30 lines
1.6 KiB
Text
30 lines
1.6 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
-- Released under Apache 2.0 license as described in the file LICENSE.
|
|
-- Author: Leonardo de Moura
|
|
import logic 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 → Bool) : Bool
|
|
:= ∀ 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 → Bool} {P : A → Bool} (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 [inline] := λ 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,
|
|
show false, from absurd s2 s3)
|