46 lines
1.5 KiB
Text
46 lines
1.5 KiB
Text
-- 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 data.nat.order logic.wf
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open nat eq.ops
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namespace nat
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inductive pred_rel : nat → nat → Prop :=
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intro : Π (n : nat), pred_rel n (succ n)
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definition not_pred_rel_zero (n : nat) : ¬ pred_rel n zero :=
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have aux : ∀{m}, pred_rel n m → succ n = m, from
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λm H, pred_rel.rec_on H (take n, rfl),
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assume H : pred_rel n zero,
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absurd (aux H) !succ_ne_zero
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definition pred_rel_succ {a b : nat} (H : pred_rel a (succ b)) : b = a :=
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have aux : pred (succ b) = a, from
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pred_rel.rec_on H (λn, rfl),
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aux
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-- Predecessor relation is well-founded
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definition pred_rel.wf : well_founded pred_rel :=
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well_founded.intro
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(λn, induction_on n
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(acc.intro zero (λy (H : pred_rel y zero), absurd H (not_pred_rel_zero y)))
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(λa (iH : acc pred_rel a),
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acc.intro (succ a) (λy (H : pred_rel y (succ a)),
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have aux : a = y, from pred_rel_succ H,
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eq.rec_on aux iH)))
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-- Less-than relation is well-founded
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definition lt.wf [instance] : well_founded lt :=
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well_founded.intro
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(λn, induction_on n
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(acc.intro zero (λ (y : nat) (H : y < 0),
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absurd H !not_lt_zero))
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(λ (n : nat) (iH : acc lt n),
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acc.intro (succ n) (λ (m : nat) (H : m < succ n),
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have H₁ : m < n ∨ m = n, from le_imp_lt_or_eq (succ_le_cancel (lt_imp_le_succ H)),
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or.elim H₁
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(assume Hlt : m < n, acc.inv iH Hlt)
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(assume Heq : m = n, Heq⁻¹ ▸ iH))))
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end nat
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