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