Commit 5c8cba5a authored by Jean-Christophe Filliâtre's avatar Jean-Christophe Filliâtre
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proof session updated

parent 3b938adc
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import ZArith.
Require Import Rbase.
Require int.Int.
(* Why3 assumption *)
Definition unit := unit.
Parameter qtmark : Type.
Parameter at1: forall (a:Type), a -> qtmark -> a.
Implicit Arguments at1.
Parameter old: forall (a:Type), a -> a.
Implicit Arguments old.
(* Why3 assumption *)
Definition implb(x:bool) (y:bool): bool := match (x,
y) with
| (true, false) => false
| (_, _) => true
end.
Parameter set : forall (a:Type), Type.
Parameter mem: forall (a:Type), a -> (set a) -> Prop.
Implicit Arguments mem.
(* Why3 assumption *)
Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
forall (x:a), (mem x s1) <-> (mem x s2).
Implicit Arguments infix_eqeq.
Axiom extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
(infix_eqeq s1 s2) -> (s1 = s2).
(* Why3 assumption *)
Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
(mem x s1) -> (mem x s2).
Implicit Arguments subset.
Axiom subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
(s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
Parameter empty: forall (a:Type), (set a).
Set Contextual Implicit.
Implicit Arguments empty.
Unset Contextual Implicit.
(* Why3 assumption *)
Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
Implicit Arguments is_empty.
Axiom empty_def1 : forall (a:Type), (is_empty (empty :(set a))).
Parameter add: forall (a:Type), a -> (set a) -> (set a).
Implicit Arguments add.
Axiom add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
(mem x (add y s)) <-> ((x = y) \/ (mem x s)).
Parameter remove: forall (a:Type), a -> (set a) -> (set a).
Implicit Arguments remove.
Axiom remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
(remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
Axiom subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
(subset (remove x s) s).
Parameter union: forall (a:Type), (set a) -> (set a) -> (set a).
Implicit Arguments union.
Axiom union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
(mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
Parameter inter: forall (a:Type), (set a) -> (set a) -> (set a).
Implicit Arguments inter.
Axiom inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
(mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
Parameter diff: forall (a:Type), (set a) -> (set a) -> (set a).
Implicit Arguments diff.
Axiom diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
(mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
Axiom subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
(subset (diff s1 s2) s1).
Parameter choose: forall (a:Type), (set a) -> a.
Implicit Arguments choose.
Axiom choose_def : forall (a:Type), forall (s:(set a)), (~ (is_empty s)) ->
(mem (choose s) s).
Parameter all: forall (a:Type), (set a).
Set Contextual Implicit.
Implicit Arguments all.
Unset Contextual Implicit.
Axiom all_def : forall (a:Type), forall (x:a), (mem x (all :(set a))).
Parameter cardinal: forall (a:Type), (set a) -> Z.
Implicit Arguments cardinal.
Axiom cardinal_nonneg : forall (a:Type), forall (s:(set a)),
(0%Z <= (cardinal s))%Z.
Axiom cardinal_empty : forall (a:Type), forall (s:(set a)),
((cardinal s) = 0%Z) <-> (is_empty s).
Axiom cardinal_add : forall (a:Type), forall (x:a), forall (s:(set a)),
(~ (mem x s)) -> ((cardinal (add x s)) = (1%Z + (cardinal s))%Z).
Axiom cardinal_remove : forall (a:Type), forall (x:a), forall (s:(set a)),
(mem x s) -> ((cardinal s) = (1%Z + (cardinal (remove x s)))%Z).
Axiom cardinal_subset : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
(subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.
Axiom cardinal1 : forall (a:Type), forall (s:(set a)),
((cardinal s) = 1%Z) -> forall (x:a), (mem x s) -> (x = (choose s)).
Parameter nth: forall (a:Type), Z -> (set a) -> a.
Implicit Arguments nth.
Axiom nth_injective : forall (a:Type), forall (s:(set a)) (i:Z) (j:Z),
((0%Z <= i)%Z /\ (i < (cardinal s))%Z) -> (((0%Z <= j)%Z /\
(j < (cardinal s))%Z) -> (((nth i s) = (nth j s)) -> (i = j))).
Axiom nth_surjective : forall (a:Type), forall (s:(set a)) (x:a), (mem x
s) -> exists i:Z, ((0%Z <= i)%Z /\ (i < (cardinal s))%Z) -> (x = (nth i
s)).
Parameter min_elt: (set Z) -> Z.
Axiom min_elt_def1 : forall (s:(set Z)), (~ (is_empty s)) -> (mem (min_elt s)
s).
Axiom min_elt_def2 : forall (s:(set Z)), (~ (is_empty s)) -> forall (x:Z),
(mem x s) -> ((min_elt s) <= x)%Z.
Parameter max_elt: (set Z) -> Z.
Axiom max_elt_def1 : forall (s:(set Z)), (~ (is_empty s)) -> (mem (max_elt s)
s).
Axiom max_elt_def2 : forall (s:(set Z)), (~ (is_empty s)) -> forall (x:Z),
(mem x s) -> (x <= (max_elt s))%Z.
Parameter below: Z -> (set Z).
Axiom below_def : forall (x:Z) (n:Z), (mem x (below n)) <-> ((0%Z <= x)%Z /\
(x < n)%Z).
Axiom cardinal_below : forall (n:Z), ((0%Z <= n)%Z ->
((cardinal (below n)) = n)) /\ ((~ (0%Z <= n)%Z) ->
((cardinal (below n)) = 0%Z)).
Parameter succ: (set Z) -> (set Z).
Axiom succ_def : forall (s:(set Z)) (i:Z), (mem i (succ s)) <->
((1%Z <= i)%Z /\ (mem (i - 1%Z)%Z s)).
Parameter pred: (set Z) -> (set Z).
Axiom pred_def : forall (s:(set Z)) (i:Z), (mem i (pred s)) <->
((0%Z <= i)%Z /\ (mem (i + 1%Z)%Z s)).
(* Why3 assumption *)
Inductive ref (a:Type) :=
| mk_ref : a -> ref a.
Implicit Arguments mk_ref.
(* Why3 assumption *)
Definition contents (a:Type)(v:(ref a)): a :=
match v with
| (mk_ref x) => x
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 set1: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
Implicit Arguments set1.
Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set1 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 (set1 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).
Parameter n: Z.
(* Why3 assumption *)
Definition solution := (map Z Z).
(* Why3 assumption *)
Definition eq_prefix (a:Type)(t:(map Z a)) (u:(map Z a)) (i:Z): Prop :=
forall (k:Z), ((0%Z <= k)%Z /\ (k < i)%Z) -> ((get t k) = (get u k)).
Implicit Arguments eq_prefix.
(* Why3 assumption *)
Definition partial_solution(k:Z) (s:(map Z Z)): Prop := forall (i:Z),
((0%Z <= i)%Z /\ (i < k)%Z) -> (((0%Z <= (get s i))%Z /\ ((get s
i) < n)%Z) /\ forall (j:Z), ((0%Z <= j)%Z /\ (j < i)%Z) -> ((~ ((get s
i) = (get s j))) /\ ((~ (((get s i) - (get s j))%Z = (i - j)%Z)) /\
~ (((get s i) - (get s j))%Z = (j - i)%Z)))).
Axiom partial_solution_eq_prefix : forall (u:(map Z Z)) (t:(map Z Z)) (k:Z),
(partial_solution k t) -> ((eq_prefix t u k) -> (partial_solution k u)).
(* Why3 assumption *)
Definition lt_sol(s1:(map Z Z)) (s2:(map Z Z)): Prop := exists i:Z,
((0%Z <= i)%Z /\ (i < n)%Z) /\ ((eq_prefix s1 s2 i) /\ ((get s1
i) < (get s2 i))%Z).
(* Why3 assumption *)
Definition solutions := (map Z (map Z Z)).
(* Why3 assumption *)
Definition sorted(s:(map Z (map Z Z))) (a:Z) (b:Z): Prop := forall (i:Z)
(j:Z), (((a <= i)%Z /\ (i < j)%Z) /\ (j < b)%Z) -> (lt_sol (get s i)
(get s j)).
Axiom no_duplicate : forall (s:(map Z (map Z Z))) (a:Z) (b:Z), (sorted s a
b) -> forall (i:Z) (j:Z), (((a <= i)%Z /\ (i < j)%Z) /\ (j < b)%Z) ->
~ (eq_prefix (get s i) (get s j) n).
Parameter col: (ref (map Z Z)).
Parameter k: (ref Z).
Parameter sol: (ref (map Z (map Z Z))).
Parameter s: (ref Z).
Require Import Why3. Ltac ae := why3 "alt-ergo".
(* Why3 goal *)
Theorem WP_parameter_t3 : forall (a:(set Z)), forall (b:(set Z)),
forall (c:(set Z)), forall (s1:Z), forall (sol1:(map Z (map Z Z))),
forall (k1:Z), forall (col1:(map Z Z)), (((0%Z < k1)%Z \/ (0%Z = k1)) /\
(((k1 + (cardinal a))%Z = n) /\ ((0%Z <= s1)%Z /\ ((forall (i:Z), (mem i
a) <-> ((((0%Z < i)%Z \/ (0%Z = i)) /\ (i < n)%Z) /\ forall (j:Z),
(((0%Z < j)%Z \/ (0%Z = j)) /\ (j < k1)%Z) -> ~ ((get col1 j) = i))) /\
((forall (i:Z), (0%Z <= i)%Z -> ((~ (mem i b)) <-> forall (j:Z),
(((0%Z < j)%Z \/ (0%Z = j)) /\ (j < k1)%Z) -> ~ ((get col1
j) = ((i + j)%Z + (-k1)%Z)%Z))) /\ ((forall (i:Z), (0%Z <= i)%Z ->
((~ (mem i c)) <-> forall (j:Z), (((0%Z < j)%Z \/ (0%Z = j)) /\
(j < k1)%Z) -> ~ ((get col1 j) = ((i + k1)%Z + (-j)%Z)%Z))) /\
forall (i:Z), ((0%Z <= i)%Z /\ (i < k1)%Z) -> (((0%Z <= (get col1 i))%Z /\
((get col1 i) < n)%Z) /\ forall (j:Z), ((0%Z <= j)%Z /\ (j < i)%Z) ->
((~ ((get col1 i) = (get col1 j))) /\ ((~ (((get col1 i) - (get col1
j))%Z = (i - j)%Z)) /\ ~ (((get col1 i) - (get col1
j))%Z = (j - i)%Z)))))))))) -> ((~ forall (x:Z), ~ (mem x a)) ->
forall (f:Z), forall (e:(set Z)), forall (s2:Z), forall (sol2:(map Z (map Z
Z))), forall (k2:Z), forall (col2:(map Z Z)), (((f = (s2 + (-s1)%Z)%Z) /\
(0%Z <= (s2 - s1)%Z)%Z) /\ ((k2 = k1) /\ ((forall (x:Z), (mem x e) ->
(mem x (diff (diff a b) c))) /\ ((forall (i:Z), ((0%Z <= i)%Z /\
(i < k2)%Z) -> (((0%Z <= (get col2 i))%Z /\ ((get col2 i) < n)%Z) /\
forall (j:Z), ((0%Z <= j)%Z /\ (j < i)%Z) -> ((~ ((get col2 i) = (get col2
j))) /\ ((~ (((get col2 i) - (get col2 j))%Z = (i - j)%Z)) /\ ~ (((get col2
i) - (get col2 j))%Z = (j - i)%Z))))) /\ ((forall (i:Z) (j:Z),
(((s1 <= i)%Z /\ (i < j)%Z) /\ (j < s2)%Z) -> (lt_sol (get sol2 i)
(get sol2 j))) /\ ((forall (i:Z) (j:Z), (mem i (diff (diff (diff a b) c)
e)) -> ((mem j e) -> (i < j)%Z)) /\ ((forall (t:(map Z Z)),
((partial_solution n t) /\ ((forall (k3:Z), ((0%Z <= k3)%Z /\
(k3 < k2)%Z) -> ((get col2 k3) = (get t k3))) /\ (mem (get t k2)
(diff (diff (diff a b) c) e)))) <-> exists i:Z, (((s1 < i)%Z \/
(s1 = i)) /\ (i < s2)%Z) /\ (eq_prefix t (get sol2 i) n)) /\
((forall (k3:Z), ((0%Z <= k3)%Z /\ (k3 < k2)%Z) -> ((get col1
k3) = (get col2 k3))) /\ forall (k3:Z), ((0%Z <= k3)%Z /\ (k3 < s1)%Z) ->
((get sol1 k3) = (get sol2 k3)))))))))) -> ((~ forall (x:Z), ~ (mem x
e)) -> forall (col3:(map Z Z)), (col3 = (set1 col2 k2 (min_elt e))) ->
forall (k3:Z), (k3 = (k2 + 1%Z)%Z) -> forall (i:Z), ((0%Z <= i)%Z /\
(i < k3)%Z) -> forall (j:Z), ((0%Z <= j)%Z /\ (j < i)%Z) -> ~ ((get col3
i) = (get col3 j)))).
intuition.
assert (case: (i < k2 \/ i = k2)%Z) by omega. destruct case.
ae.
subst.
ae.
Qed.
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import ZArith.
Require Import Rbase.
Require int.Int.
(* Why3 assumption *)
Definition unit := unit.
Parameter qtmark : Type.
Parameter at1: forall (a:Type), a -> qtmark -> a.
Implicit Arguments at1.
Parameter old: forall (a:Type), a -> a.
Implicit Arguments old.
(* Why3 assumption *)
Definition implb(x:bool) (y:bool): bool := match (x,
y) with
| (true, false) => false
| (_, _) => true
end.
Parameter set : forall (a:Type), Type.
Parameter mem: forall (a:Type), a -> (set a) -> Prop.
Implicit Arguments mem.
(* Why3 assumption *)
Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
forall (x:a), (mem x s1) <-> (mem x s2).
Implicit Arguments infix_eqeq.
Axiom extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
(infix_eqeq s1 s2) -> (s1 = s2).
(* Why3 assumption *)
Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
(mem x s1) -> (mem x s2).
Implicit Arguments subset.
Axiom subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
(s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
Parameter empty: forall (a:Type), (set a).
Set Contextual Implicit.
Implicit Arguments empty.
Unset Contextual Implicit.
(* Why3 assumption *)
Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
Implicit Arguments is_empty.
Axiom empty_def1 : forall (a:Type), (is_empty (empty :(set a))).
Parameter add: forall (a:Type), a -> (set a) -> (set a).
Implicit Arguments add.
Axiom add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
(mem x (add y s)) <-> ((x = y) \/ (mem x s)).
Parameter remove: forall (a:Type), a -> (set a) -> (set a).
Implicit Arguments remove.
Axiom remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
(remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
Axiom subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
(subset (remove x s) s).
Parameter union: forall (a:Type), (set a) -> (set a) -> (set a).
Implicit Arguments union.
Axiom union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
(mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
Parameter inter: forall (a:Type), (set a) -> (set a) -> (set a).
Implicit Arguments inter.
Axiom inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
(mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
Parameter diff: forall (a:Type), (set a) -> (set a) -> (set a).
Implicit Arguments diff.
Axiom diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
(mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
Axiom subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
(subset (diff s1 s2) s1).
Parameter choose: forall (a:Type), (set a) -> a.
Implicit Arguments choose.
Axiom choose_def : forall (a:Type), forall (s:(set a)), (~ (is_empty s)) ->
(mem (choose s) s).
Parameter all: forall (a:Type), (set a).
Set Contextual Implicit.
Implicit Arguments all.
Unset Contextual Implicit.
Axiom all_def : forall (a:Type), forall (x:a), (mem x (all :(set a))).
Parameter cardinal: forall (a:Type), (set a) -> Z.
Implicit Arguments cardinal.
Axiom cardinal_nonneg : forall (a:Type), forall (s:(set a)),
(0%Z <= (cardinal s))%Z.
Axiom cardinal_empty : forall (a:Type), forall (s:(set a)),
((cardinal s) = 0%Z) <-> (is_empty s).
Axiom cardinal_add : forall (a:Type), forall (x:a), forall (s:(set a)),
(~ (mem x s)) -> ((cardinal (add x s)) = (1%Z + (cardinal s))%Z).
Axiom cardinal_remove : forall (a:Type), forall (x:a), forall (s:(set a)),
(mem x s) -> ((cardinal s) = (1%Z + (cardinal (remove x s)))%Z).
Axiom cardinal_subset : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
(subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.
Axiom cardinal1 : forall (a:Type), forall (s:(set a)),
((cardinal s) = 1%Z) -> forall (x:a), (mem x s) -> (x = (choose s)).
Parameter nth: forall (a:Type), Z -> (set a) -> a.
Implicit Arguments nth.
Axiom nth_injective : forall (a:Type), forall (s:(set a)) (i:Z) (j:Z),
((0%Z <= i)%Z /\ (i < (cardinal s))%Z) -> (((0%Z <= j)%Z /\
(j < (cardinal s))%Z) -> (((nth i s) = (nth j s)) -> (i = j))).
Axiom nth_surjective : forall (a:Type), forall (s:(set a)) (x:a), (mem x
s) -> exists i:Z, ((0%Z <= i)%Z /\ (i < (cardinal s))%Z) -> (x = (nth i
s)).
Parameter min_elt: (set Z) -> Z.
Axiom min_elt_def1 : forall (s:(set Z)), (~ (is_empty s)) -> (mem (min_elt s)
s).
Axiom min_elt_def2 : forall (s:(set Z)), (~ (is_empty s)) -> forall (x:Z),
(mem x s) -> ((min_elt s) <= x)%Z.
Parameter max_elt: (set Z) -> Z.
Axiom max_elt_def1 : forall (s:(set Z)), (~ (is_empty s)) -> (mem (max_elt s)
s).
Axiom max_elt_def2 : forall (s:(set Z)), (~ (is_empty s)) -> forall (x:Z),
(mem x s) -> (x <= (max_elt s))%Z.
Parameter below: Z -> (set Z).
Axiom below_def : forall (x:Z) (n:Z), (mem x (below n)) <-> ((0%Z <= x)%Z /\
(x < n)%Z).
Axiom cardinal_below : forall (n:Z), ((0%Z <= n)%Z ->
((cardinal (below n)) = n)) /\ ((~ (0%Z <= n)%Z) ->
((cardinal (below n)) = 0%Z)).
Parameter succ: (set Z) -> (set Z).
Axiom succ_def : forall (s:(set Z)) (i:Z), (mem i (succ s)) <->
((1%Z <= i)%Z /\ (mem (i - 1%Z)%Z s)).
Parameter pred: (set Z) -> (set Z).
Axiom pred_def : forall (s:(set Z)) (i:Z), (mem i (pred s)) <->
((0%Z <= i)%Z /\ (mem (i + 1%Z)%Z s)).
(* Why3 assumption *)
Inductive ref (a:Type) :=
| mk_ref : a -> ref a.
Implicit Arguments mk_ref.
(* Why3 assumption *)
Definition contents (a:Type)(v:(ref a)): a :=
match v with
| (mk_ref x) => x
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 set1: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
Implicit Arguments set1.
Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set1 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 (set1 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).
Parameter n: Z.
(* Why3 assumption *)
Definition solution := (map Z Z).
(* Why3 assumption *)
Definition eq_prefix (a:Type)(t:(map Z a)) (u:(map Z a)) (i:Z): Prop :=
forall (k:Z), ((0%Z <= k)%Z /\ (k < i)%Z) -> ((get t k) = (get u k)).
Implicit Arguments eq_prefix.
(* Why3 assumption *)
Definition partial_solution(k:Z) (s:(map Z Z)): Prop := forall (i:Z),
((0%Z <= i)%Z /\ (i < k)%Z) -> (((0%Z <= (get s i))%Z /\ ((get s
i) < n)%Z) /\ forall (j:Z), ((0%Z <= j)%Z /\ (j < i)%Z) -> ((~ ((get s
i) = (get s j))) /\ ((~ (((get s i) - (get s j))%Z = (i - j)%Z)) /\
~ (((get s i) - (get s j))%Z = (j - i)%Z)))).