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(* This file was originally generated by why.
It can be modified; only the generated parts will be overwritten. *)
Require Import Why.
Require Export Match.
(*Why*) Parameter OUTPUT : forall (j: Z), unit.
(* Why obligation from file "brute_force.c", characters 369-373 *)
Lemma BF_po_1 :
forall (m: Z),
forall (n: Z),
forall (x: (array Z)),
forall (y: (array Z)),
forall (Pre10: (array_length x) = m /\ (array_length y) = n /\ 0 <= n /\
0 <= m),
forall (j1: Z),
forall (Post1: j1 = 0),
forall (Variant1: Z),
forall (j2: Z),
forall (Pre9: Variant1 = (n - m + 1 - j2)),
forall (Pre8: 0 <= j2),
forall (Test8: j2 <= (n - m)),
forall (i2: Z),
forall (Post2: i2 = 0),
forall (Variant3: Z),
forall (i3: Z),
forall (Pre6: Variant3 = (m - i3)),
forall (Pre5: (0 <= i3 /\ i3 <= m) /\ (match_ x 0 y j2 i3)),
forall (Test5: true = true),
forall (Test4: i3 < m),
0 <= i3 /\ i3 < (array_length x).
Proof.
auto with *.
Qed.
(* Why obligation from file "brute_force.c", characters 377-385 *)
Lemma BF_po_2 :
forall (m: Z),
forall (n: Z),
forall (x: (array Z)),
forall (y: (array Z)),
forall (Pre10: (array_length x) = m /\ (array_length y) = n /\ 0 <= n /\
0 <= m),
forall (j1: Z),
forall (Post1: j1 = 0),
forall (Variant1: Z),
forall (j2: Z),
forall (Pre9: Variant1 = (n - m + 1 - j2)),
forall (Pre8: 0 <= j2),
forall (Test8: j2 <= (n - m)),
forall (i2: Z),
forall (Post2: i2 = 0),
forall (Variant3: Z),
forall (i3: Z),
forall (Pre6: Variant3 = (m - i3)),
forall (Pre5: (0 <= i3 /\ i3 <= m) /\ (match_ x 0 y j2 i3)),
forall (Test5: true = true),
forall (Test4: i3 < m),
forall (Pre4: 0 <= i3 /\ i3 < (array_length x)),
forall (c_aux_1: Z),
forall (Post4: c_aux_1 = (access x i3)),
0 <= (i3 + j2) /\ (i3 + j2) < (array_length y).
Proof.
auto with *.
Qed.
(* Why obligation from file "brute_force.c", characters 369-385 *)
Lemma BF_po_3 :
forall (m: Z),
forall (n: Z),
forall (x: (array Z)),
forall (y: (array Z)),
forall (Pre10: (array_length x) = m /\ (array_length y) = n /\ 0 <= n /\
0 <= m),
forall (j1: Z),
forall (Post1: j1 = 0),
forall (Variant1: Z),
forall (j2: Z),
forall (Pre9: Variant1 = (n - m + 1 - j2)),
forall (Pre8: 0 <= j2),
forall (Test8: j2 <= (n - m)),
forall (i2: Z),
forall (Post2: i2 = 0),
forall (Variant3: Z),
forall (i3: Z),
forall (Pre6: Variant3 = (m - i3)),
forall (Pre5: (0 <= i3 /\ i3 <= m) /\ (match_ x 0 y j2 i3)),
forall (Test5: true = true),
forall (Test4: i3 < m),
forall (Pre4: 0 <= i3 /\ i3 < (array_length x)),
forall (c_aux_1: Z),
forall (Post4: c_aux_1 = (access x i3)),
forall (Pre3: 0 <= (i3 + j2) /\ (i3 + j2) < (array_length y)),
forall (c_aux_2: Z),
forall (Post3: c_aux_2 = (access y (i3 + j2))),
forall (result4: bool),
forall (Post25: (if result4 then c_aux_1 = c_aux_2 else c_aux_1 <> c_aux_2)),
(if result4
then (forall (i:Z),
(i = (i3 + 1) -> ((0 <= i /\ i <= m) /\ (match_ x 0 y j2 i)) /\
(Zwf 0 (m - i) (m - i3))))
else ((i3 >= m ->
(forall (j:Z),
(j = (j2 + 1) -> 0 <= j /\
(Zwf 0 (n - m + 1 - j) (n - m + 1 - j2)))) /\
(match_ x 0 y j2 (array_length x)))) /\
((i3 < m ->
(forall (j:Z),
(j = (j2 + 1) -> 0 <= j /\ (Zwf 0 (n - m + 1 - j) (n - m + 1 - j2))))))).
Proof.
simple_destruct result4; intuition.
subst i.
apply match_right_extension; auto with *.
subst c_aux_1 c_aux_2; ring (0 + i3)%Z; ring (j2 + i3)%Z; assumption.
assert (i3 = array_length x).
omega.
subst i3; assumption.
Qed.
(* Why obligation from file "brute_force.c", characters 360-385 *)
Lemma BF_po_4 :
forall (m: Z),
forall (n: Z),
forall (x: (array Z)),
forall (y: (array Z)),
forall (Pre10: (array_length x) = m /\ (array_length y) = n /\ 0 <= n /\
0 <= m),
forall (j1: Z),
forall (Post1: j1 = 0),
forall (Variant1: Z),
forall (j2: Z),
forall (Pre9: Variant1 = (n - m + 1 - j2)),
forall (Pre8: 0 <= j2),
forall (Test8: j2 <= (n - m)),
forall (i2: Z),
forall (Post2: i2 = 0),
forall (Variant3: Z),
forall (i3: Z),
forall (Pre6: Variant3 = (m - i3)),
forall (Pre5: (0 <= i3 /\ i3 <= m) /\ (match_ x 0 y j2 i3)),
forall (Test5: true = true),
forall (Test3: i3 >= m),
((i3 >= m ->
(forall (j:Z),
(j = (j2 + 1) -> 0 <= j /\ (Zwf 0 (n - m + 1 - j) (n - m + 1 - j2)))) /\
(match_ x 0 y j2 (array_length x)))) /\
((i3 < m ->
(forall (j:Z),
(j = (j2 + 1) -> 0 <= j /\ (Zwf 0 (n - m + 1 - j) (n - m + 1 - j2)))))).
Proof.
intuition.
assert (i3 = array_length x).
omega.
subst i3; assumption.
Qed.
(* Why obligation from file "brute_force.c", characters 412-445 *)
Lemma BF_po_5 :
forall (m: Z),
forall (n: Z),
forall (x: (array Z)),
forall (y: (array Z)),
forall (Pre10: (array_length x) = m /\ (array_length y) = n /\ 0 <= n /\
0 <= m),
forall (j1: Z),
forall (Post1: j1 = 0),
forall (Variant1: Z),
forall (j2: Z),
forall (Pre9: Variant1 = (n - m + 1 - j2)),
forall (Pre8: 0 <= j2),
forall (Test8: j2 <= (n - m)),
forall (i2: Z),
forall (Post2: i2 = 0),
(0 <= i2 /\ i2 <= m) /\ (match_ x 0 y j2 i2).
Proof.
intuition.
subst i2; apply match_empty; auto with *.
Qed.
(* Why obligation from file "brute_force.c", characters 308-314 *)
Lemma BF_po_6 :
forall (m: Z),
forall (n: Z),
forall (x: (array Z)),
forall (y: (array Z)),
forall (Pre10: (array_length x) = m /\ (array_length y) = n /\ 0 <= n /\
0 <= m),
forall (j1: Z),
forall (Post1: j1 = 0),
0 <= j1.
Proof.
intuition.
Qed.
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