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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 Import Omega.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_1 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
(0 <= 0 /\ 0 <= ((array_length t) - 1)) /\ (sorted_array t 0 (0 - 1)) /\
(permut t t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < 0 ->
(0 <= l /\ l < (array_length t) -> (access t k) <= (access t l))))).
Proof.
unfold sorted_array; intuition.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_2 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
((i + 1) <= (i + 1) /\ (i + 1) <= (array_length t0)) /\ (i <= i /\ i <
(array_length t0)) /\
(forall (k:Z), (i <= k /\ k < (i + 1) -> (access t0 i) <= (access t0 k))).
Proof.
intuition.
assert (h: k = i).
omega.
subst result k; omega.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_3 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_7: j < result0),
0 <= j /\ j < (array_length t0).
Proof.
intuition.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_4 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_7: j < result0),
forall (HW_8: 0 <= j /\ j < (array_length t0)),
forall (result1: Z),
forall (HW_9: result1 = (access t0 j)),
0 <= min /\ min < (array_length t0).
Proof.
intuition.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_5 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_7: j < result0),
forall (HW_8: 0 <= j /\ j < (array_length t0)),
forall (result1: Z),
forall (HW_9: result1 = (access t0 j)),
forall (HW_10: 0 <= min /\ min < (array_length t0)),
forall (result2: Z),
forall (HW_11: result2 = (access t0 min)),
forall (HW_12: result1 < result2),
forall (min0: Z),
forall (HW_13: min0 = j),
forall (j0: Z),
forall (HW_14: j0 = (j + 1)),
(((i + 1) <= j0 /\ j0 <= (array_length t0)) /\ (i <= min0 /\ min0 <
(array_length t0)) /\
(forall (k:Z), (i <= k /\ k < j0 -> (access t0 min0) <= (access t0 k)))) /\
(Zwf 0 ((array_length t0) - j0) ((array_length t0) - j)).
Proof.
intuition.
assert (h: (k < j) \/ k = j).
omega.
intuition.
apply Zle_trans with (access t0 min).
subst min0; omega.
auto with *.
subst min0 k; omega.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_6 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_7: j < result0),
forall (HW_8: 0 <= j /\ j < (array_length t0)),
forall (result1: Z),
forall (HW_9: result1 = (access t0 j)),
forall (HW_10: 0 <= min /\ min < (array_length t0)),
forall (result2: Z),
forall (HW_11: result2 = (access t0 min)),
forall (HW_15: result1 >= result2),
forall (j0: Z),
forall (HW_16: j0 = (j + 1)),
(((i + 1) <= j0 /\ j0 <= (array_length t0)) /\ (i <= min /\ min <
(array_length t0)) /\
(forall (k:Z), (i <= k /\ k < j0 -> (access t0 min) <= (access t0 k)))) /\
(Zwf 0 ((array_length t0) - j0) ((array_length t0) - j)).
Proof.
intuition.
assert (h: (k < j) \/ k = j).
omega.
intuition.
subst k; omega.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_7 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_17: j >= result0),
0 <= min /\ min < (array_length t0).
Proof.
intuition.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_8 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_17: j >= result0),
forall (HW_18: 0 <= min /\ min < (array_length t0)),
forall (result1: Z),
forall (HW_19: result1 = (access t0 min)),
0 <= i /\ i < (array_length t0).
Proof.
intuition.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_9 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_17: j >= result0),
forall (HW_18: 0 <= min /\ min < (array_length t0)),
forall (result1: Z),
forall (HW_19: result1 = (access t0 min)),
forall (HW_20: 0 <= i /\ i < (array_length t0)),
forall (result2: Z),
forall (HW_21: result2 = (access t0 i)),
forall (HW_22: 0 <= min /\ min < (array_length t0)),
forall (t1: (array Z)),
forall (HW_23: t1 = (update t0 min result2)),
0 <= i /\ i < (array_length t1).
Proof.
intuition.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_10 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_4: i < (result - 1)),
forall (j: Z),
forall (min: Z),
forall (HW_5: ((i + 1) <= j /\ j <= (array_length t0)) /\ (i <= min /\
min < (array_length t0)) /\
(forall (k:Z),
(i <= k /\ k < j -> (access t0 min) <= (access t0 k)))),
forall (result0: Z),
forall (HW_6: result0 = (array_length t0)),
forall (HW_17: j >= result0),
forall (HW_18: 0 <= min /\ min < (array_length t0)),
forall (result1: Z),
forall (HW_19: result1 = (access t0 min)),
forall (HW_20: 0 <= i /\ i < (array_length t0)),
forall (result2: Z),
forall (HW_21: result2 = (access t0 i)),
forall (HW_22: 0 <= min /\ min < (array_length t0)),
forall (t1: (array Z)),
forall (HW_23: t1 = (update t0 min result2)),
forall (HW_24: 0 <= i /\ i < (array_length t1)),
forall (t2: (array Z)),
forall (HW_25: t2 = (update t1 i result1)),
forall (i0: Z),
forall (HW_26: i0 = (i + 1)),
((0 <= i0 /\ i0 <= ((array_length t2) - 1)) /\
(sorted_array t2 0 (i0 - 1)) /\ (permut t2 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i0 ->
(i0 <= l /\ l < (array_length t2) -> (access t2 k) <= (access t2 l)))))) /\
(Zwf 0 ((array_length t2) - i0) ((array_length t0) - i)).
Proof.
intuition.
ArraySubst t1.
Save.
(* Why obligation from file "", line 0, characters 0-0: *)
(*Why goal*) Lemma selection_po_11 :
forall (t: (array Z)),
forall (HW_1: (array_length t) >= 1),
forall (i: Z),
forall (t0: (array Z)),
forall (HW_2: (0 <= i /\ i <= ((array_length t0) - 1)) /\
(sorted_array t0 0 (i - 1)) /\ (permut t0 t) /\
(forall (k:Z),
(forall (l:Z),
(0 <= k /\ k < i ->
(i <= l /\ l < (array_length t0) -> (access t0 k) <=
(access t0 l)))))),
forall (result: Z),
forall (HW_3: result = (array_length t0)),
forall (HW_27: i >= (result - 1)),
(sorted_array t0 0 ((array_length t0) - 1)) /\ (permut t0 t).
Proof.
intuition.
ArraySubst t2.
ArraySubst t1.
assert (h: i = 0%Z \/ (0 < i)).
omega.
intuition.
replace (i0 - 1)%Z with 0%Z.
unfold sorted_array; intros; omega.
omega.
replace (i0 - 1)%Z with (i - 1 + 1)%Z.
apply right_extension.
omega.
ArraySubst t2.
ArraySubst t1.
apply sorted_array_id with t0.
assumption.
unfold array_id; intros.
subst t2; do 2 AccessOther.
replace (i - 1 + 1)%Z with i.
subst t2 t1; do 2 AccessOther.
subst result1.
auto with *.
omega.
omega.
apply permut_trans with t0.
subst t2; subst t1; subst result1 result2.
apply exchange_is_permut with min i; auto with *.
assumption.
assert (h: k = i \/ (k < i)).
omega.
intuition.
subst k.
subst t2.
AccessSame.
AccessOther.
subst t1.
assert (h: l = min \/ min <> l).
omega.
intuition.
subst l; AccessSame.
subst result1 result2; auto with *.
assert (h: l < array_length (update (update t0 min result2) i result1)).
assumption.
do 2 rewrite array_length_update in h.
AccessOther.
subst.
auto with *.
ArraySubst t1.
assert (h: i < array_length (update t0 min result2)). assumption.
rewrite array_length_update in h.
assert (h': l < array_length (update (update t0 min result2) i result1)). assumption.
do 2 rewrite array_length_update in h'.
omega.
subst.
AccessOther.
AccessOther.
AccessOther.
assert (h: l < array_length (update (update t0 min (access t0 i)) i (access t0 min))). assumption.
do 2 rewrite array_length_update in h.
assert (l = min \/ min <> l).
omega.
intuition.
subst l; AccessSame.
auto with *.
AccessOther.
auto with *.
assert (h: l < array_length (update (update t0 min (access t0 i)) i (access t0 min))). assumption.
do 2 rewrite array_length_update in h.
omega.
subst t2 t1; simpl; unfold Zwf; omega.
Save.
Proof.
intuition.
assert (i=0 \/ 0<i). omega. intuition.
unfold sorted_array; intros; omega.
replace (array_length t0 - 1)%Z with (i - 1 + 1)%Z.
apply right_extension; try omega.
assumption.
auto with *.
omega.
Qed.
(*Why*) Parameter selection_valid :
forall (_: unit), forall (t: (array Z)), forall (_: (array_length t) >= 1),
(sig_2 (array Z) unit
(fun (t0: (array Z)) (result: unit) =>
((sorted_array t0 0 ((array_length t0) - 1)) /\ (permut t0 t)))).
|
