-- 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)