From 45b453873b7d90a35734cfe2ecabb23e9beed225 Mon Sep 17 00:00:00 2001 From: Leonardo de Moura Date: Thu, 30 Jan 2014 12:33:08 -0800 Subject: [PATCH] doc(examples/lean): add well-founded induction theorem Signed-off-by: Leonardo de Moura --- examples/lean/wf.lean | 26 ++++++++++++++++++++++++++ 1 file changed, 26 insertions(+) create mode 100644 examples/lean/wf.lean diff --git a/examples/lean/wf.lean b/examples/lean/wf.lean new file mode 100644 index 000000000..b6121354a --- /dev/null +++ b/examples/lean/wf.lean @@ -0,0 +1,26 @@ +import macros + +-- 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 U)} (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 U)} {R : A → A → Bool} {P : A → Bool} (Hwf : wf R) (iH : ∀ x, (∀ y, R y x → P y) → P x) + : ∀ x, P x +:= refute (λ N : ¬ ∀ x, P x, + obtain (w : A) (Hw : ¬ P w), from not_forall_elim N, + -- The main "trick" is to define Q x and ¬ 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 : A → Bool := λ x, ¬ P x, + Qw : ∃ w, Q w := exists_intro w Hw, + Qwf : ∃ min, Q min ∧ ∀ b, R b min → ¬ Q b := Hwf Q Qw + in 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. + let s1 : ∀ b, R b r → P b := take b : A, assume H : R b r, + -- We are using Hr to derive ¬ ¬ P b + not_not_elim (and_elimr Hr b H), + s2 : P r := iH r s1, + s3 : ¬ P r := and_eliml Hr + in absurd s2 s3)