lean2/library/struc/wf.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
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import logic.axioms.classical
-- Well-founded relation definition
-- We are essentially saying that a relation R is well-founded
-- if every non-empty "set" P, has a R-minimal element
definition wf {A : Type} (R : A → A → Prop) : Prop :=
∀P, (∃w, P w) → ∃min, P min ∧ ∀b, R b min → ¬P b
-- Well-founded induction theorem
theorem wf_induction {A : Type} {R : A → A → Prop} {P : A → Prop} (Hwf : wf R) (iH : ∀x, (∀y, R y x → P y) → P x)
: ∀x, P x :=
by_contradiction (assume N : ¬∀x, P x,
obtain (w : A) (Hw : ¬P w), from not_forall_exists N,
-- The main "trick" is to define Q x as ¬P x.
-- Since R is well-founded, there must be a R-minimal element r s.t. Q r (which is ¬P r)
let Q x := ¬P x in
have Qw : ∃w, Q w, from exists_intro w Hw,
have Qwf : ∃min, Q min ∧ ∀b, R b min → ¬Q b, from Hwf Q Qw,
obtain (r : A) (Hr : Q r ∧ ∀b, R b r → ¬Q b), from Qwf,
-- Using the inductive hypothesis iH and Hr, we show P r, and derive the contradiction.
have s1 : ∀b, R b r → P b, from
take b : A, assume H : R b r,
-- We are using Hr to derive ¬¬P b
not_not_elim (and_elim_right Hr b H),
have s2 : P r, from iH r s1,
have s3 : ¬P r, from and_elim_left Hr,
absurd s2 s3)