find: updated proof

examples/programs/find/find_WP_FIND_WP_parameter_find_1.v

-(* This file is generated by Why3's Coq driver *)
-(* Beware! Only edit allowed sections below    *)
-Require Import ZArith.
-Require Import Rbase.
-Definition unit  := unit.
-
-Parameter mark : Type.
-
-Parameter at1: forall (a:Type), a -> mark  -> a.
-
-Implicit Arguments at1.
-
-Parameter old: forall (a:Type), a  -> a.
-
-Implicit Arguments old.
-
-Inductive ref (a:Type) :=
-  | mk_ref : a -> ref a.
-Implicit Arguments mk_ref.
-
-Definition contents (a:Type)(u:(ref a)): a :=
-  match u with
-  | mk_ref contents1 => contents1
-  end.
-Implicit Arguments contents.
-
-Parameter map : forall (a:Type) (b:Type), Type.
-
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a  -> b.
-
-Implicit Arguments get.
-
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b  -> (map a b).
-
-Implicit Arguments set.
-
-Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
-  forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set m a1 b1)
-  a2) = b1).
-
-Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)),
-  forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1)
-  a2) = (get m a2)).
-
-Parameter const: forall (b:Type) (a:Type), b  -> (map a b).
-
-Set Contextual Implicit.
-Implicit Arguments const.
-Unset Contextual Implicit.
-
-Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
-  b1):(map a b)) a1) = b1).
-
-Inductive array (a:Type) :=
-  | mk_array : Z -> (map Z a) -> array a.
-Implicit Arguments mk_array.
-
-Definition elts (a:Type)(u:(array a)): (map Z a) :=
-  match u with
-  | mk_array _ elts1 => elts1
-  end.
-Implicit Arguments elts.
-
-Definition length (a:Type)(u:(array a)): Z :=
-  match u with
-  | mk_array length1 _ => length1
-  end.
-Implicit Arguments length.
-
-Definition get1 (a:Type)(a1:(array a)) (i:Z): a := (get (elts a1) i).
-Implicit Arguments get1.
-
-Definition set1 (a:Type)(a1:(array a)) (i:Z) (v:a): (array a) :=
-  match a1 with
-  | mk_array xcl0 _ => (mk_array xcl0 (set (elts a1) i v))
-  end.
-Implicit Arguments set1.
-
-Definition map_eq_sub (a:Type)(a1:(map Z a)) (a2:(map Z a)) (l:Z)
-  (u:Z): Prop := forall (i:Z), ((l <= i)%Z /\ (i <  u)%Z) -> ((get a1
-  i) = (get a2 i)).
-Implicit Arguments map_eq_sub.
-
-Definition exchange (a:Type)(a1:(map Z a)) (a2:(map Z a)) (i:Z)
-  (j:Z): Prop := ((get a1 i) = (get a2 j)) /\ (((get a2 i) = (get a1 j)) /\
-  forall (k:Z), ((~ (k = i)) /\ ~ (k = j)) -> ((get a1 k) = (get a2 k))).
-Implicit Arguments exchange.
-
-Axiom exchange_set : forall (a:Type), forall (a1:(map Z a)), forall (i:Z)
-  (j:Z), (exchange a1 (set (set a1 i (get a1 j)) j (get a1 i)) i j).
-
-Inductive permut_sub{a:Type}  : (map Z a) -> (map Z a) -> Z -> Z -> Prop :=
-  | permut_refl : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
-      (map_eq_sub a1 a2 l u) -> (permut_sub a1 a2 l u)
-  | permut_sym : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
-      (permut_sub a1 a2 l u) -> (permut_sub a2 a1 l u)
-  | permut_trans : forall (a1:(map Z a)) (a2:(map Z a)) (a3:(map Z a)),
-      forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> ((permut_sub a2 a3 l
-      u) -> (permut_sub a1 a3 l u))
-  | permut_exchange : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z)
-      (u:Z) (i:Z) (j:Z), ((l <= i)%Z /\ (i <  u)%Z) -> (((l <= j)%Z /\
-      (j <  u)%Z) -> ((exchange a1 a2 i j) -> (permut_sub a1 a2 l u))).
-Implicit Arguments permut_sub.
-
-Axiom permut_weakening : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z
-  a)), forall (l1:Z) (r1:Z) (l2:Z) (r2:Z), (((l1 <= l2)%Z /\ (l2 <= r2)%Z) /\
-  (r2 <= r1)%Z) -> ((permut_sub a1 a2 l2 r2) -> (permut_sub a1 a2 l1 r1)).
-
-Axiom permut_eq : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z a)),
-  forall (l:Z) (u:Z), (l <= u)%Z -> ((permut_sub a1 a2 l u) -> forall (i:Z),
-  ((i <  l)%Z \/ (u <= i)%Z) -> ((get a2 i) = (get a1 i))).
-
-Axiom permut_exists : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z a)),
-  forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> forall (i:Z), ((l <= i)%Z /\
-  (i <  u)%Z) -> exists j:Z, ((l <= j)%Z /\ (j <  u)%Z) /\ ((get a2
-  i) = (get a1 j)).
-
-Definition exchange1 (a:Type)(a1:(array a)) (a2:(array a)) (i:Z)
-  (j:Z): Prop := (exchange (elts a1) (elts a2) i j).
-Implicit Arguments exchange1.
-
-Definition permut_sub1 (a:Type)(a1:(array a)) (a2:(array a)) (l:Z)
-  (u:Z): Prop := (permut_sub (elts a1) (elts a2) l u).
-Implicit Arguments permut_sub1.
-
-Definition permut (a:Type)(a1:(array a)) (a2:(array a)): Prop :=
-  ((length a1) = (length a2)) /\ (permut_sub (elts a1) (elts a2) 0%Z
-  (length a1)).
-Implicit Arguments permut.
-
-Axiom exchange_permut : forall (a:Type), forall (a1:(array a)) (a2:(array a))
-  (i:Z) (j:Z), (exchange1 a1 a2 i j) -> (((length a1) = (length a2)) ->
-  (((0%Z <= i)%Z /\ (i <  (length a1))%Z) -> (((0%Z <= j)%Z /\
-  (j <  (length a1))%Z) -> (permut a1 a2)))).
-
-Definition array_eq_sub (a:Type)(a1:(array a)) (a2:(array a)) (l:Z)
-  (u:Z): Prop := (map_eq_sub (elts a1) (elts a2) l u).
-Implicit Arguments array_eq_sub.
-
-Definition array_eq (a:Type)(a1:(array a)) (a2:(array a)): Prop :=
-  ((length a1) = (length a2)) /\ (array_eq_sub a1 a2 0%Z (length a1)).
-Implicit Arguments array_eq.
-
-Axiom array_eq_sub_permut : forall (a:Type), forall (a1:(array a)) (a2:(array
-  a)) (l:Z) (u:Z), (array_eq_sub a1 a2 l u) -> (permut_sub1 a1 a2 l u).
-
-Axiom array_eq_permut : forall (a:Type), forall (a1:(array a)) (a2:(array
-  a)), (array_eq a1 a2) -> (permut a1 a2).
-
-Parameter usN:  Z.
-
-
-Parameter f:  Z.
-
-
-Axiom f_N_range : (1%Z <= (f ))%Z /\ ((f ) <= (usN ))%Z.
-
-Definition found(a:(array Z)): Prop := forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\
-  (p <= (f ))%Z) /\ ((f ) <= q)%Z) /\ (q <= (usN ))%Z) -> (((get1 a
-  p) <= (get1 a (f )))%Z /\ ((get1 a (f )) <= (get1 a q))%Z).
-
-Definition m_invariant(m:Z) (a:(array Z)): Prop := (m <= (f ))%Z /\
-  forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\ (p <  m)%Z) /\ (m <= q)%Z) /\
-  (q <= (usN ))%Z) -> ((get1 a p) <= (get1 a q))%Z.
-
-Definition n_invariant(n:Z) (a:(array Z)): Prop := ((f ) <= n)%Z /\
-  forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\ (p <= n)%Z) /\ (n <  q)%Z) /\
-  (q <= (usN ))%Z) -> ((get1 a p) <= (get1 a q))%Z.
-
-Definition i_invariant(m:Z) (n:Z) (i:Z) (r:Z) (a:(array Z)): Prop :=
-  (m <= i)%Z /\ ((forall (p:Z), ((1%Z <= p)%Z /\ (p <  i)%Z) -> ((get1 a
-  p) <= r)%Z) /\ ((i <= n)%Z -> exists p:Z, ((i <= p)%Z /\ (p <= n)%Z) /\
-  (r <= (get1 a p))%Z)).
-
-Definition j_invariant(m:Z) (n:Z) (j:Z) (r:Z) (a:(array Z)): Prop :=
-  (j <= n)%Z /\ ((forall (q:Z), ((j <  q)%Z /\ (q <= (usN ))%Z) ->
-  (r <= (get1 a q))%Z) /\ ((m <= j)%Z -> exists q:Z, ((m <= q)%Z /\
-  (q <= j)%Z) /\ ((get1 a q) <= r)%Z)).
-
-Definition termination(i:Z) (j:Z) (i0:Z) (j0:Z) (r:Z) (a:(array Z)): Prop :=
-  ((i0 <  i)%Z /\ (j <  j0)%Z) \/ (((i <= (f ))%Z /\ ((f ) <= j)%Z) /\
-  ((get1 a (f )) = r)).
-
-Theorem WP_parameter_find : forall (a:Z), forall (a1:(map Z Z)), let a2 :=
-  (mk_array a a1) in ((a = ((usN ) + 1%Z)%Z) -> forall (n:Z), forall (m:Z),
-  forall (a3:(map Z Z)), let a4 := (mk_array a a3) in (((m_invariant m a4) /\
-  ((n_invariant n a4) /\ ((permut a4 a2) /\ ((1%Z <= m)%Z /\
-  (n <= (usN ))%Z)))) -> ((m <  n)%Z -> (((0%Z <= (f ))%Z /\
-  ((f ) <  a)%Z) -> let result := (get a3 (f )) in forall (j:Z),
-  forall (i:Z), forall (a5:(map Z Z)), let a6 := (mk_array a a5) in
-  (((i_invariant m n i result a6) /\ ((j_invariant m n j result a6) /\
-  ((m_invariant m a6) /\ ((n_invariant n a6) /\ ((0%Z <= j)%Z /\
-  ((i <= ((usN ) + 1%Z)%Z)%Z /\ ((termination i j m n result a6) /\
-  (permut a6 a2)))))))) -> ((i <= j)%Z -> forall (i1:Z), ((i_invariant m n i1
-  result a6) /\ (((i <= i1)%Z /\ (i1 <= n)%Z) /\ (termination i1 j m n result
-  a6))) -> (((0%Z <= i1)%Z /\ (i1 <  a)%Z) -> ((~ ((get a5
-  i1) <  result)%Z) -> forall (j1:Z), ((j_invariant m n j1 result a6) /\
-  ((j1 <= j)%Z /\ ((m <= j1)%Z /\ (termination i1 j1 m n result a6)))) ->
-  (((0%Z <= j1)%Z /\ (j1 <  a)%Z) -> ((~ (result <  (get a5 j1))%Z) ->
-  ((((get a5 j1) <= result)%Z /\ (result <= (get a5 i1))%Z) ->
-  ((i1 <= j1)%Z -> (((0%Z <= i1)%Z /\ (i1 <  a)%Z) -> (((0%Z <= j1)%Z /\
-  (j1 <  a)%Z) -> (((0%Z <= i1)%Z /\ (i1 <  a)%Z) -> forall (a7:(map Z Z)),
-  (a7 = (set a5 i1 (get a5 j1))) -> (((0%Z <= j1)%Z /\ (j1 <  a)%Z) ->
-  forall (a8:(map Z Z)), (a8 = (set a7 j1 (get a5 i1))) -> ((exchange a8 a5
-  i1 j1) -> (((get a8 i1) <= result)%Z -> ((result <= (get a8 j1))%Z ->
-  forall (i2:Z), (i2 = (i1 + 1%Z)%Z) -> forall (j2:Z), (j2 = (j1 - 1%Z)%Z) ->
-  (permut (mk_array a a8) a2)))))))))))))))))))).
-(* YOU MAY EDIT THE PROOF BELOW *)
-intuition.
-intuition.
-unfold permut; simpl.
-intuition.
-apply permut_trans with a5.
-apply permut_exchange with i1 j1; intuition.
-red in H17.
-simpl in H17.
-intuition.
-Qed.
-(* DO NOT EDIT BELOW *)
-
-

examples/programs/find/find_WP_FIND_WP_parameter_find_2.v

-(* This file is generated by Why3's Coq driver *)
-(* Beware! Only edit allowed sections below    *)
-Require Import ZArith.
-Require Import Rbase.
-Definition unit  := unit.
-
-Parameter mark : Type.
-
-Parameter at1: forall (a:Type), a -> mark  -> a.
-
-Implicit Arguments at1.
-
-Parameter old: forall (a:Type), a  -> a.
-
-Implicit Arguments old.
-
-Inductive ref (a:Type) :=
-  | mk_ref : a -> ref a.
-Implicit Arguments mk_ref.
-
-Definition contents (a:Type)(u:(ref a)): a :=
-  match u with
-  | mk_ref contents1 => contents1
-  end.
-Implicit Arguments contents.
-
-Parameter map : forall (a:Type) (b:Type), Type.
-
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a  -> b.
-
-Implicit Arguments get.
-
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b  -> (map a b).
-
-Implicit Arguments set.
-
-Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
-  forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set m a1 b1)
-  a2) = b1).
-
-Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)),
-  forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1)
-  a2) = (get m a2)).
-
-Parameter const: forall (b:Type) (a:Type), b  -> (map a b).
-
-Set Contextual Implicit.
-Implicit Arguments const.
-Unset Contextual Implicit.
-
-Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
-  b1):(map a b)) a1) = b1).
-
-Inductive array (a:Type) :=
-  | mk_array : Z -> (map Z a) -> array a.
-Implicit Arguments mk_array.
-
-Definition elts (a:Type)(u:(array a)): (map Z a) :=
-  match u with
-  | mk_array _ elts1 => elts1
-  end.
-Implicit Arguments elts.
-
-Definition length (a:Type)(u:(array a)): Z :=
-  match u with
-  | mk_array length1 _ => length1
-  end.
-Implicit Arguments length.
-
-Definition get1 (a:Type)(a1:(array a)) (i:Z): a := (get (elts a1) i).
-Implicit Arguments get1.
-
-Definition set1 (a:Type)(a1:(array a)) (i:Z) (v:a): (array a) :=
-  match a1 with
-  | mk_array xcl0 _ => (mk_array xcl0 (set (elts a1) i v))
-  end.
-Implicit Arguments set1.
-
-Definition map_eq_sub (a:Type)(a1:(map Z a)) (a2:(map Z a)) (l:Z)
-  (u:Z): Prop := forall (i:Z), ((l <= i)%Z /\ (i <  u)%Z) -> ((get a1
-  i) = (get a2 i)).
-Implicit Arguments map_eq_sub.
-
-Definition exchange (a:Type)(a1:(map Z a)) (a2:(map Z a)) (i:Z)
-  (j:Z): Prop := ((get a1 i) = (get a2 j)) /\ (((get a2 i) = (get a1 j)) /\
-  forall (k:Z), ((~ (k = i)) /\ ~ (k = j)) -> ((get a1 k) = (get a2 k))).
-Implicit Arguments exchange.
-
-Axiom exchange_set : forall (a:Type), forall (a1:(map Z a)), forall (i:Z)
-  (j:Z), (exchange a1 (set (set a1 i (get a1 j)) j (get a1 i)) i j).
-
-Inductive permut_sub{a:Type}  : (map Z a) -> (map Z a) -> Z -> Z -> Prop :=
-  | permut_refl : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
-      (map_eq_sub a1 a2 l u) -> (permut_sub a1 a2 l u)
-  | permut_sym : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
-      (permut_sub a1 a2 l u) -> (permut_sub a2 a1 l u)
-  | permut_trans : forall (a1:(map Z a)) (a2:(map Z a)) (a3:(map Z a)),
-      forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> ((permut_sub a2 a3 l
-      u) -> (permut_sub a1 a3 l u))
-  | permut_exchange : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z)
-      (u:Z) (i:Z) (j:Z), ((l <= i)%Z /\ (i <  u)%Z) -> (((l <= j)%Z /\
-      (j <  u)%Z) -> ((exchange a1 a2 i j) -> (permut_sub a1 a2 l u))).
-Implicit Arguments permut_sub.
-
-Axiom permut_weakening : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z
-  a)), forall (l1:Z) (r1:Z) (l2:Z) (r2:Z), (((l1 <= l2)%Z /\ (l2 <= r2)%Z) /\
-  (r2 <= r1)%Z) -> ((permut_sub a1 a2 l2 r2) -> (permut_sub a1 a2 l1 r1)).
-
-Axiom permut_eq : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z a)),
-  forall (l:Z) (u:Z), (l <= u)%Z -> ((permut_sub a1 a2 l u) -> forall (i:Z),
-  ((i <  l)%Z \/ (u <= i)%Z) -> ((get a2 i) = (get a1 i))).
-
-Axiom permut_exists : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z a)),
-  forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> forall (i:Z), ((l <= i)%Z /\
-  (i <  u)%Z) -> exists j:Z, ((l <= j)%Z /\ (j <  u)%Z) /\ ((get a2
-  i) = (get a1 j)).
-
-Definition exchange1 (a:Type)(a1:(array a)) (a2:(array a)) (i:Z)
-  (j:Z): Prop := (exchange (elts a1) (elts a2) i j).
-Implicit Arguments exchange1.
-
-Definition permut_sub1 (a:Type)(a1:(array a)) (a2:(array a)) (l:Z)
-  (u:Z): Prop := (permut_sub (elts a1) (elts a2) l u).
-Implicit Arguments permut_sub1.
-
-Definition permut (a:Type)(a1:(array a)) (a2:(array a)): Prop :=
-  ((length a1) = (length a2)) /\ (permut_sub (elts a1) (elts a2) 0%Z
-  (length a1)).
-Implicit Arguments permut.
-
-Axiom exchange_permut : forall (a:Type), forall (a1:(array a)) (a2:(array a))
-  (i:Z) (j:Z), (exchange1 a1 a2 i j) -> (((length a1) = (length a2)) ->
-  (((0%Z <= i)%Z /\ (i <  (length a1))%Z) -> (((0%Z <= j)%Z /\
-  (j <  (length a1))%Z) -> (permut a1 a2)))).
-
-Definition array_eq_sub (a:Type)(a1:(array a)) (a2:(array a)) (l:Z)
-  (u:Z): Prop := (map_eq_sub (elts a1) (elts a2) l u).
-Implicit Arguments array_eq_sub.
-
-Definition array_eq (a:Type)(a1:(array a)) (a2:(array a)): Prop :=
-  ((length a1) = (length a2)) /\ (array_eq_sub a1 a2 0%Z (length a1)).
-Implicit Arguments array_eq.
-
-Axiom array_eq_sub_permut : forall (a:Type), forall (a1:(array a)) (a2:(array
-  a)) (l:Z) (u:Z), (array_eq_sub a1 a2 l u) -> (permut_sub1 a1 a2 l u).
-
-Axiom array_eq_permut : forall (a:Type), forall (a1:(array a)) (a2:(array
-  a)), (array_eq a1 a2) -> (permut a1 a2).
-
-Parameter usN:  Z.
-
-
-Parameter f:  Z.
-
-
-Axiom f_N_range : (1%Z <= (f ))%Z /\ ((f ) <= (usN ))%Z.
-
-Definition found(a:(array Z)): Prop := forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\
-  (p <= (f ))%Z) /\ ((f ) <= q)%Z) /\ (q <= (usN ))%Z) -> (((get1 a
-  p) <= (get1 a (f )))%Z /\ ((get1 a (f )) <= (get1 a q))%Z).
-
-Definition m_invariant(m:Z) (a:(array Z)): Prop := (m <= (f ))%Z /\
-  forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\ (p <  m)%Z) /\ (m <= q)%Z) /\
-  (q <= (usN ))%Z) -> ((get1 a p) <= (get1 a q))%Z.
-
-Definition n_invariant(n:Z) (a:(array Z)): Prop := ((f ) <= n)%Z /\
-  forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\ (p <= n)%Z) /\ (n <  q)%Z) /\
-  (q <= (usN ))%Z) -> ((get1 a p) <= (get1 a q))%Z.
-
-Definition i_invariant(m:Z) (n:Z) (i:Z) (r:Z) (a:(array Z)): Prop :=
-  (m <= i)%Z /\ ((forall (p:Z), ((1%Z <= p)%Z /\ (p <  i)%Z) -> ((get1 a
-  p) <= r)%Z) /\ ((i <= n)%Z -> exists p:Z, ((i <= p)%Z /\ (p <= n)%Z) /\
-  (r <= (get1 a p))%Z)).
-
-Definition j_invariant(m:Z) (n:Z) (j:Z) (r:Z) (a:(array Z)): Prop :=
-  (j <= n)%Z /\ ((forall (q:Z), ((j <  q)%Z /\ (q <= (usN ))%Z) ->
-  (r <= (get1 a q))%Z) /\ ((m <= j)%Z -> exists q:Z, ((m <= q)%Z /\
-  (q <= j)%Z) /\ ((get1 a q) <= r)%Z)).
-
-Definition termination(i:Z) (j:Z) (i0:Z) (j0:Z) (r:Z) (a:(array Z)): Prop :=
-  ((i0 <  i)%Z /\ (j <  j0)%Z) \/ (((i <= (f ))%Z /\ ((f ) <= j)%Z) /\
-  ((get1 a (f )) = r)).
-
-Theorem WP_parameter_find : forall (a:Z), forall (a1:(map Z Z)), let a2 :=
-  (mk_array a a1) in ((a = ((usN ) + 1%Z)%Z) -> forall (n:Z), forall (m:Z),
-  forall (a3:(map Z Z)), let a4 := (mk_array a a3) in ((((m <= (f ))%Z /\
-  forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\ (p <  m)%Z) /\ (m <= q)%Z) /\
-  (q <= (usN ))%Z) -> ((get1 a4 p) <= (get1 a4 q))%Z) /\ ((((f ) <= n)%Z /\
-  forall (p:Z) (q:Z), ((((1%Z <= p)%Z /\ (p <= n)%Z) /\ (n <  q)%Z) /\
-  (q <= (usN ))%Z) -> ((get1 a4 p) <= (get1 a4 q))%Z) /\
-  ((((length a4) = (length a2)) /\ (permut_sub (elts a4) (elts a2) 0%Z
-  (length a4))) /\ (((1%Z <  m)%Z \/ (1%Z = m)) /\ ((n <  (usN ))%Z \/
-  (n = (usN ))))))) -> ((m <  n)%Z -> ((((0%Z <  (f ))%Z \/ (0%Z = (f ))) /\
-  ((f ) <  a)%Z) -> let result := (get a3 (f )) in forall (j:Z),
-  forall (i:Z), forall (a5:(map Z Z)), let a6 := (mk_array a a5) in
-  ((((m <= i)%Z /\ ((forall (p:Z), ((1%Z <= p)%Z /\ (p <  i)%Z) -> ((get1 a6
-  p) <= result)%Z) /\ ((i <= n)%Z -> exists p:Z, ((i <= p)%Z /\
-  (p <= n)%Z) /\ (result <= (get1 a6 p))%Z))) /\ (((j <= n)%Z /\
-  ((forall (q:Z), ((j <  q)%Z /\ (q <= (usN ))%Z) -> (result <= (get1 a6
-  q))%Z) /\ ((m <= j)%Z -> exists q:Z, ((m <= q)%Z /\ (q <= j)%Z) /\
-  ((get1 a6 q) <= result)%Z))) /\ (((m <= (f ))%Z /\ forall (p:Z) (q:Z),
-  ((((1%Z <= p)%Z /\ (p <  m)%Z) /\ (m <= q)%Z) /\ (q <= (usN ))%Z) ->
-  ((get1 a6 p) <= (get1 a6 q))%Z) /\ ((((f ) <= n)%Z /\ forall (p:Z) (q:Z),
-  ((((1%Z <= p)%Z /\ (p <= n)%Z) /\ (n <  q)%Z) /\ (q <= (usN ))%Z) ->
-  ((get1 a6 p) <= (get1 a6 q))%Z) /\ (((0%Z <  j)%Z \/ (0%Z = j)) /\
-  (((i <  ((usN ) + 1%Z)%Z)%Z \/ (i = ((usN ) + 1%Z)%Z)) /\ ((((m <  i)%Z /\
-  (j <  n)%Z) \/ (((i <= (f ))%Z /\ ((f ) <= j)%Z) /\ ((get1 a6
-  (f )) = result))) /\ (((length a6) = (length a2)) /\ (permut_sub (elts a6)
-  (elts a2) 0%Z (length a6)))))))))) -> (((i <  j)%Z \/ (i = j)) ->
-  forall (i1:Z), (((m <= i1)%Z /\ ((forall (p:Z), ((1%Z <= p)%Z /\
-  (p <  i1)%Z) -> ((get1 a6 p) <= result)%Z) /\ ((i1 <= n)%Z -> exists p:Z,
-  ((i1 <= p)%Z /\ (p <= n)%Z) /\ (result <= (get1 a6 p))%Z))) /\
-  ((((i <  i1)%Z \/ (i = i1)) /\ ((i1 <  n)%Z \/ (i1 = n))) /\
-  (((m <  i1)%Z /\ (j <  n)%Z) \/ (((i1 <= (f ))%Z /\ ((f ) <= j)%Z) /\
-  ((get1 a6 (f )) = result))))) -> ((((0%Z <  i1)%Z \/ (0%Z = i1)) /\
-  (i1 <  a)%Z) -> ((~ ((get a5 i1) <  result)%Z) -> forall (j1:Z),
-  (((j1 <= n)%Z /\ ((forall (q:Z), ((j1 <  q)%Z /\ (q <= (usN ))%Z) ->
-  (result <= (get1 a6 q))%Z) /\ ((m <= j1)%Z -> exists q:Z, ((m <= q)%Z /\
-  (q <= j1)%Z) /\ ((get1 a6 q) <= result)%Z))) /\ (((j1 <  j)%Z \/
-  (j1 = j)) /\ (((m <  j1)%Z \/ (m = j1)) /\ (((m <  i1)%Z /\ (j1 <  n)%Z) \/
-  (((i1 <= (f ))%Z /\ ((f ) <= j1)%Z) /\ ((get1 a6 (f )) = result)))))) ->
-  ((((0%Z <  j1)%Z \/ (0%Z = j1)) /\ (j1 <  a)%Z) -> ((~ (result <  (get a5
-  j1))%Z) -> (((((get a5 j1) <  result)%Z \/ ((get a5 j1) = result)) /\
-  ((result <  (get a5 i1))%Z \/ (result = (get a5 i1)))) -> (((i1 <  j1)%Z \/
-  (i1 = j1)) -> ((((0%Z <  i1)%Z \/ (0%Z = i1)) /\ (i1 <  a)%Z) ->
-  ((((0%Z <  j1)%Z \/ (0%Z = j1)) /\ (j1 <  a)%Z) -> ((((0%Z <  i1)%Z \/
-  (0%Z = i1)) /\ (i1 <  a)%Z) -> forall (a7:(map Z Z)), (a7 = (set a5 i1
-  (get a5 j1))) -> ((((0%Z <  j1)%Z \/ (0%Z = j1)) /\ (j1 <  a)%Z) ->
-  forall (a8:(map Z Z)), (a8 = (set a7 j1 (get a5 i1))) -> ((((get a8
-  i1) = (get a5 j1)) /\ (((get a5 i1) = (get a8 j1)) /\ forall (k:Z),
-  ((~ (k = i1)) /\ ~ (k = j1)) -> ((get a8 k) = (get a5 k)))) -> ((((get a8
-  i1) <  result)%Z \/ ((get a8 i1) = result)) -> (((result <  (get a8
-  j1))%Z \/ (result = (get a8 j1))) -> forall (i2:Z), (i2 = (i1 + 1%Z)%Z) ->
-  forall (j2:Z), (j2 = (j1 + (-1%Z)%Z)%Z) -> forall (p:Z), ((1%Z <= p)%Z /\
-  (p <  i2)%Z) -> ((get1 (mk_array a a8) p) <= result)%Z))))))))))))))))))).
-(* YOU MAY EDIT THE PROOF BELOW *)
-unfold get1; simpl; intros.
-destruct H19.
-rewrite H19 in *; clear H19.
-destruct H25.
-rewrite <- H19 in *; clear H19.
-
-
-Qed.
-(* DO NOT EDIT BELOW *)
-
-

examples/programs/find/find_WP_FIND_WP_parameter_find_3.v → examples/programs/find/find_WP_FIND_WP_parameter_find_4.v

@@ -6,11 +6,11 @@ Definition unit  := unit.
 
 Parameter mark : Type.
 
-Parameter at1: forall (a:Type), a -> mark  -> a.
+Parameter at1: forall (a:Type), a -> mark -> a.
 
 Implicit Arguments at1.
 
-Parameter old: forall (a:Type), a  -> a.
+Parameter old: forall (a:Type), a -> a.
 
 Implicit Arguments old.
 
@@ -26,11 +26,11 @@ Implicit Arguments contents.
 
 Parameter map : forall (a:Type) (b:Type), Type.
 
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a  -> b.
+Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
 
 Implicit Arguments get.
 
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b  -> (map a b).
+Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
 
 Implicit Arguments set.
 
@@ -42,7 +42,7 @@ Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)),
   forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1)
   a2) = (get m a2)).
 
-Parameter const: forall (b:Type) (a:Type), b  -> (map a b).
+Parameter const: forall (b:Type) (a:Type), b -> (map a b).
 
 Set Contextual Implicit.
 Implicit Arguments const.
@@ -107,8 +107,8 @@ Axiom permut_weakening : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z
   (r2 <= r1)%Z) -> ((permut_sub a1 a2 l2 r2) -> (permut_sub a1 a2 l1 r1)).
 
 Axiom permut_eq : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z a)),
-  forall (l:Z) (u:Z), (l <= u)%Z -> ((permut_sub a1 a2 l u) -> forall (i:Z),
-  ((i <  l)%Z \/ (u <= i)%Z) -> ((get a2 i) = (get a1 i))).