20 lines
779 B
Coq
20 lines
779 B
Coq
Require Import Coq.Logic.FunctionalExtensionality.
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(* This is a simplified version of the lemma [Fix_eq], which is defined in
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[Coq.Init.Wf]. We use functional extensionality to remove one hypothesis.
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Furthermore, we introduce the auxiliary equality [f = Fix Rwf P F] so as
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to avoid duplicating the (usually large) term [F] in the right-hand side
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of the conclusion. *)
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Lemma Fix_eq_simplified
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(A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
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(P : A -> Type)
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(F : forall x : A, (forall y : A, R y x -> P y) -> P x)
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(f : forall x, P x) :
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f = Fix Rwf P F ->
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forall x : A,
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f x = F x (fun (y : A) (_ : R y x) => f y).
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Proof.
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intros. subst. eapply Fix_eq. intros. f_equal.
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eauto using functional_extensionality_dep, functional_extensionality.
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Qed.
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