From 14f41f622196bafcd9ad40543f21b72d88e01e7d Mon Sep 17 00:00:00 2001 From: Robert Sloan Date: Thu, 23 Jun 2016 19:31:11 -0400 Subject: Remove vestigal BoundedWord machinery --- src/Assembly/BoundedWord.v | 421 ---------------------------------------- src/Assembly/MultiBoundedWord.v | 252 ------------------------ 2 files changed, 673 deletions(-) delete mode 100644 src/Assembly/BoundedWord.v delete mode 100644 src/Assembly/MultiBoundedWord.v (limited to 'src/Assembly') diff --git a/src/Assembly/BoundedWord.v b/src/Assembly/BoundedWord.v deleted file mode 100644 index 3c5684793..000000000 --- a/src/Assembly/BoundedWord.v +++ /dev/null @@ -1,421 +0,0 @@ - -Require Import Bedrock.Word Bedrock.Nomega. -Require Import NArith PArith Ndigits Compare_dec Arith. -Require Import ProofIrrelevance. -Require Import Ring. -Require Import Wordize. - -Section BoundedWord. - - Local Open Scope wordize_scope. - - Context {n: nat}. - - (* Word Operations *) - - Definition shiftr (w: word n) (bits: nat): word n. - destruct (le_dec bits n). - - - replace n with (bits + (n - bits)) in * by (abstract intuition). - refine (zext (split1 bits (n - bits) w) (n - bits)). - - - exact (wzero n). - Defined. - - Lemma shiftr_spec: forall (w : word n) (bits: nat), - wordToN (shiftr w bits) = N.shiftr (wordToN w) (N.of_nat bits). - intros; unfold shiftr; destruct (le_dec bits n). - - - admit. - - - replace (wordToN (wzero n)) with 0%N by admit. - unfold N.shiftr. - induction bits. - - + replace (N.of_nat 0) with 0%N by intuition. - assert (n = 0) by intuition; clear n0; subst. - replace w with WO; intuition. - - + induction bits; admit. - Qed. - - Definition mask (m: nat) (w: word n): word n. - destruct (le_dec m n). - - - replace n with (m + (n - m)) in * by (abstract intuition). - refine (w ^& (zext (wones m) (n - m))). - - - exact w. - Defined. - - (* Definitions of Inequality and simple bounds. *) - - Lemma le_ge : forall n m, (n <= m -> m >= n)%nat. - Proof. - intros; omega. - Qed. - - Lemma ge_le : forall n m, (n >= m -> m <= n)%nat. - Proof. - intros; omega. - Qed. - - Ltac ge_to_le := - try apply N.ge_le; - repeat match goal with - | [ H : _ |- _ ] => apply N.le_ge in H - end. - - Ltac ge_to_le_nat := - try apply le_ge; - repeat match goal with - | [ H : _ |- _ ] => apply ge_le in H - end. - - Ltac preomega := unfold wordLeN; intros; ge_to_le; pre_nomega. - - Hint Rewrite wordToN_nat Nat2N.inj_add N2Nat.inj_add Nat2N.inj_mul N2Nat.inj_mul Npow2_nat : N. - - Theorem word_size_bound : forall (w: word n), - w <= Npow2 n - 1. - Proof. - intros; unfold wordLeN; rewrite wordToN_nat. - - assert (B := wordToNat_bound w); - rewrite <- Npow2_nat in B; - apply nat_compare_lt in B. - - unfold N.le; intuition; - rewrite N2Nat.inj_compare in H; - rewrite Nat2N.id in H. - - apply nat_compare_lt in B. - apply nat_compare_gt in H. - - replace (N.to_nat (Npow2 n)) with (S (N.to_nat (Npow2 n - 1))) in * by admit. - intuition. - Qed. - - Theorem constant_bound_N : forall k, - NToWord n k <= k. - Proof. - preomega. - rewrite NToWord_nat. - destruct (le_lt_dec (pow2 n) (N.to_nat k)). - - specialize (wordToNat_bound (natToWord n (N.to_nat k))); nomega. - - rewrite wordToNat_natToWord_idempotent; nomega. - Qed. - - Theorem constant_bound_nat : forall k, - natToWord n k <= N.of_nat k. - Proof. - preomega. - destruct (le_lt_dec (pow2 n) k). - - specialize (wordToNat_bound (natToWord n k)); nomega. - - rewrite wordToNat_natToWord_idempotent; nomega. - Qed. - - Lemma let_bound : forall (x: word n) (f: word n -> word n) xb fb, x <= xb - -> (forall x', x' <= xb -> f x' <= fb) - -> (let k := x in f k) <= fb. - eauto. - Qed. - - Theorem wplus_bound : forall (w1 w2 : word n) b1 b2, - w1 <= b1 - -> w2 <= b2 - -> w1 ^+ w2 <= b1 + b2. - Proof. - preomega. - destruct (le_lt_dec (pow2 n) (N.to_nat b1 + N.to_nat b2)). - - specialize (wordToNat_bound (w1 ^+ w2)); nomega. - - rewrite wplus_alt. - unfold wplusN, wordBinN. - rewrite wordToNat_natToWord_idempotent; nomega. - Qed. - - Theorem wmult_bound : forall (w1 w2 : word n) b1 b2, - w1 <= b1 - -> w2 <= b2 - -> w1 ^* w2 <= b1 * b2. - Proof. - preomega. - destruct (le_lt_dec (pow2 n) (N.to_nat b1 * N.to_nat b2)). - - specialize (wordToNat_bound (w1 ^* w2)); nomega. - - rewrite wmult_alt. - unfold wmultN, wordBinN. - rewrite wordToNat_natToWord_idempotent. - ge_to_le_nat. - - apply Mult.mult_le_compat; nomega. - pre_nomega. - apply Lt.le_lt_trans with (N.to_nat b1 * N.to_nat b2); auto. - apply Mult.mult_le_compat; nomega. - Qed. - - Theorem shiftr_bound : forall (w : word n) b bits, - w <= b - -> shiftr w bits <= N.shiftr b (N.of_nat bits). - Proof. - admit. - Qed. - - Theorem mask_bound : forall (w : word n) m, - mask m w <= Npow2 m - 1. - Proof. - admit. - Qed. - - Theorem mask_update_bound : forall (w : word n) b m, - w <= b - -> mask m w <= (N.min b (Npow2 m - 1)). - Proof. - admit. - Qed. - - - Ltac word_bound := - repeat ( - eassumption - || apply wplus_bound - || apply wmult_bound - || apply mask_update_bound - || apply mask_bound - || apply shiftr_bound - || apply constant_bound_N - || apply constant_bound_nat - || apply word_size_bound - ). - - Notation "$" := (natToWord _). - - Lemma example1 : forall (w1 w2 w3 w4 : word n) b1 b2 b3 b4, - w1 <= b1 - -> w2 <= b2 - -> w3 <= b3 - -> w4 <= b4 - -> { b | w1 ^+ (w2 ^* w3) ^* w4 <= b }. - Proof. - eexists. - word_bound. - Defined. - - (* Eval simpl in fun (w1 w2 w3 w4 : word n) (b1 b2 b3 b4 : N) - (H1 : w1 <= b1) (H2 : w2 <= b2) (H3 : w3 <= b3) (H4 : w4 <= b4) => - projT1 (example1 H1 H2 H3 H4). *) - - Lemma example2 : forall (w1 w2 w3 w4 : word n) b1 b2 b3 b4, - w1 <= b1 - -> w2 <= b2 - -> w3 <= b3 - -> w4 <= b4 - -> { b | w1 ^+ (w2 ^* $7 ^* w3) ^* w4 ^+ $8 ^+ w2 <= b }. - Proof. - eexists. - word_bound. - Defined. - - (*Eval simpl in fun (w1 w2 w3 w4 : word n) (b1 b2 b3 b4 : N) - (H1 : w1 <= b1) (H2 : w2 <= b2) (H3 : w3 <= b3) (H4 : w4 <= b4) => - projT1 (example2 H1 H2 H3 H4). *) - - Lemma example3 : forall (w1 w2 w3 w4 : word n), - w1 <= Npow2 3 - -> w2 <= Npow2 4 - -> w3 <= Npow2 8 - -> w4 <= Npow2 16 - -> { b | w1 ^+ (w2 ^* $7 ^* w3) ^* w4 ^+ $8 ^+ w2 <= b }. - Proof. - eexists. - word_bound. - Defined. - - (* Eval simpl in fun (w1 w2 w3 w4 : word n) - (H1 : w1 <= _) (H2 : w2 <= _) (H3 : w3 <= _) (H4 : w4 <= _) => - projT1 (example3 H1 H2 H3 H4). *) - -End BoundedWord. - -Section MulmodExamples. - - Notation "A <= B" := (wordLeN A B) (at level 70). - Notation "$" := (natToWord _). - - Lemma mask_wand : forall (n: nat) (x: word n) m b, - mask (N.to_nat m) x <= b - -> x ^& (@NToWord n (N.ones m)) <= b. - Proof. - Admitted. - - Ltac word_bound_step := - idtac; match goal with - | [ H: ?x <= _ |- ?x <= _] => eexact H - | [|- (let x := ?y in @?z x) <= ?b ] => refine (@let_bound _ y z _ b _ _); [ | intros ? ? ] - | [|- (let x := ?y in (?a <= ?b)) ] => change ((let x := y in a) <= b) - | [|- (let x := ?y in (?a <= @?b x)) ] => change ((let x := y in a) <= b y); cbv beta - | [|- mask _ _ <= _] => apply mask_bound - | [|- _ ^+ _ <= _] => apply wplus_bound - | [|- _ ^* _ <= _] => apply wmult_bound - | [|- shiftr _ _ <= _] => apply shiftr_bound - | [|- $ _ <= _] => apply constant_bound_nat - | [|- NToWord _ _ <= _] => apply constant_bound_N - | [|- _ <= Npow2 _ - 1] => apply word_size_bound - | [|- _ ^& (@NToWord _ (N.ones _)) <= _] => apply mask_wand - end. - - Ltac simpl_hyps := - match goal with - | [ H: ?x <= _ |- context[?x]] => - unfold Npow, Pos.pow, Npow2, N.shiftr in H; - simpl in H - | [ H: _ |- _ ] => clear H - | _ => idtac - end. - - Ltac word_bound := repeat (word_bound_step; simpl_hyps). - - Ltac word_bound_danger := - word_bound; try eassumption; try apply word_size_bound. - - Lemma example_and : forall x : word 32, - wand x (NToWord 32 (N.ones 10)) <= 1023. - intros. - replace (wand x (NToWord 32 (N.ones 10))) with (mask 10 x) by admit. - word_bound. - Qed. - - Lemma example_shiftr : forall x : word 32, shiftr x 30 <= 3. - intros. - replace 3%N with (N.shiftr (Npow2 32 - 1) (N.of_nat 30)) by (simpl; intuition). - word_bound. - Qed. - - Lemma example_shiftr2 : forall x : word 32, x <= 1023 -> shiftr x 5 <= 31. - intros. - replace 31%N with (N.shiftr 1023%N 5%N) by (simpl; intuition). - word_bound. - Qed. - - Variable f0 f1 f2 f3 f4 f5 f6 f7 f8 f9 : word 32. - Variable g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 : word 32. - Hypothesis Hf0 : f0 <= 2^26. - Hypothesis Hf1 : f1 <= 2^25. - Hypothesis Hf2 : f2 <= 2^26. - Hypothesis Hf3 : f3 <= 2^25. - Hypothesis Hf4 : f4 <= 2^26. - Hypothesis Hf5 : f5 <= 2^25. - Hypothesis Hf6 : f6 <= 2^26. - Hypothesis Hf7 : f7 <= 2^25. - Hypothesis Hf8 : f8 <= 2^26. - Hypothesis Hf9 : f9 <= 2^25. - Hypothesis Hg0 : g0 <= 2^26. - Hypothesis Hg1 : g1 <= 2^25. - Hypothesis Hg2 : g2 <= 2^26. - Hypothesis Hg3 : g3 <= 2^25. - Hypothesis Hg4 : g4 <= 2^26. - Hypothesis Hg5 : g5 <= 2^25. - Hypothesis Hg6 : g6 <= 2^26. - Hypothesis Hg7 : g7 <= 2^25. - - Hypothesis Hg8 : g8 <= 2^26. - Hypothesis Hg9 : g9 <= 2^25. - - Lemma example_mulmod_s_ppt : { b | f0 ^* g0 <= b}. - eexists. - word_bound. - Defined. - - Lemma example_mulmod_s_pp : { b | f0 ^* g0 ^+ $19 ^* (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ f2 ^* g8 ^+ f1 ^* g9 ^* $2) <= b}. - eexists. - word_bound. - Defined. - - Lemma example_mulmod_s_pp_shiftr : - { b | shiftr (f0 ^* g0 ^+ $19 ^* (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ f2 ^* g8 ^+ f1 ^* g9 ^* $2)) 26 <= b}. - eexists. - word_bound. - Defined. - - Lemma example_mulmod_u_fg1 : { b | - (let y : word 32 := (* the type declarations on the let-s make type inference not take forever *) - (f0 ^* g0 ^+ - $19 ^* - (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ f2 ^* g8 ^+ - f1 ^* g9 ^* $2)) in - let y0 : word 32 := - (shiftr y 26 ^+ - (f1 ^* g0 ^+ f0 ^* g1 ^+ - $19 ^* (f9 ^* g2 ^+ f8 ^* g3 ^+ f7 ^* g4 ^+ f6 ^* g5 ^+ f5 ^* g6 ^+ f4 ^* g7 ^+ f3 ^* g8 ^+ f2 ^* g9))) in - let y1 : word 32 := - (shiftr y0 25 ^+ - (f2 ^* g0 ^+ f1 ^* g1 ^* $2 ^+ f0 ^* g2 ^+ - $19 ^* (f9 ^* g3 ^* $2 ^+ f8 ^* g4 ^+ f7 ^* g5 ^* $2 ^+ f6 ^* g6 ^+ f5 ^* g7 ^* $2 ^+ f4 ^* g8 ^+ f3 ^* g9 ^* $2))) in - let y2 : word 32 := - (shiftr y1 26 ^+ - (f3 ^* g0 ^+ f2 ^* g1 ^+ f1 ^* g2 ^+ f0 ^* g3 ^+ - $19 ^* (f9 ^* g4 ^+ f8 ^* g5 ^+ f7 ^* g6 ^+ f6 ^* g7 ^+ f5 ^* g8 ^+ f4 ^* g9))) in - let y3 : word 32 := - (shiftr y2 25 ^+ - (f4 ^* g0 ^+ f3 ^* g1 ^* $2 ^+ f2 ^* g2 ^+ f1 ^* g3 ^* $2 ^+ f0 ^* g4 ^+ - $19 ^* (f9 ^* g5 ^* $2 ^+ f8 ^* g6 ^+ f7 ^* g7 ^* $2 ^+ f6 ^* g8 ^+ f5 ^* g9 ^* $2))) in - let y4 : word 32 := - (shiftr y3 26 ^+ - (f5 ^* g0 ^+ f4 ^* g1 ^+ f3 ^* g2 ^+ f2 ^* g3 ^+ f1 ^* g4 ^+ f0 ^* g5 ^+ - $19 ^* (f9 ^* g6 ^+ f8 ^* g7 ^+ f7 ^* g8 ^+ f6 ^* g9))) in - let y5 : word 32 := - (shiftr y4 25 ^+ - (f6 ^* g0 ^+ f5 ^* g1 ^* $2 ^+ f4 ^* g2 ^+ f3 ^* g3 ^* $2 ^+ f2 ^* g4 ^+ f1 ^* g5 ^* $2 ^+ f0 ^* g6 ^+ - $19 ^* (f9 ^* g7 ^* $2 ^+ f8 ^* g8 ^+ f7 ^* g9 ^* $2))) in - let y6 : word 32 := - (shiftr y5 26 ^+ - (f7 ^* g0 ^+ f6 ^* g1 ^+ f5 ^* g2 ^+ f4 ^* g3 ^+ f3 ^* g4 ^+ f2 ^* g5 ^+ f1 ^* g6 ^+ f0 ^* g7 ^+ - $19 ^* (f9 ^* g8 ^+ f8 ^* g9))) in - let y7 : word 32 := - (shiftr y6 25 ^+ - (f8 ^* g0 ^+ f7 ^* g1 ^* $2 ^+ f6 ^* g2 ^+ f5 ^* g3 ^* $2 ^+ f4 ^* g4 ^+ f3 ^* g5 ^* $2 ^+ f2 ^* g6 ^+ f1 ^* g7 ^* $2 ^+ - f0 ^* g8 ^+ $19 ^* f9 ^* g9 ^* $2)) in - let y8 : word 32 := - (shiftr y7 26 ^+ - (f9 ^* g0 ^+ f8 ^* g1 ^+ f7 ^* g2 ^+ f6 ^* g3 ^+ f5 ^* g4 ^+ f4 ^* g5 ^+ f3 ^* g6 ^+ f2 ^* g7 ^+ f1 ^* g8 ^+ - f0 ^* g9)) in - let y9 : word 32 := - ($19 ^* shiftr y8 25 ^+ - wand - (f0 ^* g0 ^+ - $19 ^* - (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ - f2 ^* g8 ^+ f1 ^* g9 ^* $2)) (@NToWord 32 (N.ones 26%N))) in - let fg1 : word 32 := (shiftr y9 26 ^+ - wand - (shiftr y 26 ^+ - (f1 ^* g0 ^+ f0 ^* g1 ^+ - $19 ^* (f9 ^* g2 ^+ f8 ^* g3 ^+ f7 ^* g4 ^+ f6 ^* g5 ^+ f5 ^* g6 ^+ f4 ^* g7 ^+ f3 ^* g8 ^+ f2 ^* g9))) - (@NToWord 32 (N.ones 26%N))) in - fg1) <= b }. - Proof. - eexists. - (* Time word_bound. *) (* <- It works, but don't do this in the build! *) - Abort. - - Require Import ZArith. - Variable shiftra : forall {l}, word l -> nat -> word l. (* "arithmetic" aka "signed" bitshift *) - Hypothesis shiftra_spec : forall {l} (w : word l) (n:nat), wordToZ (shiftra l w n) = Z.shiftr (wordToZ w) (Z.of_nat n). - - Lemma example_shiftra : forall x : word 4, shiftra 4 x 2 <= 15. - Abort. - - Lemma example_shiftra : forall x : word 4, x <= 7 -> shiftra 4 x 2 <= 1. - Abort. - - Lemma example_mulmod_s_pp_shiftra : - { b | shiftra 32 (f0 ^* g0 ^+ $19 ^* (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ f2 ^* g8 ^+ f1 ^* g9 ^* $2)) 26 <= b}. - Abort. -End MulmodExamples. diff --git a/src/Assembly/MultiBoundedWord.v b/src/Assembly/MultiBoundedWord.v deleted file mode 100644 index add2aa1a2..000000000 --- a/src/Assembly/MultiBoundedWord.v +++ /dev/null @@ -1,252 +0,0 @@ - -Require Import Bedrock.Word Bedrock.Nomega. -Require Import NArith PArith Ndigits Compare_dec Arith. -Require Import ProofIrrelevance Ring. -Require Import BoundedWord. - -Import BoundedWord. - -(* Parameters of boundedness calculations *) -Notation "A <= B" := (wordLeN A B) (at level 70). -Notation "$" := (natToWord _). - -(* Hypothesis-based word-bound tactic *) -Ltac multi_apply0 A L := pose proof (L A). - -Ltac multi_apply1 A L := - match goal with - | [ H: A <= ?b |- _] => pose proof (L A b H) - end. - -Ltac multi_apply2 A B L := - match goal with - | [ H1: A <= ?b1, H2: B <= ?b2 |- _] => pose proof (L A B b1 b2 H1 H2) - end. - -Ltac multi_recurse n T := - match goal with - | [ H: T <= _ |- _] => idtac - | _ => - is_var T; - let T' := (eval cbv delta [T] in T) in multi_recurse n T'; - match goal with - | [ H : T' <= ?x |- _ ] => - pose proof (H : T <= x) - end - - | _ => - match T with - | ?W1 ^+ ?W2 => - multi_recurse n W1; multi_recurse n W2; - multi_apply2 W1 W2 (@wplus_bound n) - - | ?W1 ^* ?W2 => - multi_recurse n W1; multi_recurse n W2; - multi_apply2 W1 W2 (@wmult_bound n) - - | mask ?m ?w => - multi_recurse n w; - multi_apply1 w (fun b => @mask_update_bound n w b) - - | mask ?m ?w => - multi_recurse n w; - pose proof (@mask_bound n w m) - - | ?x ^& (@NToWord _ (N.ones ?m)) => - multi_recurse n (mask (N.to_nat m) x); - match goal with - | [ H: (mask (N.to_nat m) x) <= ?b |- _] => - pose proof (@mask_wand n x m b H) - end - - | shiftr ?w ?bits => - multi_recurse n w; - match goal with - | [ H: w <= ?b |- _] => - pose proof (@shiftr_bound n w b bits H) - end - - | NToWord _ ?k => pose proof (@constant_bound_N n k) - | natToWord _ ?k => pose proof (@constant_bound_nat n k) - | ($ ?k) => pose proof (@constant_bound_nat n k) - | _ => pose proof (@word_size_bound n T) - end - end. - -Lemma unwrap_let: forall {n} (y: word n) (f: word n -> word n) (b: N), - (let x := y in f x) <= b <-> let x := y in (f x <= b). -Proof. intuition. Qed. - -Ltac multi_bound n := - match goal with - | [|- let A := ?B in _] => - multi_recurse n B; intro; multi_bound n - | [|- (let A := _ in _) <= _] => - apply unwrap_let; multi_bound n - | [|- ?W <= _ ] => - multi_recurse n W - end. - -(* Examples *) -Lemma example1 : forall {n} (w1 w2 w3 w4 : word n) b1 b2 b3 b4, - w1 <= b1 - -> w2 <= b2 - -> w3 <= b3 - -> w4 <= b4 - -> { b | let w5 := w2 ^* w3 in w1 ^+ w5 ^* w4 <= b }. -Proof. - eexists. - multi_bound n. - eassumption. -Defined. - -Lemma example2 : forall {n} (w1 w2 w3 w4 : word n) b1 b2 b3 b4, - w1 <= b1 - -> w2 <= b2 - -> w3 <= b3 - -> w4 <= b4 - -> { b | (let w5 := (w2 ^* $7 ^* w3) in w1 ^+ w5 ^* w4 ^+ $8 ^+ w2) <= b }. -Proof. - eexists. - multi_bound n. - eassumption. -Defined. - -Lemma example3 : forall {n} (w1 w2 w3 w4 : word n), - w1 <= Npow2 3 - -> w2 <= Npow2 4 - -> w3 <= Npow2 8 - -> w4 <= Npow2 16 - -> { b | w1 ^+ (w2 ^* $7 ^* w3) ^* w4 ^+ $8 ^+ w2 <= b }. -Proof. - eexists. - multi_bound n. - eassumption. -Defined. - -Section MulmodExamples. - - Notation "A <= B" := (wordLeN A B) (at level 70). - Notation "$" := (natToWord 32). - - Lemma example_and : forall x : word 32, wand x (NToWord 32 (N.ones 10)) <= 1023. - intros. - replace (wand x (NToWord 32 (N.ones 10))) with (mask 10 x) by admit. - multi_bound 32; eassumption. - Qed. - - Lemma example_shiftr : forall x : word 32, shiftr x 30 <= 3. - intros. - replace 3%N with (N.shiftr (Npow2 32 - 1) (N.of_nat 30)) by (simpl; intuition). - multi_bound 32; eassumption. - Qed. - - Lemma example_shiftr2 : forall x : word 32, x <= 1023 -> shiftr x 5 <= 31. - intros. - replace 31%N with (N.shiftr 1023%N 5%N) by (simpl; intuition). - multi_bound 32; eassumption. - Qed. - - Variable f0 f1 f2 f3 f4 f5 f6 f7 f8 f9 : word 32. - Variable g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 : word 32. - Hypothesis Hf0 : f0 <= 2^26. - Hypothesis Hf1 : f1 <= 2^25. - Hypothesis Hf2 : f2 <= 2^26. - Hypothesis Hf3 : f3 <= 2^25. - Hypothesis Hf4 : f4 <= 2^26. - Hypothesis Hf5 : f5 <= 2^25. - Hypothesis Hf6 : f6 <= 2^26. - Hypothesis Hf7 : f7 <= 2^25. - Hypothesis Hf8 : f8 <= 2^26. - Hypothesis Hf9 : f9 <= 2^25. - Hypothesis Hg0 : g0 <= 2^26. - Hypothesis Hg1 : g1 <= 2^25. - Hypothesis Hg2 : g2 <= 2^26. - Hypothesis Hg3 : g3 <= 2^25. - Hypothesis Hg4 : g4 <= 2^26. - Hypothesis Hg5 : g5 <= 2^25. - Hypothesis Hg6 : g6 <= 2^26. - Hypothesis Hg7 : g7 <= 2^25. - Hypothesis Hg8 : g8 <= 2^26. - Hypothesis Hg9 : g9 <= 2^25. - - Lemma example_mulmod_s_ppt : { b | f0 ^* g0 <= b}. - eexists. - multi_bound 32; eassumption. - Defined. - - Lemma example_mulmod_s_pp : { b | f0 ^* g0 ^+ $19 ^* (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ f2 ^* g8 ^+ f1 ^* g9 ^* $2) <= b}. - eexists. - multi_bound 32; eassumption. - Defined. - - Lemma example_mulmod_s_pp_shiftr : - { b | shiftr (f0 ^* g0 ^+ $19 ^* (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ f2 ^* g8 ^+ f1 ^* g9 ^* $2)) 26 <= b}. - eexists. - multi_bound 32; eassumption. - Defined. - - Lemma example_mulmod_u_fg1 : { b | - (let y : word 32 := - (f0 ^* g0 ^+ - $19 ^* - (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ f2 ^* g8 ^+ - f1 ^* g9 ^* $2)) in - let y0 : word 32 := - (shiftr y 26 ^+ - (f1 ^* g0 ^+ f0 ^* g1 ^+ - $19 ^* (f9 ^* g2 ^+ f8 ^* g3 ^+ f7 ^* g4 ^+ f6 ^* g5 ^+ f5 ^* g6 ^+ f4 ^* g7 ^+ f3 ^* g8 ^+ f2 ^* g9))) in - let y1 : word 32 := - (shiftr y0 25 ^+ - (f2 ^* g0 ^+ f1 ^* g1 ^* $2 ^+ f0 ^* g2 ^+ - $19 ^* (f9 ^* g3 ^* $2 ^+ f8 ^* g4 ^+ f7 ^* g5 ^* $2 ^+ f6 ^* g6 ^+ f5 ^* g7 ^* $2 ^+ f4 ^* g8 ^+ f3 ^* g9 ^* $2))) in - let y2 : word 32 := - (shiftr y1 26 ^+ - (f3 ^* g0 ^+ f2 ^* g1 ^+ f1 ^* g2 ^+ f0 ^* g3 ^+ - $19 ^* (f9 ^* g4 ^+ f8 ^* g5 ^+ f7 ^* g6 ^+ f6 ^* g7 ^+ f5 ^* g8 ^+ f4 ^* g9))) in - let y3 : word 32 := - (shiftr y2 25 ^+ - (f4 ^* g0 ^+ f3 ^* g1 ^* $2 ^+ f2 ^* g2 ^+ f1 ^* g3 ^* $2 ^+ f0 ^* g4 ^+ - $19 ^* (f9 ^* g5 ^* $2 ^+ f8 ^* g6 ^+ f7 ^* g7 ^* $2 ^+ f6 ^* g8 ^+ f5 ^* g9 ^* $2))) in - let y4 : word 32 := - (shiftr y3 26 ^+ - (f5 ^* g0 ^+ f4 ^* g1 ^+ f3 ^* g2 ^+ f2 ^* g3 ^+ f1 ^* g4 ^+ f0 ^* g5 ^+ - $19 ^* (f9 ^* g6 ^+ f8 ^* g7 ^+ f7 ^* g8 ^+ f6 ^* g9))) in - let y5 : word 32 := - (shiftr y4 25 ^+ - (f6 ^* g0 ^+ f5 ^* g1 ^* $2 ^+ f4 ^* g2 ^+ f3 ^* g3 ^* $2 ^+ f2 ^* g4 ^+ f1 ^* g5 ^* $2 ^+ f0 ^* g6 ^+ - $19 ^* (f9 ^* g7 ^* $2 ^+ f8 ^* g8 ^+ f7 ^* g9 ^* $2))) in - let y6 : word 32 := - (shiftr y5 26 ^+ - (f7 ^* g0 ^+ f6 ^* g1 ^+ f5 ^* g2 ^+ f4 ^* g3 ^+ f3 ^* g4 ^+ f2 ^* g5 ^+ f1 ^* g6 ^+ f0 ^* g7 ^+ - $19 ^* (f9 ^* g8 ^+ f8 ^* g9))) in - let y7 : word 32 := - (shiftr y6 25 ^+ - (f8 ^* g0 ^+ f7 ^* g1 ^* $2 ^+ f6 ^* g2 ^+ f5 ^* g3 ^* $2 ^+ f4 ^* g4 ^+ f3 ^* g5 ^* $2 ^+ f2 ^* g6 ^+ f1 ^* g7 ^* $2 ^+ - f0 ^* g8 ^+ $19 ^* f9 ^* g9 ^* $2)) in - let y8 : word 32 := - (shiftr y7 26 ^+ - (f9 ^* g0 ^+ f8 ^* g1 ^+ f7 ^* g2 ^+ f6 ^* g3 ^+ f5 ^* g4 ^+ f4 ^* g5 ^+ f3 ^* g6 ^+ f2 ^* g7 ^+ f1 ^* g8 ^+ - f0 ^* g9)) in - let y9 : word 32 := - ($19 ^* shiftr y8 25 ^+ - wand - (f0 ^* g0 ^+ - $19 ^* - (f9 ^* g1 ^* $2 ^+ f8 ^* g2 ^+ f7 ^* g3 ^* $2 ^+ f6 ^* g4 ^+ f5 ^* g5 ^* $2 ^+ f4 ^* g6 ^+ f3 ^* g7 ^* $2 ^+ - f2 ^* g8 ^+ f1 ^* g9 ^* $2)) (@NToWord 32 (N.ones 26%N))) in - let fg1 : word 32 := (shiftr y9 26 ^+ - wand - (shiftr y 26 ^+ - (f1 ^* g0 ^+ f0 ^* g1 ^+ - $19 ^* (f9 ^* g2 ^+ f8 ^* g3 ^+ f7 ^* g4 ^+ f6 ^* g5 ^+ f5 ^* g6 ^+ f4 ^* g7 ^+ f3 ^* g8 ^+ f2 ^* g9))) - (@NToWord 32 (N.ones 26%N))) in - fg1) <= b }. - Proof. - eexists; multi_bound 32; eassumption. - - Defined. - - Eval simpl in (proj1_sig example_mulmod_u_fg1). - -End MulmodExamples. -- cgit v1.2.3