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Require Import TestSuite.admit.
Section Foo.
Variable a : nat.
Lemma l1 : True.
Fail Proof using non_existing.
Proof using a.
exact I.
Qed.
Lemma l2 : True.
Proof using a.
Admitted.
Lemma l3 : True.
Proof using a.
admit.
Qed.
End Foo.
Check (l1 3).
Check (l2 3).
Check (l3 3).
Section Bar.
Variable T : Type.
Variable a b : T.
Variable H : a = b.
Lemma l4 : a = b.
Proof using H.
exact H.
Qed.
End Bar.
Check (l4 _ 1 1 _ : 1 = 1).
Section S1.
Variable v1 : nat.
Section S2.
Variable v2 : nat.
Lemma deep : v1 = v2.
Proof using v1 v2.
admit.
Qed.
Lemma deep2 : v1 = v2.
Proof using v1 v2.
Admitted.
End S2.
Check (deep 3 : v1 = 3).
Check (deep2 3 : v1 = 3).
End S1.
Check (deep 3 4 : 3 = 4).
Check (deep2 3 4 : 3 = 4).
Section P1.
Variable x : nat.
Variable y : nat.
Variable z : nat.
Collection TOTO := x y.
Collection TITI := TOTO - x.
Lemma t1 : True. Proof using TOTO. trivial. Qed.
Lemma t2 : True. Proof using TITI. trivial. Qed.
Section P2.
Collection TOTO := x.
Lemma t3 : True. Proof using TOTO. trivial. Qed.
End P2.
Lemma t4 : True. Proof using TOTO. trivial. Qed.
End P1.
Lemma t5 : True. Fail Proof using TOTO. trivial. Qed.
Check (t1 1 2 : True).
Check (t2 1 : True).
Check (t3 1 : True).
Check (t4 1 2 : True).
Section T1.
Variable x : nat.
Hypothesis px : 1 = x.
Let w := x + 1.
Set Suggest Proof Using.
Set Default Proof Using "Type".
Lemma bla : 2 = w.
Proof.
admit.
Qed.
End T1.
Check (bla 7 : 2 = 8).
Section A.
Variable a : nat.
Variable b : nat.
Variable c : nat.
Variable H1 : a = 3.
Variable H2 : a = 3 -> b = 7.
Variable H3 : c = 3.
Lemma foo : a = a.
Proof using Type*.
pose H1 as e1.
pose H2 as e2.
reflexivity.
Qed.
Lemma bar : a = 3 -> b = 7.
Proof using b*.
exact H2.
Qed.
Lemma baz : c=3.
Proof using c*.
exact H3.
Qed.
Lemma baz2 : c=3.
Proof using c* a.
exact H3.
Qed.
End A.
Check (foo 3 7 (refl_equal 3)
(fun _ => refl_equal 7)).
Check (bar 3 7 (refl_equal 3)
(fun _ => refl_equal 7)).
Check (baz2 99 3 (refl_equal 3)).
Check (baz 3 (refl_equal 3)).
Section Let.
Variables a b : nat.
Let pa : a = a. Proof. reflexivity. Qed.
Unset Default Proof Using.
Set Suggest Proof Using.
Lemma test_let : a = a.
Proof using a.
exact pa.
Qed.
Let ppa : pa = pa. Proof. reflexivity. Qed.
Lemma test_let2 : pa = pa.
Proof using Type.
exact ppa.
Qed.
End Let.
Check (test_let 3).
(* Disabled
Section Clear.
Variable a: nat.
Hypotheses H : a = 4.
Set Proof Using Clear Unused.
Lemma test_clear : a = a.
Proof using a.
Fail rewrite H.
trivial.
Qed.
End Clear.
*)
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