290 lines
9.7 KiB
Text
290 lines
9.7 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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Authors: Floris van Doorn, Leonardo de Moura
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-/
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prelude
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import init.wf init.tactic init.num
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open eq.ops decidable or
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namespace nat
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notation `ℕ` := nat
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/- basic definitions on natural numbers -/
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inductive le (a : ℕ) : ℕ → Prop :=
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| refl : le a a
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| step : Π {b}, le a b → le a (succ b)
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infix `≤` := le
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attribute le.refl [refl]
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definition lt [reducible] (n m : ℕ) := succ n ≤ m
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definition ge [reducible] (n m : ℕ) := m ≤ n
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definition gt [reducible] (n m : ℕ) := succ m ≤ n
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infix `<` := lt
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infix `≥` := ge
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infix `>` := gt
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definition pred [unfold-c 1] (a : nat) : nat :=
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nat.cases_on a zero (λ a₁, a₁)
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-- add is defined in init.num
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definition sub (a b : nat) : nat :=
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nat.rec_on b a (λ b₁ r, pred r)
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definition mul (a b : nat) : nat :=
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nat.rec_on b zero (λ b₁ r, r + a)
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notation a - b := sub a b
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notation a * b := mul a b
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/- properties of ℕ -/
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protected definition is_inhabited [instance] : inhabited nat :=
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inhabited.mk zero
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protected definition has_decidable_eq [instance] : ∀ x y : nat, decidable (x = y)
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| has_decidable_eq zero zero := inl rfl
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| has_decidable_eq (succ x) zero := inr (by contradiction)
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| has_decidable_eq zero (succ y) := inr (by contradiction)
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| has_decidable_eq (succ x) (succ y) :=
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match has_decidable_eq x y with
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| inl xeqy := inl (by rewrite xeqy)
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| inr xney := inr (λ h : succ x = succ y, by injection h with xeqy; exact absurd xeqy xney)
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end
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/- properties of inequality -/
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theorem le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ le.refl n
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theorem le_succ (n : ℕ) : n ≤ succ n := by repeat constructor
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theorem pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
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theorem le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
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by induction H2 with n H2 IH;exact H1;exact le.step IH
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theorem le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
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theorem le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
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theorem le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
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theorem succ_le_succ [unfold-c 3] {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
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by induction H;reflexivity;exact le.step v_0
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theorem pred_le_pred [unfold-c 3] {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
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by induction H;reflexivity;cases b;exact v_0;exact le.step v_0
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theorem le_of_succ_le_succ [unfold-c 3] {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
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pred_le_pred H
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theorem le_succ_of_pred_le [unfold-c 1] {n m : ℕ} (H : pred n ≤ m) : n ≤ succ m :=
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by cases n;exact le.step H;exact succ_le_succ H
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theorem not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
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by induction n with n IH;all_goals intros;cases a;apply IH;exact le_of_succ_le_succ a
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theorem zero_le (n : ℕ) : 0 ≤ n :=
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by induction n with n IH;apply le.refl;exact le.step IH
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theorem lt.step {n m : ℕ} (H : n < m) : n < succ m :=
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le.step H
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theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
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by induction n with n IH;apply le.refl;exact le.step IH
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theorem lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
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le.trans (le.step H1) H2
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theorem lt_of_le_of_lt [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
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le.trans (succ_le_succ H1) H2
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theorem lt_of_lt_of_le [trans] {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
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le.trans H1 H2
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theorem le.antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
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begin
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cases H1 with m' H1',
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{ reflexivity},
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{ cases H2 with n' H2',
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{ reflexivity},
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{ exfalso, apply not_succ_le_self, exact lt.trans H1' H2'}},
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end
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theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
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by intro H; cases H
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theorem lt.irrefl (n : ℕ) : ¬n < n := not_succ_le_self
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theorem self_lt_succ (n : ℕ) : n < succ n := !le.refl
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theorem lt.base (n : ℕ) : n < succ n := !le.refl
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theorem le_lt_antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m < n) : false :=
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!lt.irrefl (lt_of_le_of_lt H1 H2)
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theorem lt_le_antisymm {n m : ℕ} (H1 : n < m) (H2 : m ≤ n) : false :=
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le_lt_antisymm H2 H1
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theorem lt.asymm {n m : ℕ} (H1 : n < m) (H2 : m < n) : false :=
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le_lt_antisymm (le_of_lt H1) H2
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definition lt.by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
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begin
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revert b H1 H2 H3, induction a with a IH,
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{ intros, cases b,
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exact H2 rfl,
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exact H1 !zero_lt_succ},
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{ intros, cases b with b,
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exact H3 !zero_lt_succ,
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{ apply IH,
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intro H, exact H1 (succ_le_succ H),
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intro H, exact H2 (congr rfl H),
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intro H, exact H3 (succ_le_succ H)}}
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end
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theorem lt.trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
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lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
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definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
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lt.by_cases H1 (λH, H2 (le_of_eq H⁻¹)) (λH, H2 (le_of_lt H))
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theorem lt_or_ge (a b : ℕ) : (a < b) ∨ (a ≥ b) :=
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lt_ge_by_cases inl inr
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definition not_lt_zero (a : ℕ) : ¬ a < zero :=
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by intro H; cases H
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-- less-than is well-founded
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definition lt.wf [instance] : well_founded lt :=
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begin
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constructor, intro n, induction n with n IH,
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{ constructor, intros n H, exfalso, exact !not_lt_zero H},
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{ constructor, intros m H,
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assert aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m,
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{ intros n₁ hlt, induction hlt,
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{ intro p, injection p with q, exact q ▸ IH},
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{ intro p, injection p with q, exact (acc.inv (q ▸ IH) a)}},
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apply aux H rfl},
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end
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definition measure {A : Type} (f : A → ℕ) : A → A → Prop :=
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inv_image lt f
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definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
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inv_image.wf f lt.wf
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theorem succ_lt_succ {a b : ℕ} (H : a < b) : succ a < succ b :=
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succ_le_succ H
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theorem lt_of_succ_lt {a b : ℕ} (H : succ a < b) : a < b :=
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le_of_succ_le H
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theorem lt_of_succ_lt_succ {a b : ℕ} (H : succ a < succ b) : a < b :=
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le_of_succ_le_succ H
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definition decidable_le [instance] : decidable_rel le :=
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begin
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intros n, induction n with n IH,
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{ intro m, left, apply zero_le},
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{ intro m, cases m with m,
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{ right, apply not_succ_le_zero},
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{ let H := IH m, clear IH,
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cases H with H H,
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left, exact succ_le_succ H,
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right, intro H2, exact H (le_of_succ_le_succ H2)}}
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end
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definition decidable_lt [instance] : decidable_rel lt := _
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definition decidable_gt [instance] : decidable_rel gt := _
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definition decidable_ge [instance] : decidable_rel ge := _
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theorem eq_or_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ∨ a < b :=
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by cases H with b' H; exact inl rfl; exact inr (succ_le_succ H)
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theorem le_of_eq_or_lt {a b : ℕ} (H : a = b ∨ a < b) : a ≤ b :=
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by cases H with H H; exact le_of_eq H; exact le_of_lt H
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theorem eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ∨ b < a :=
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or.rec_on (lt.trichotomy a b)
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(λ hlt, absurd hlt hnlt)
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(λ h, h)
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theorem lt_succ_of_le {a b : ℕ} (h : a ≤ b) : a < succ b :=
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succ_le_succ h
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theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
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theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
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definition max (a b : ℕ) : ℕ := if a < b then b else a
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definition min (a b : ℕ) : ℕ := if a < b then a else b
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theorem max_self (a : ℕ) : max a a = a :=
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eq.rec_on !if_t_t rfl
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theorem max_eq_right {a b : ℕ} (H : a < b) : max a b = b :=
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if_pos H
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theorem max_eq_left {a b : ℕ} (H : ¬ a < b) : max a b = a :=
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if_neg H
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theorem eq_max_right {a b : ℕ} (H : a < b) : b = max a b :=
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eq.rec_on (max_eq_right H) rfl
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theorem eq_max_left {a b : ℕ} (H : ¬ a < b) : a = max a b :=
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eq.rec_on (max_eq_left H) rfl
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theorem le_max_left (a b : ℕ) : a ≤ max a b :=
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by_cases
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(λ h : a < b, le_of_lt (eq.rec_on (eq_max_right h) h))
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(λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
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theorem le_max_right (a b : ℕ) : b ≤ max a b :=
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by_cases
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(λ h : a < b, eq.rec_on (eq_max_right h) !le.refl)
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(λ h : ¬ a < b, or.rec_on (eq_or_lt_of_not_lt h)
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(λ heq, eq.rec_on heq (eq.rec_on (eq.symm (max_self a)) !le.refl))
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(λ h : b < a,
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have aux : a = max a b, from eq_max_left (lt.asymm h),
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eq.rec_on aux (le_of_lt h)))
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theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
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by induction b with b IH;reflexivity; apply congr (eq.refl pred) IH
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theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
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eq.rec_on (succ_sub_succ_eq_sub a b) rfl
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theorem zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
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nat.rec_on a
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rfl
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(λ a₁ (ih : zero - a₁ = zero), congr (eq.refl pred) ih)
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theorem zero_eq_zero_sub (a : ℕ) : zero = zero - a :=
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eq.rec_on (zero_sub_eq_zero a) rfl
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theorem sub_lt {a b : ℕ} : zero < a → zero < b → a - b < a :=
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have aux : Π {a}, zero < a → Π {b}, zero < b → a - b < a, from
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λa h₁, le.rec_on h₁
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(λb h₂, le.cases_on h₂
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(lt.base zero)
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(λ b₁ bpos,
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eq.rec_on (sub_eq_succ_sub_succ zero b₁)
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(eq.rec_on (zero_eq_zero_sub b₁) (lt.base zero))))
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(λa₁ apos ih b h₂, le.cases_on h₂
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(lt.base a₁)
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(λ b₁ bpos,
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eq.rec_on (sub_eq_succ_sub_succ a₁ b₁)
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(lt.trans (@ih b₁ bpos) (lt.base a₁)))),
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λ h₁ h₂, aux h₁ h₂
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theorem sub_le (a b : ℕ) : a - b ≤ a :=
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nat.rec_on b
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(le.refl a)
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(λ b₁ ih, le.trans !pred_le ih)
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lemma sub_lt_succ (a b : ℕ) : a - b < succ a := lt_succ_of_le (sub_le a b)
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end nat
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