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Diffstat (limited to 'theories/Numbers/Natural/Peano/NPeano.v')
| -rw-r--r-- | theories/Numbers/Natural/Peano/NPeano.v | 84 |
1 files changed, 82 insertions, 2 deletions
diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v index 58b1b018..d0168bd9 100644 --- a/theories/Numbers/Natural/Peano/NPeano.v +++ b/theories/Numbers/Natural/Peano/NPeano.v @@ -8,8 +8,88 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -Require Import PeanoNat NAxioms. +Require Import PeanoNat Even NAxioms. -(** * [PeanoNat.Nat] already implements [NAxiomSig] *) +(** This file is DEPRECATED ! Use [PeanoNat] (or [Arith]) instead. *) + +(** [PeanoNat.Nat] already implements [NAxiomSig] *) Module Nat <: NAxiomsSig := Nat. + +(** Compat notations for stuff that used to be at the beginning of NPeano. *) + +Notation leb := Nat.leb (compat "8.4"). +Notation ltb := Nat.ltb (compat "8.4"). +Notation leb_le := Nat.leb_le (compat "8.4"). +Notation ltb_lt := Nat.ltb_lt (compat "8.4"). +Notation pow := Nat.pow (compat "8.4"). +Notation pow_0_r := Nat.pow_0_r (compat "8.4"). +Notation pow_succ_r := Nat.pow_succ_r (compat "8.4"). +Notation square := Nat.square (compat "8.4"). +Notation square_spec := Nat.square_spec (compat "8.4"). +Notation Even := Nat.Even (compat "8.4"). +Notation Odd := Nat.Odd (compat "8.4"). +Notation even := Nat.even (compat "8.4"). +Notation odd := Nat.odd (compat "8.4"). +Notation even_spec := Nat.even_spec (compat "8.4"). +Notation odd_spec := Nat.odd_spec (compat "8.4"). + +Lemma Even_equiv n : Even n <-> Even.even n. +Proof. symmetry. apply Even.even_equiv. Qed. +Lemma Odd_equiv n : Odd n <-> Even.odd n. +Proof. symmetry. apply Even.odd_equiv. Qed. + +Notation divmod := Nat.divmod (compat "8.4"). +Notation div := Nat.div (compat "8.4"). +Notation modulo := Nat.modulo (compat "8.4"). +Notation divmod_spec := Nat.divmod_spec (compat "8.4"). +Notation div_mod := Nat.div_mod (compat "8.4"). +Notation mod_bound_pos := Nat.mod_bound_pos (compat "8.4"). +Notation sqrt_iter := Nat.sqrt_iter (compat "8.4"). +Notation sqrt := Nat.sqrt (compat "8.4"). +Notation sqrt_iter_spec := Nat.sqrt_iter_spec (compat "8.4"). +Notation sqrt_spec := Nat.sqrt_spec (compat "8.4"). +Notation log2_iter := Nat.log2_iter (compat "8.4"). +Notation log2 := Nat.log2 (compat "8.4"). +Notation log2_iter_spec := Nat.log2_iter_spec (compat "8.4"). +Notation log2_spec := Nat.log2_spec (compat "8.4"). +Notation log2_nonpos := Nat.log2_nonpos (compat "8.4"). +Notation gcd := Nat.gcd (compat "8.4"). +Notation divide := Nat.divide (compat "8.4"). +Notation gcd_divide := Nat.gcd_divide (compat "8.4"). +Notation gcd_divide_l := Nat.gcd_divide_l (compat "8.4"). +Notation gcd_divide_r := Nat.gcd_divide_r (compat "8.4"). +Notation gcd_greatest := Nat.gcd_greatest (compat "8.4"). +Notation testbit := Nat.testbit (compat "8.4"). +Notation shiftl := Nat.shiftl (compat "8.4"). +Notation shiftr := Nat.shiftr (compat "8.4"). +Notation bitwise := Nat.bitwise (compat "8.4"). +Notation land := Nat.land (compat "8.4"). +Notation lor := Nat.lor (compat "8.4"). +Notation ldiff := Nat.ldiff (compat "8.4"). +Notation lxor := Nat.lxor (compat "8.4"). +Notation double_twice := Nat.double_twice (compat "8.4"). +Notation testbit_0_l := Nat.testbit_0_l (compat "8.4"). +Notation testbit_odd_0 := Nat.testbit_odd_0 (compat "8.4"). +Notation testbit_even_0 := Nat.testbit_even_0 (compat "8.4"). +Notation testbit_odd_succ := Nat.testbit_odd_succ (compat "8.4"). +Notation testbit_even_succ := Nat.testbit_even_succ (compat "8.4"). +Notation shiftr_spec := Nat.shiftr_spec (compat "8.4"). +Notation shiftl_spec_high := Nat.shiftl_spec_high (compat "8.4"). +Notation shiftl_spec_low := Nat.shiftl_spec_low (compat "8.4"). +Notation div2_bitwise := Nat.div2_bitwise (compat "8.4"). +Notation odd_bitwise := Nat.odd_bitwise (compat "8.4"). +Notation div2_decr := Nat.div2_decr (compat "8.4"). +Notation testbit_bitwise_1 := Nat.testbit_bitwise_1 (compat "8.4"). +Notation testbit_bitwise_2 := Nat.testbit_bitwise_2 (compat "8.4"). +Notation land_spec := Nat.land_spec (compat "8.4"). +Notation ldiff_spec := Nat.ldiff_spec (compat "8.4"). +Notation lor_spec := Nat.lor_spec (compat "8.4"). +Notation lxor_spec := Nat.lxor_spec (compat "8.4"). + +Infix "<=?" := Nat.leb (at level 70) : nat_scope. +Infix "<?" := Nat.ltb (at level 70) : nat_scope. +Infix "^" := Nat.pow : nat_scope. +Infix "/" := Nat.div : nat_scope. +Infix "mod" := Nat.modulo (at level 40, no associativity) : nat_scope. +Notation "( x | y )" := (Nat.divide x y) (at level 0) : nat_scope. |