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|
(* This file was originally generated by why.
It can be modified; only the generated parts will be overwritten. *)
Require Export binary_search_spec_why.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 14, characters 6-147: *)
(*Why goal*) Lemma binary_search_impl_po_1 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
(* CADUCEUS_4 *) ((((0 <= 0 /\ (n - 1) <= (n - 1)) /\ (Zopp 1) <= (Zopp
1) /\ (Zopp 1) <= (n - 1)) /\
(((Zopp 1) = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> 0 <= k /\ k <= (n - 1))))))) /\
(((Zopp 1) >= 0 -> (acc intM_global (shift t (Zopp 1))) = v))).
Proof.
intuition.
(* FILL PROOF HERE *)
Save.
(* Why obligation from file "why/binary_search.why", line 37, characters 25-48: *)
(*Why goal*) Lemma binary_search_impl_po_2 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
2 <> 0.
Proof.
intuition.
Save.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 21, characters 15-26: *)
(*Why goal*) Lemma binary_search_impl_po_3 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
(* CADUCEUS_7 *) (* CADUCEUS_7 *) (l <= result /\ result <= u).
Proof.
intros.
generalize (mean_1 l u); intuition.
Save.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 22, characters 8-12: *)
(*Why goal*) Lemma binary_search_impl_po_4 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
forall (HW_6: (* CADUCEUS_7 *) (l <= result /\ result <= u)),
forall (result0: (pointer global)),
forall (HW_7: result0 = (shift t result)),
(valid alloc result0).
Proof.
intuition.
subst.
apply valid_range_valid_shift with 0 (n-1); intuition.
Save.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 14, characters 6-147: *)
(*Why goal*) Lemma binary_search_impl_po_5 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
forall (HW_6: (* CADUCEUS_7 *) (l <= result /\ result <= u)),
forall (result0: (pointer global)),
forall (HW_7: result0 = (shift t result)),
forall (HW_8: (valid alloc result0)),
forall (result1: Z),
forall (HW_9: result1 = (acc intM_global result0)),
forall (HW_10: result1 < v),
forall (l0: Z),
forall (HW_11: l0 = (result + 1)),
(* CADUCEUS_4 *) ((((0 <= l0 /\ u <= (n - 1)) /\ (Zopp 1) <= p /\ p <=
(n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l0 <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v))).
Proof.
intros; subst.
generalize (mean_1 l u); intuition.
assert (h: k <= (l+u)/2 \/ (l + u) / 2 + 1 <= k). omega. destruct h.
assert (shift t k # intM_global <= shift t ((l+u)/2) # intM_global).
apply H1; intuition.
absurd (shift t k # intM_global = v); omega.
assumption.
admit. (*TODO*)
Save.
(* Why obligation from file "why/binary_search.why", line 34, characters 24-37: *)
(*Why goal*) Lemma binary_search_impl_po_6 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
forall (HW_6: (* CADUCEUS_7 *) (l <= result /\ result <= u)),
forall (result0: (pointer global)),
forall (HW_7: result0 = (shift t result)),
forall (HW_8: (valid alloc result0)),
forall (result1: Z),
forall (HW_9: result1 = (acc intM_global result0)),
forall (HW_10: result1 < v),
forall (l0: Z),
forall (HW_11: l0 = (result + 1)),
(Zwf 0 (u - l0) (u - l)).
Proof.
intuition.
subst.
apply valid_range_valid_shift with 0 (n-1); intuition.
Save.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 24, characters 13-17: *)
(*Why goal*) Lemma binary_search_impl_po_7 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
forall (HW_6: (* CADUCEUS_7 *) (l <= result /\ result <= u)),
forall (result0: (pointer global)),
forall (HW_7: result0 = (shift t result)),
forall (HW_8: (valid alloc result0)),
forall (result1: Z),
forall (HW_9: result1 = (acc intM_global result0)),
forall (HW_12: result1 >= v),
forall (result2: (pointer global)),
forall (HW_13: result2 = (shift t result)),
(valid alloc result2).
Proof.
admit. (*TODO*)
Save.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 14, characters 6-147: *)
(*Why goal*) Lemma binary_search_impl_po_8 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
forall (HW_6: (* CADUCEUS_7 *) (l <= result /\ result <= u)),
forall (result0: (pointer global)),
forall (HW_7: result0 = (shift t result)),
forall (HW_8: (valid alloc result0)),
forall (result1: Z),
forall (HW_9: result1 = (acc intM_global result0)),
forall (HW_12: result1 >= v),
forall (result2: (pointer global)),
forall (HW_13: result2 = (shift t result)),
forall (HW_14: (valid alloc result2)),
forall (result3: Z),
forall (HW_15: result3 = (acc intM_global result2)),
forall (HW_16: result3 > v),
forall (u0: Z),
forall (HW_17: u0 = (result - 1)),
(* CADUCEUS_4 *) ((((0 <= l /\ u0 <= (n - 1)) /\ (Zopp 1) <= p /\ p <=
(n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u0)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v))).
Proof.
intuition.
admit. (*TODO*)
Save.
(* Why obligation from file "why/binary_search.why", line 34, characters 24-37: *)
(*Why goal*) Lemma binary_search_impl_po_9 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
forall (HW_6: (* CADUCEUS_7 *) (l <= result /\ result <= u)),
forall (result0: (pointer global)),
forall (HW_7: result0 = (shift t result)),
forall (HW_8: (valid alloc result0)),
forall (result1: Z),
forall (HW_9: result1 = (acc intM_global result0)),
forall (HW_12: result1 >= v),
forall (result2: (pointer global)),
forall (HW_13: result2 = (shift t result)),
forall (HW_14: (valid alloc result2)),
forall (result3: Z),
forall (HW_15: result3 = (acc intM_global result2)),
forall (HW_16: result3 > v),
forall (u0: Z),
forall (HW_17: u0 = (result - 1)),
(Zwf 0 (u0 - l) (u - l)).
Proof.
intuition.
admit. (*TODO*)
Save.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 8, characters 4-102: *)
(*Why goal*) Lemma binary_search_impl_po_10 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_3: l <= u),
forall (HW_4: 2 <> 0),
forall (result: Z),
forall (HW_5: result = ((Zdiv (l + u) 2))),
forall (HW_6: (* CADUCEUS_7 *) (l <= result /\ result <= u)),
forall (result0: (pointer global)),
forall (HW_7: result0 = (shift t result)),
forall (HW_8: (valid alloc result0)),
forall (result1: Z),
forall (HW_9: result1 = (acc intM_global result0)),
forall (HW_12: result1 >= v),
forall (result2: (pointer global)),
forall (HW_13: result2 = (shift t result)),
forall (HW_14: (valid alloc result2)),
forall (result3: Z),
forall (HW_15: result3 = (acc intM_global result2)),
forall (HW_18: result3 <= v),
forall (p0: Z),
forall (HW_19: p0 = result),
(* CADUCEUS_3 *) (p0 >= 0 /\ (acc intM_global (shift t p0)) = v \/ p0 =
(Zopp 1) /\
(forall (k:Z), (0 <= k /\ k < n -> (acc intM_global (shift t k)) <> v))).
Proof.
intuition.
(* FILL PROOF HERE *)
Save.
(* Why obligation from file "/home/jcf/soft/why/examples-c/tutorial/binary_search.c", line 8, characters 4-102: *)
(*Why goal*) Lemma binary_search_impl_po_11 :
forall (t: (pointer global)),
forall (n: Z),
forall (v: Z),
forall (alloc: alloc_table),
forall (intM_global: (memory Z global)),
forall (HW_1: (* CADUCEUS_2 *) ((n >= 0 /\
(valid_range alloc t 0 (n - 1))) /\
(forall (k1:Z),
(forall (k2:Z),
((0 <= k1 /\ k1 <= k2) /\ k2 <= (n - 1) ->
(acc intM_global (shift t k1)) <=
(acc intM_global (shift t k2))))))),
forall (l: Z),
forall (p: Z),
forall (u: Z),
forall (HW_2: (* CADUCEUS_4 *) ((((0 <= l /\ u <= (n - 1)) /\ (Zopp 1) <=
p /\ p <= (n - 1)) /\
((p = (Zopp 1) ->
(forall (k:Z),
(0 <= k /\ k < n ->
((acc intM_global (shift t k)) = v -> l <= k /\ k <= u)))))) /\
((p >= 0 -> (acc intM_global (shift t p)) = v)))),
forall (HW_20: l > u),
(* CADUCEUS_3 *) (p >= 0 /\ (acc intM_global (shift t p)) = v \/ p = (Zopp
1) /\
(forall (k:Z), (0 <= k /\ k < n -> (acc intM_global (shift t k)) <> v))).
Proof.
intuition.
(* FILL PROOF HERE *)
Save.
|
