n) as [ [?|?] | ? ]; + subst; + rewrite ?Z.pow_neg_r by omega; + autorewrite with zsimplify_const; + [ | | simpl; omega ]. + { rewrite !Z.shiftl_mul_pow2 by omega. + nia. } + { rewrite !Z.shiftl_div_pow2 by omega. + rewrite Z.div_opp_l_complete by auto with zarith. + reflexivity. } + Qed. + Hint Rewrite shiftl_opp_l : push_Zshift. + Hint Rewrite <- shiftl_opp_l : pull_Zshift. + + Lemma shiftr_opp_l a n + : Z.shiftr (-a) n = - Z.shiftr a n - (if Z_zerop (a mod 2 ^ n) then 0 else 1). + Proof. + unfold Z.shiftr; rewrite shiftl_opp_l at 1; rewrite Z.opp_involutive. + reflexivity. + Qed. + Hint Rewrite shiftr_opp_l : push_Zshift. + Hint Rewrite <- shiftr_opp_l : pull_Zshift. + + Lemma shl_shr_lt x y n m (Hx : 0 <= x < 2^n) (Hy : 0 <= y < 2^n) (Hm : 0 <= m <= n) + : 0 <= (x >> (n - m)) + ((y << m) mod 2^n) < 2^n. + Proof. + cut (0 <= (x >> (n - m)) + ((y << m) mod 2^n) <= 2^n - 1); [ omega | ]. + assert (0 <= x <= 2^n - 1) by omega. + assert (0 <= y <= 2^n - 1) by omega. + assert (0 < 2 ^ (n - m)) by auto with zarith. + assert (0 <= y mod 2 ^ (n - m) < 2^(n-m)) by auto with zarith. + assert (0 <= y mod 2 ^ (n - m) <= 2 ^ (n - m) - 1) by omega. + assert (0 <= (y mod 2 ^ (n - m)) * 2^m <= (2^(n-m) - 1)*2^m) by auto with zarith. + assert (0 <= x / 2^(n-m) < 2^n / 2^(n-m)). + { split; Z.zero_bounds. + apply Z.div_lt_upper_bound; autorewrite with pull_Zpow zsimplify; nia. } + autorewrite with Zshift_to_pow. + split; Z.zero_bounds. + replace (2^n) with (2^(n-m) * 2^m) by (autorewrite with pull_Zpow; f_equal; omega). + rewrite Zmult_mod_distr_r. + autorewrite with pull_Zpow zsimplify push_Zmul in * |- . + nia. + Qed. + + Lemma add_shift_mod x y n m + (Hx : 0 <= x < 2^n) (Hy : 0 <= y) + (Hn : 0 <= n) (Hm : 0 < m) + : (x + y << n) mod (m * 2^n) = x + (y mod m) << n. + Proof. + pose proof (Z.mod_bound_pos y m). + specialize_by omega. + assert (0 < 2^n) by auto with zarith. + autorewrite with Zshift_to_pow. + rewrite Zplus_mod, !Zmult_mod_distr_r. + rewrite Zplus_mod, !Zmod_mod, <- Zplus_mod. + rewrite !(Zmod_eq (_ + _)) by nia. + etransitivity; [ | apply Z.add_0_r ]. + rewrite <- !Z.add_opp_r, <- !Z.add_assoc. + repeat apply f_equal. + ring_simplify. + cut (((x + y mod m * 2 ^ n) / (m * 2 ^ n)) = 0); [ nia | ]. + apply Z.div_small; split; nia. + Qed. + + Lemma add_mul_mod x y n m + (Hx : 0 <= x < 2^n) (Hy : 0 <= y) + (Hn : 0 <= n) (Hm : 0 < m) + : (x + y * 2^n) mod (m * 2^n) = x + (y mod m) * 2^n. + Proof. + generalize (add_shift_mod x y n m). + autorewrite with Zshift_to_pow; auto. + Qed. + + Lemma lt_pow_2_shiftr : forall a n, 0 <= a < 2 ^ n -> a >> n = 0. + Proof. + intros a n H. + destruct (Z_le_dec 0 n). + + rewrite Z.shiftr_div_pow2 by assumption. + auto using Z.div_small. + + assert (2 ^ n = 0) by (apply Z.pow_neg_r; omega). + omega. + Qed. + + Hint Rewrite Z.pow2_bits_eqb using zutil_arith : Ztestbit. + Lemma pow_2_shiftr : forall n, 0 <= n -> (2 ^ n) >> n = 1. + Proof. + intros; apply Z.bits_inj'; intros. + replace 1 with (2 ^ 0) by ring. + repeat match goal with + | |- _ => progress intros + | |- _ => progress rewrite ?Z.eqb_eq, ?Z.eqb_neq in * + | |- _ => progress autorewrite with Ztestbit + | |- context[Z.eqb ?a ?b] => case_eq (Z.eqb a b) + | |- _ => reflexivity || omega + end. + Qed. + + Lemma lt_mul_2_pow_2_shiftr : forall a n, 0 <= a < 2 * 2 ^ n -> + a >> n = if Z_lt_dec a (2 ^ n) then 0 else 1. + Proof. + intros a n H; break_match; [ apply lt_pow_2_shiftr; omega | ]. + destruct (Z_le_dec 0 n). + + replace (2 * 2 ^ n) with (2 ^ (n + 1)) in * + by (rewrite Z.pow_add_r; try omega; ring). + pose proof (Z.shiftr_ones a (n + 1) n H). + pose proof (Z.shiftr_le (2 ^ n) a n). + specialize_by omega. + replace (n + 1 - n) with 1 in * by ring. + replace (Z.ones 1) with 1 in * by reflexivity. + rewrite pow_2_shiftr in * by omega. + omega. + + assert (2 ^ n = 0) by (apply Z.pow_neg_r; omega). + omega. + Qed. + + Lemma shiftr_nonneg_le : forall a n, 0 <= a -> 0 <= n -> a >> n <= a. + Proof. + intros. + repeat match goal with + | [ H : _ <= _ |- _ ] + => rewrite Z.lt_eq_cases in H + | [ H : _ \/ _ |- _ ] => destruct H + | _ => progress subst + | _ => progress autorewrite with zsimplify Zshift_to_pow + | _ => solve [ auto with zarith omega ] + end. + Qed. + Hint Resolve shiftr_nonneg_le : zarith. +End Z. diff --git a/src/Util/ZUtil/Stabilization.v b/src/Util/ZUtil/Stabilization.v index 4df0300da..7e89ea1b4 100644 --- a/src/Util/ZUtil/Stabilization.v +++ b/src/Util/ZUtil/Stabilization.v @@ -1,7 +1,10 @@ Require Import Coq.ZArith.ZArith. Require Import Coq.micromega.Lia. Require Import Coq.Classes.Morphisms. -Require Import Crypto.Util.ZUtil. +Require Import Crypto.Util.ZUtil.Hints.Core. +Require Import Crypto.Util.ZUtil.Hints.ZArith. +Require Import Crypto.Util.ZUtil.Tactics.ReplaceNegWithPos. +Require Import Crypto.Util.ZUtil.Testbit. Require Import Crypto.Util.Tactics.DestructHead. Require Import Crypto.Util.Tactics.SpecializeBy. diff --git a/src/Util/ZUtil/Tactics/PullPush/Modulo.v b/src/Util/ZUtil/Tactics/PullPush/Modulo.v index 55889cbf0..fe0c3224c 100644 --- a/src/Util/ZUtil/Tactics/PullPush/Modulo.v +++ b/src/Util/ZUtil/Tactics/PullPush/Modulo.v @@ -3,89 +3,92 @@ Require Import Crypto.Util.ZUtil.Hints.Core. Require Import Crypto.Util.ZUtil.Modulo.PullPush. Local Open Scope Z_scope. -Ltac push_Zmod := - repeat match goal with - | _ => progress autorewrite with push_Zmod - | [ |- context[(?x * ?y) mod ?z] ] - => first [ rewrite (Z.mul_mod_push x y z) by Z.NoZMod - | rewrite (Z.mul_mod_l_push x y z) by Z.NoZMod - | rewrite (Z.mul_mod_r_push x y z) by Z.NoZMod ] - | [ |- context[(?x + ?y) mod ?z] ] - => first [ rewrite (Z.add_mod_push x y z) by Z.NoZMod - | rewrite (Z.add_mod_l_push x y z) by Z.NoZMod - | rewrite (Z.add_mod_r_push x y z) by Z.NoZMod ] - | [ |- context[(?x - ?y) mod ?z] ] - => first [ rewrite (Z.sub_mod_push x y z) by Z.NoZMod - | rewrite (Z.sub_mod_l_push x y z) by Z.NoZMod - | rewrite (Z.sub_mod_r_push x y z) by Z.NoZMod ] - | [ |- context[(-?y) mod ?z] ] - => rewrite (Z.opp_mod_mod_push y z) by Z.NoZMod - | [ |- context[(?p^?q) mod ?z] ] - => rewrite (Z.pow_mod_push p q z) by Z.NoZMod - end. +Ltac push_Zmod_step := + match goal with + | _ => progress autorewrite with push_Zmod + | [ |- context[(?x * ?y) mod ?z] ] + => first [ rewrite (Z.mul_mod_push x y z) by Z.NoZMod + | rewrite (Z.mul_mod_l_push x y z) by Z.NoZMod + | rewrite (Z.mul_mod_r_push x y z) by Z.NoZMod ] + | [ |- context[(?x + ?y) mod ?z] ] + => first [ rewrite (Z.add_mod_push x y z) by Z.NoZMod + | rewrite (Z.add_mod_l_push x y z) by Z.NoZMod + | rewrite (Z.add_mod_r_push x y z) by Z.NoZMod ] + | [ |- context[(?x - ?y) mod ?z] ] + => first [ rewrite (Z.sub_mod_push x y z) by Z.NoZMod + | rewrite (Z.sub_mod_l_push x y z) by Z.NoZMod + | rewrite (Z.sub_mod_r_push x y z) by Z.NoZMod ] + | [ |- context[(-?y) mod ?z] ] + => rewrite (Z.opp_mod_mod_push y z) by Z.NoZMod + | [ |- context[(?p^?q) mod ?z] ] + => rewrite (Z.pow_mod_push p q z) by Z.NoZMod + end. +Ltac push_Zmod := repeat push_Zmod_step. -Ltac push_Zmod_hyps := - repeat match goal with - | _ => progress autorewrite with push_Zmod in * |- - | [ H : context[(?x * ?y) mod ?z] |- _ ] - => first [ rewrite (Z.mul_mod_push x y z) in H by Z.NoZMod - | rewrite (Z.mul_mod_l_push x y z) in H by Z.NoZMod - | rewrite (Z.mul_mod_r_push x y z) in H by Z.NoZMod ] - | [ H : context[(?x + ?y) mod ?z] |- _ ] - => first [ rewrite (Z.add_mod_push x y z) in H by Z.NoZMod - | rewrite (Z.add_mod_l_push x y z) in H by Z.NoZMod - | rewrite (Z.add_mod_r_push x y z) in H by Z.NoZMod ] - | [ H : context[(?x - ?y) mod ?z] |- _ ] - => first [ rewrite (Z.sub_mod_push x y z) in H by Z.NoZMod - | rewrite (Z.sub_mod_l_push x y z) in H by Z.NoZMod - | rewrite (Z.sub_mod_r_push x y z) in H by Z.NoZMod ] - | [ H : context[(-?y) mod ?z] |- _ ] - => rewrite (Z.opp_mod_mod_push y z) in H by Z.NoZMod - | [ H : context[(?p^?q) mod ?z] |- _ ] - => rewrite (Z.pow_mod_push p q z) in H by Z.NoZMod - end. +Ltac push_Zmod_hyps_step := + match goal with + | _ => progress autorewrite with push_Zmod in * |- + | [ H : context[(?x * ?y) mod ?z] |- _ ] + => first [ rewrite (Z.mul_mod_push x y z) in H by Z.NoZMod + | rewrite (Z.mul_mod_l_push x y z) in H by Z.NoZMod + | rewrite (Z.mul_mod_r_push x y z) in H by Z.NoZMod ] + | [ H : context[(?x + ?y) mod ?z] |- _ ] + => first [ rewrite (Z.add_mod_push x y z) in H by Z.NoZMod + | rewrite (Z.add_mod_l_push x y z) in H by Z.NoZMod + | rewrite (Z.add_mod_r_push x y z) in H by Z.NoZMod ] + | [ H : context[(?x - ?y) mod ?z] |- _ ] + => first [ rewrite (Z.sub_mod_push x y z) in H by Z.NoZMod + | rewrite (Z.sub_mod_l_push x y z) in H by Z.NoZMod + | rewrite (Z.sub_mod_r_push x y z) in H by Z.NoZMod ] + | [ H : context[(-?y) mod ?z] |- _ ] + => rewrite (Z.opp_mod_mod_push y z) in H by Z.NoZMod + | [ H : context[(?p^?q) mod ?z] |- _ ] + => rewrite (Z.pow_mod_push p q z) in H by Z.NoZMod + end. +Ltac push_Zmod_hyps := repeat push_Zmod_hyps_step. Ltac has_no_mod x z := lazymatch x with | context[_ mod z] => fail | _ => idtac end. -Ltac pull_Zmod := - repeat match goal with - | [ |- context[((?x mod ?z) * (?y mod ?z)) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.mul_mod_full x y z) - | [ |- context[((?x mod ?z) * ?y) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.mul_mod_l x y z) - | [ |- context[(?x * (?y mod ?z)) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.mul_mod_r x y z) - | [ |- context[((?x mod ?z) + (?y mod ?z)) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.add_mod_full x y z) - | [ |- context[((?x mod ?z) + ?y) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.add_mod_l x y z) - | [ |- context[(?x + (?y mod ?z)) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.add_mod_r x y z) - | [ |- context[((?x mod ?z) - (?y mod ?z)) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.sub_mod_full x y z) - | [ |- context[((?x mod ?z) - ?y) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.sub_mod_l x y z) - | [ |- context[(?x - (?y mod ?z)) mod ?z] ] - => has_no_mod x z; has_no_mod y z; - rewrite <- (Z.sub_mod_r x y z) - | [ |- context[(((-?y) mod ?z)) mod ?z] ] - => has_no_mod y z; - rewrite <- (Z.opp_mod_mod y z) - | [ |- context[((?x mod ?z)^?y) mod ?z] ] - => has_no_mod x z; - rewrite <- (Z.pow_mod_full x y z) - | [ |- context[(?x mod ?z) mod ?z] ] - => rewrite (Zmod_mod x z) - | _ => progress autorewrite with pull_Zmod - end. +Ltac pull_Zmod_step := + match goal with + | [ |- context[((?x mod ?z) * (?y mod ?z)) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.mul_mod_full x y z) + | [ |- context[((?x mod ?z) * ?y) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.mul_mod_l x y z) + | [ |- context[(?x * (?y mod ?z)) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.mul_mod_r x y z) + | [ |- context[((?x mod ?z) + (?y mod ?z)) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.add_mod_full x y z) + | [ |- context[((?x mod ?z) + ?y) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.add_mod_l x y z) + | [ |- context[(?x + (?y mod ?z)) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.add_mod_r x y z) + | [ |- context[((?x mod ?z) - (?y mod ?z)) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.sub_mod_full x y z) + | [ |- context[((?x mod ?z) - ?y) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.sub_mod_l x y z) + | [ |- context[(?x - (?y mod ?z)) mod ?z] ] + => has_no_mod x z; has_no_mod y z; + rewrite <- (Z.sub_mod_r x y z) + | [ |- context[(-(?y mod ?z)) mod ?z] ] + => has_no_mod y z; + rewrite <- (Z.opp_mod_mod y z) + | [ |- context[((?x mod ?z)^?y) mod ?z] ] + => has_no_mod x z; + rewrite <- (Z.pow_mod_full x y z) + | [ |- context[(?x mod ?z) mod ?z] ] + => rewrite (Zmod_mod x z) + | _ => progress autorewrite with pull_Zmod + end. +Ltac pull_Zmod := repeat pull_Zmod_step. diff --git a/src/Util/ZUtil/Testbit.v b/src/Util/ZUtil/Testbit.v index 175d07b02..f8ef5465a 100644 --- a/src/Util/ZUtil/Testbit.v +++ b/src/Util/ZUtil/Testbit.v @@ -1,7 +1,12 @@ +Require Import Coq.micromega.Lia. Require Import Coq.ZArith.ZArith. Require Import Crypto.Util.ZUtil.Definitions. Require Import Crypto.Util.ZUtil.Hints. Require Import Crypto.Util.ZUtil.Notations. +Require Import Crypto.Util.ZUtil.Lnot. +Require Import Crypto.Util.ZUtil.Div. +Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds. +Require Import Crypto.Util.ZUtil.Tactics.LtbToLt. Require Import Crypto.Util.Tactics.BreakMatch. Local Open Scope Z_scope. @@ -87,4 +92,39 @@ Module Z. auto using Z.mod_pow2_bits_low. Qed. Hint Rewrite testbit_add_shiftl_low using zutil_arith : Ztestbit. + + Lemma testbit_sub_pow2 n i x (i_range:0 <= i < n) (x_range:0 < x < 2 ^ n) : + Z.testbit (2 ^ n - x) i = negb (Z.testbit (x - 1) i). + Proof. + rewrite <-Z.lnot_spec, Z.lnot_sub1 by omega. + rewrite <-(Z.mod_pow2_bits_low (-x) _ _ (proj2 i_range)). + f_equal. + rewrite Z.mod_opp_l_nz; autorewrite with zsimplify; omega. + Qed. + + Lemma testbit_false_bound : forall a x, 0 <= x -> + (forall n, ~ (n < x) -> Z.testbit a n = false) -> + a < 2 ^ x. + Proof. + intros a x H H0. + assert (H1 : a = Z.pow2_mod a x). { + apply Z.bits_inj'; intros. + rewrite Z.testbit_pow2_mod by omega; break_match; auto. + } + rewrite H1. + cbv [Z.pow2_mod]; rewrite Z.land_ones by auto. + try apply Z.mod_pos_bound; Z.zero_bounds. + Qed. + + Lemma testbit_neg_eq_if x n : + 0 <= n -> + - (2 ^ n) <= x < 2 ^ n -> + Z.b2z (if x <? 0 then true else Z.testbit x n) = - (x / 2 ^ n) mod 2. + Proof. + intros. break_match; Z.ltb_to_lt. + { autorewrite with zsimplify. reflexivity. } + { autorewrite with zsimplify. + rewrite Z.bits_above_pow2 by omega. + reflexivity. } + Qed. End Z. diff --git a/src/Util/ZUtil/Z2Nat.v b/src/Util/ZUtil/Z2Nat.v index d6dd49a41..75d27dcaf 100644 --- a/src/Util/ZUtil/Z2Nat.v +++ b/src/Util/ZUtil/Z2Nat.v @@ -7,3 +7,41 @@ Module Z2Nat. destruct n; try reflexivity; lia. Qed. End Z2Nat. + +Module Z. + Lemma pos_pow_nat_pos : forall x n, + Z.pos x ^ Z.of_nat n > 0. + Proof. intros; apply Z.lt_gt, Z.pow_pos_nonneg; lia. Qed. + + Lemma pow_Z2N_Zpow : forall a n, 0 <= a -> + ((Z.to_nat a) ^ n = Z.to_nat (a ^ Z.of_nat n)%Z)%nat. + Proof. + intros a n H; induction n as [|n IHn]; try reflexivity. + rewrite Nat2Z.inj_succ. + rewrite Nat.pow_succ_r by apply le_0_n. + rewrite Z.pow_succ_r by apply Zle_0_nat. + rewrite IHn. + rewrite Z2Nat.inj_mul; auto using Z.pow_nonneg. + Qed. + + Lemma pow_Zpow : forall a n : nat, Z.of_nat (a ^ n) = Z.of_nat a ^ Z.of_nat n. + Proof with auto using Zle_0_nat, Z.pow_nonneg. + intros; apply Z2Nat.inj... + rewrite <- pow_Z2N_Zpow, !Nat2Z.id... + Qed. + Hint Rewrite pow_Zpow : push_Zof_nat. + Hint Rewrite <- pow_Zpow : pull_Zof_nat. + + Lemma Zpow_sub_1_nat_pow a v + : (Z.pos a^Z.of_nat v - 1 = Z.of_nat (Z.to_nat (Z.pos a)^v - 1))%Z. + Proof. + rewrite <- (Z2Nat.id (Z.pos a)) at 1 by lia. + change 2%Z with (Z.of_nat 2); change 1%Z with (Z.of_nat 1); + autorewrite with pull_Zof_nat. + rewrite Nat2Z.inj_sub + by (change 1%nat with (Z.to_nat (Z.pos a)^0)%nat; apply Nat.pow_le_mono_r; simpl; lia). + reflexivity. + Qed. + Hint Rewrite Zpow_sub_1_nat_pow : pull_Zof_nat. + Hint Rewrite <- Zpow_sub_1_nat_pow : push_Zof_nat. +End Z. |