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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).
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