Commit 056832f1 by MARCHE Claude

### Example hamming_sequence in progress + realization of number.Coprime

parent e25459b0
 ... ... @@ -839,7 +839,7 @@ COQLIBS_BOOL = \$(addprefix lib/coq/bool/, \$(COQLIBS_BOOL_FILES)) COQLIBS_REAL_FILES = Abs ExpLog FromInt MinMax PowerInt Real Square RealInfix COQLIBS_REAL = \$(addprefix lib/coq/real/, \$(COQLIBS_REAL_FILES)) COQLIBS_NUMBER_FILES = Divisibility Gcd Parity Prime COQLIBS_NUMBER_FILES = Divisibility Gcd Parity Prime Coprime COQLIBS_NUMBER = \$(addprefix lib/coq/number/, \$(COQLIBS_NUMBER_FILES)) COQLIBS_SET_FILES = Set ... ...
 ... ... @@ -14,34 +14,75 @@ module HammingSequence use import int.Int use import int.MinMax use import number.Divisibility use import number.Prime use import number.Coprime (* for Euclid's lemma *) function hamming (n:int) : int predicate is_hamming (n:int) = forall d:int. prime d /\ divides d n -> d = 2 \/ d = 3 \/ d = 5 lemma is_hamming_times2 : forall n:int. n >= 1 -> is_hamming n -> is_hamming (2*n) lemma is_hamming_times3 : forall n:int. n >= 1 -> is_hamming n -> is_hamming (3*n) lemma is_hamming_times5 : forall n:int. n >= 1 -> is_hamming n -> is_hamming (5*n) use import array.Array use import ref.Ref let hamming (n:int) : array int requires { n >= 1 } ensures { forall k:int. 0 <= k < n -> result[k] = hamming k } ensures { forall k:int. 0 <= k < n -> is_hamming result[k] } ensures { forall k:int. 0 < k < n -> result[k-1] < result[k] } ensures { forall k m:int. 0 < k < n -> result[k-1] < m < result[k] -> not (is_hamming m) } = let t = make n 0 in t[0] <- 1; let x2 = ref 2 in let j2 = ref 0 in let x3 = ref 3 in let j3 = ref 0 in let x5 = ref 5 in let j5 = ref 0 in for i=0 to n-1 do invariant { !x2 = 2*t[!j2] /\ !x3 = 3*t[!j3] /\ !x5 = 5*t[!j5] /\ !x2 > t[i] /\ !x3 > t[i] /\ !x5 > t[i] } for i=1 to n-1 do invariant { forall k:int. 0 <= k < i -> t[k] > 0 } invariant { forall k:int. 0 < k < i -> t[k-1] < t[k] } invariant { forall k:int. 0 <= k < i -> is_hamming t[k] } invariant { 0 <= !j2 < i } invariant { 0 <= !j3 < i } invariant { 0 <= !j5 < i } invariant { !x2 = 2*t[!j2] } invariant { !x3 = 3*t[!j3] } invariant { !x5 = 5*t[!j5] } invariant { !x2 > t[i-1] } invariant { !x3 > t[i-1] } invariant { !x5 > t[i-1] } let next = min !x2 (min !x3 !x5) in t[i] <- next; while !x2 <= next do j2 := !j2+1; x2 := 2*t[!j2] done; while !x3 <= next do j3 := !j3+1; x3 := 3*t[!j3] done; while !x5 <= next do j5 := !j5+1; x5 := 5*t[!j5] done while !x2 <= next do invariant { 0 <= !j2 <= i } invariant { !x2 = 2*t[!j2] } variant { next - !x2 } assert { !j2 < i }; j2 := !j2+1; x2 := 2*t[!j2] done; while !x3 <= next do invariant { 0 <= !j3 <= i } invariant { !x3 = 3*t[!j3] } variant { next - !x3 } assert { !j3 < i }; j3 := !j3+1; x3 := 3*t[!j3] done; while !x5 <= next do invariant { 0 <= !j5 <= i } invariant { !x5 = 5*t[!j5] } variant { next - !x5 } assert { !j5 < i }; j5 := !j5+1; x5 := 5*t[!j5] done done; t let test () = hamming 30 end
This diff is collapsed.
 ... ... @@ -430,6 +430,10 @@ let built_in_theories = "infix >", None, eval_int_rel Big_int.gt_big_int; "infix >=", None, eval_int_rel Big_int.ge_big_int; ] ; ["int"],"MinMax", [], [ "min", None, eval_int_op Big_int.min_big_int; "max", None, eval_int_op Big_int.max_big_int; ] ; ["int"],"ComputerDivision", [], [ "div", None, eval_int_op computer_div_big_int; "mod", None, eval_int_op computer_mod_big_int; ... ...
 ... ... @@ -189,4 +189,12 @@ theory Coprime forall p: int. prime p <-> 2 <= p && forall n:int. 1 <= n < p -> coprime n p lemma Gauss: forall a b c:int. divides a (b*c) /\ coprime a b -> divides a c lemma Euclid: forall p a b:int. prime p /\ divides p (a*b) -> divides p a \/ divides p b end
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