diff options
Diffstat (limited to 'theories/Init/Wf.v')
| -rwxr-xr-x | theories/Init/Wf.v | 10 |
1 files changed, 5 insertions, 5 deletions
diff --git a/theories/Init/Wf.v b/theories/Init/Wf.v index 78432dcf4..745af5ef6 100755 --- a/theories/Init/Wf.v +++ b/theories/Init/Wf.v @@ -59,13 +59,13 @@ Chapter Well_founded. (P:A->Type)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a). Proof. Intros; Apply (Acc_rect P); Auto. - Save. + Qed. Theorem well_founded_induction : (P:A->Set)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a). Proof. Exact [P:A->Set](well_founded_induction_type P). - Save. + Qed. Theorem well_founded_ind : (P:A->Prop)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a). @@ -98,13 +98,13 @@ Lemma Fix_F_eq : (x:A)(r:(Acc x)) (F x [y:A][p:(R y x)](Fix_F y (Acc_inv x r y p)))=(Fix_F x r). Intros x r; Elim r using Acc_inv_dep; Auto. -Save. +Qed. Lemma Fix_F_inv : (x:A)(r,s:(Acc x))(Fix_F x r)=(Fix_F x s). Intro x; Elim (Rwf x); Intros. Case (Fix_F_eq x0 r); Case (Fix_F_eq x0 s); Intros. Apply F_ext; Auto. -Save. +Qed. Lemma fix_eq : (x:A)(fix x)=(F x [y:A][p:(R y x)](fix y)). @@ -112,7 +112,7 @@ Intro; Unfold fix. Case (Fix_F_eq x). Apply F_ext; Intros. Apply Fix_F_inv. -Save. +Qed. End FixPoint. |