144 lines
5.6 KiB
Text
144 lines
5.6 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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Show that "bounded" quantifiers: (∃x, x < n ∧ P x) and (∀x, x < n → P x)
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are decidable when P is decidable.
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This module allow us to write if-then-else expressions such as
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if (∀ x : nat, x < n → ∃ y : nat, y < n ∧ y * y = x) then t else s
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without assuming classical axioms.
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More importantly, they can be reduced inside of the Lean kernel.
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-/
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import data.nat.order data.nat.div
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namespace nat
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open subtype
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definition bex [reducible] (n : nat) (P : nat → Prop) : Prop :=
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∃ x, x < n ∧ P x
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definition bsub [reducible] (n : nat) (P : nat → Prop) : Type₁ :=
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{x | x < n ∧ P x}
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definition ball [reducible] (n : nat) (P : nat → Prop) : Prop :=
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∀ x, x < n → P x
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lemma bex_of_bsub {n : nat} {P : nat → Prop} : bsub n P → bex n P :=
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assume h, ex_of_sub h
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theorem not_bex_zero (P : nat → Prop) : ¬ bex 0 P :=
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λ H, obtain (w : nat) (Hw : w < 0 ∧ P w), from H,
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and.rec_on Hw (λ h₁ h₂, absurd h₁ (not_lt_zero w))
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theorem not_bsub_zero (P : nat → Prop) : bsub 0 P → false :=
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λ H, absurd (bex_of_bsub H) (not_bex_zero P)
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definition bsub_succ {P : nat → Prop} {n : nat} (H : bsub n P) : bsub (succ n) P :=
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obtain (w : nat) (Hw : w < n ∧ P w), from H,
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and.rec_on Hw (λ hlt hp, tag w (and.intro (lt.step hlt) hp))
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theorem bex_succ {P : nat → Prop} {n : nat} (H : bex n P) : bex (succ n) P :=
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obtain (w : nat) (Hw : w < n ∧ P w), from H,
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and.rec_on Hw (λ hlt hp, exists.intro w (and.intro (lt.step hlt) hp))
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definition bsub_succ_of_pred {P : nat → Prop} {a : nat} (H : P a) : bsub (succ a) P :=
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tag a (and.intro (lt.base a) H)
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theorem bex_succ_of_pred {P : nat → Prop} {a : nat} (H : P a) : bex (succ a) P :=
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bex_of_bsub (bsub_succ_of_pred H)
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theorem not_bex_succ {P : nat → Prop} {n : nat} (H₁ : ¬ bex n P) (H₂ : ¬ P n) : ¬ bex (succ n) P :=
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λ H, obtain (w : nat) (Hw : w < succ n ∧ P w), from H,
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and.rec_on Hw (λ hltsn hp, or.rec_on (nat.eq_or_lt_of_le (le_of_succ_le_succ hltsn))
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(λ heq : w = n, absurd (eq.rec_on heq hp) H₂)
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(λ hltn : w < n, absurd (exists.intro w (and.intro hltn hp)) H₁))
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theorem not_bsub_succ {P : nat → Prop} {n : nat} (H₁ : ¬ bex n P) (H₂ : ¬ P n) : bsub (succ n) P → false :=
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λ H, absurd (bex_of_bsub H) (not_bex_succ H₁ H₂)
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theorem ball_zero (P : nat → Prop) : ball zero P :=
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λ x Hlt, absurd Hlt !not_lt_zero
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theorem ball_of_ball_succ {n : nat} {P : nat → Prop} (H : ball (succ n) P) : ball n P :=
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λ x Hlt, H x (lt.step Hlt)
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theorem ball_succ_of_ball {n : nat} {P : nat → Prop} (H₁ : ball n P) (H₂ : P n) : ball (succ n) P :=
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λ (x : nat) (Hlt : x < succ n), or.elim (nat.eq_or_lt_of_le (le_of_succ_le_succ Hlt))
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(λ heq : x = n, eq.rec_on (eq.rec_on heq rfl) H₂)
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(λ hlt : x < n, H₁ x hlt)
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theorem not_ball_of_not {n : nat} {P : nat → Prop} (H₁ : ¬ P n) : ¬ ball (succ n) P :=
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λ (H : ball (succ n) P), absurd (H n (lt.base n)) H₁
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theorem not_ball_succ_of_not_ball {n : nat} {P : nat → Prop} (H₁ : ¬ ball n P) : ¬ ball (succ n) P :=
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λ (H : ball (succ n) P), absurd (ball_of_ball_succ H) H₁
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end nat
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section
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open nat decidable
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definition decidable_bex [instance] (n : nat) (P : nat → Prop) [H : decidable_pred P] : decidable (bex n P) :=
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nat.rec_on n
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(inr (not_bex_zero P))
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(λ a ih, decidable.rec_on ih
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(λ hpos : bex a P, inl (bex_succ hpos))
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(λ hneg : ¬ bex a P, decidable.rec_on (H a)
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(λ hpa : P a, inl (bex_succ_of_pred hpa))
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(λ hna : ¬ P a, inr (not_bex_succ hneg hna))))
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definition decidable_ball [instance] (n : nat) (P : nat → Prop) [H : decidable_pred P] : decidable (ball n P) :=
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nat.rec_on n
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(inl (ball_zero P))
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(λ n₁ ih, decidable.rec_on ih
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(λ ih_pos, decidable.rec_on (H n₁)
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(λ p_pos, inl (ball_succ_of_ball ih_pos p_pos))
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(λ p_neg, inr (not_ball_of_not p_neg)))
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(λ ih_neg, inr (not_ball_succ_of_not_ball ih_neg)))
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definition decidable_bex_le [instance] (n : nat) (P : nat → Prop) [H : decidable_pred P]
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: decidable (∃ x, x ≤ n ∧ P x) :=
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decidable_of_decidable_of_iff
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(decidable_bex (succ n) P)
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(exists_congr (λn, and_iff_and !lt_succ_iff_le !iff.refl))
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definition decidable_ball_le [instance] (n : nat) (P : nat → Prop) [H : decidable_pred P]
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: decidable (∀ x, x ≤ n → P x) :=
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decidable_of_decidable_of_iff
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(decidable_ball (succ n) P)
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(forall_congr (λn, imp_iff_imp !lt_succ_iff_le !iff.refl))
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end
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namespace nat
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open decidable
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variable {P : nat → Prop}
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variable [decP : decidable_pred P]
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include decP
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definition bsub_not_of_not_ball : ∀ {n : nat}, ¬ ball n P → {i | i < n ∧ ¬ P i}
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| 0 h := absurd (ball_zero P) h
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| (succ n) h := decidable.by_cases
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(λ hp : P n,
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have ¬ ball n P, from
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assume b : ball n P, absurd (ball_succ_of_ball b hp) h,
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have {i | i < n ∧ ¬ P i}, from bsub_not_of_not_ball this,
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bsub_succ this)
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(λ hn : ¬ P n, bsub_succ_of_pred hn)
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theorem bex_not_of_not_ball {n : nat} (H : ¬ ball n P) : bex n (λ n, ¬ P n) :=
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bex_of_bsub (bsub_not_of_not_ball H)
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theorem ball_not_of_not_bex : ∀ {n : nat}, ¬ bex n P → ball n (λ n, ¬ P n)
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| 0 h := ball_zero _
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| (succ n) h := by_cases
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(λ hp : P n, absurd (bex_succ_of_pred hp) h)
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(λ hn : ¬ P n,
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have ¬ bex n P, from
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assume b : bex n P, absurd (bex_succ b) h,
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have ball n (λ n, ¬ P n), from ball_not_of_not_bex this,
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ball_succ_of_ball this hn)
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end nat
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