414 lines
13 KiB
Text
414 lines
13 KiB
Text
--- Copyright (c) 2014 Floris van Doorn. All rights reserved.
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--- Released under Apache 2.0 license as described in the file LICENSE.
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--- Author: Floris van Doorn
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-- data.nat.basic
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-- ==============
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--
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-- Basic operations on the natural numbers.
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import logic data.num tools.tactic struc.binary tools.helper_tactics
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using num tactic binary eq_ops
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using decidable (hiding induction_on rec_on)
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using relation -- for subst_iff
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using helper_tactics
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-- Definition of the type
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-- ----------------------
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namespace nat
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inductive nat : Type :=
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zero : nat,
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succ : nat → nat
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notation `ℕ` := nat
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theorem nat_rec_zero {P : ℕ → Type} (x : P zero) (f : ∀m, P m → P (succ m)) : nat_rec x f zero = x
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theorem nat_rec_succ {P : ℕ → Type} (x : P zero) (f : ∀m, P m → P (succ m)) (n : ℕ) :
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nat_rec x f (succ n) = f n (nat_rec x f n)
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theorem induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P zero) (H2 : ∀ (n : ℕ) (IH : P n), P (succ n)) :
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P a :=
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nat_rec H1 H2 a
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definition rec_on {P : ℕ → Type} (n : ℕ) (H1 : P zero) (H2 : ∀m, P m → P (succ m)) : P n :=
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nat_rec H1 H2 n
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-- Coercion from num
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-- -----------------
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abbreviation plus (x y : ℕ) : ℕ :=
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nat_rec x (λ n r, succ r) y
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definition to_nat [coercion] [inline] (n : num) : ℕ :=
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num_rec zero
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(λ n, pos_num_rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n
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-- Successor and predecessor
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-- -------------------------
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theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
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assume H : succ n = 0,
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have H2 : true = false, from
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let f := (nat_rec false (fun a b, true)) in
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calc
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true = f (succ n) : rfl
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... = f 0 : {H}
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... = false : rfl,
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absurd H2 true_ne_false
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-- add_rewrite succ_ne_zero
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definition pred (n : ℕ) := nat_rec 0 (fun m x, m) n
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theorem pred_zero : pred 0 = 0
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theorem pred_succ (n : ℕ) : pred (succ n) = n
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opaque_hint (hiding pred)
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theorem zero_or_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
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induction_on n
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(or_inl (refl 0))
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(take m IH, or_inr
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(show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
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theorem zero_or_exists_succ (n : ℕ) : n = 0 ∨ ∃k, n = succ k :=
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or_imp_or (zero_or_succ_pred n) (assume H, H)
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(assume H : n = succ (pred n), exists_intro (pred n) H)
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theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n :=
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induction_on n H1 (take m IH, H2 m)
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theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
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or_elim (zero_or_succ_pred n)
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(take H3 : n = 0, H1 H3)
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(take H3 : n = succ (pred n), H2 (pred n) H3)
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theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m :=
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calc
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n = pred (succ n) : pred_succ n⁻¹
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... = pred (succ m) : {H}
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... = m : pred_succ m
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theorem succ_ne_self (n : ℕ) : succ n ≠ n :=
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induction_on n
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(take H : 1 = 0,
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have ne : 1 ≠ 0, from succ_ne_zero 0,
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absurd H ne)
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(take k IH H, IH (succ_inj H))
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theorem decidable_eq [instance] (n m : ℕ) : decidable (n = m) :=
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have general : ∀n, decidable (n = m), from
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rec_on m
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(take n,
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rec_on n
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(inl (refl 0))
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(λ m iH, inr (succ_ne_zero m)))
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(λ (m' : ℕ) (iH1 : ∀n, decidable (n = m')),
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take n, rec_on n
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(inr (ne_symm (succ_ne_zero m')))
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(λ (n' : ℕ) (iH2 : decidable (n' = succ m')),
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have d1 : decidable (n' = m'), from iH1 n',
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decidable.rec_on d1
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(assume Heq : n' = m', inl (congr_arg succ Heq))
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(assume Hne : n' ≠ m',
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have H1 : succ n' ≠ succ m', from
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assume Heq, absurd (succ_inj Heq) Hne,
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inr H1))),
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general n
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theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
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(H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a :=
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have stronger : P a ∧ P (succ a), from
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induction_on a
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(and_intro H1 H2)
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(take k IH,
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have IH1 : P k, from and_elim_left IH,
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have IH2 : P (succ k), from and_elim_right IH,
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and_intro IH2 (H3 k IH1 IH2)),
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and_elim_left stronger
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theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
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(H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m :=
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have general : ∀m, P n m, from induction_on n
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(take m : ℕ, H1 m)
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(take k : ℕ,
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assume IH : ∀m, P k m,
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take m : ℕ,
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discriminate
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(assume Hm : m = 0,
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Hm⁻¹ ▸ (H2 k))
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(take l : ℕ,
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assume Hm : m = succ l,
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Hm⁻¹ ▸ (H3 k l (IH l)))),
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general m
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-- Addition
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-- --------
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definition add (x y : ℕ) : ℕ := plus x y
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infixl `+` := add
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theorem add_zero_right (n : ℕ) : n + 0 = n
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theorem add_succ_right (n m : ℕ) : n + succ m = succ (n + m)
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opaque_hint (hiding add)
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theorem add_zero_left (n : ℕ) : 0 + n = n :=
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induction_on n
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(add_zero_right 0)
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(take m IH, show 0 + succ m = succ m, from
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calc
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0 + succ m = succ (0 + m) : add_succ_right _ _
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... = succ m : {IH})
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theorem add_succ_left (n m : ℕ) : (succ n) + m = succ (n + m) :=
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induction_on m
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(calc
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succ n + 0 = succ n : add_zero_right (succ n)
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... = succ (n + 0) : {symm (add_zero_right n)})
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(take k IH,
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calc
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succ n + succ k = succ (succ n + k) : add_succ_right _ _
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... = succ (succ (n + k)) : {IH}
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... = succ (n + succ k) : {symm (add_succ_right _ _)})
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theorem add_comm (n m : ℕ) : n + m = m + n :=
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induction_on m
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(trans (add_zero_right _) (symm (add_zero_left _)))
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(take k IH,
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calc
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n + succ k = succ (n+k) : add_succ_right _ _
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... = succ (k + n) : {IH}
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... = succ k + n : symm (add_succ_left _ _))
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theorem add_move_succ (n m : ℕ) : succ n + m = n + succ m :=
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calc
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succ n + m = succ (n + m) : add_succ_left n m
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... = n +succ m : symm (add_succ_right n m)
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theorem add_comm_succ (n m : ℕ) : n + succ m = m + succ n :=
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calc
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n + succ m = succ n + m : symm (add_move_succ n m)
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... = m + succ n : add_comm (succ n) m
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theorem add_assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
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induction_on k
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(calc
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(n + m) + 0 = n + m : add_zero_right _
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... = n + (m + 0) : {symm (add_zero_right m)})
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(take l IH,
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calc
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(n + m) + succ l = succ ((n + m) + l) : add_succ_right _ _
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... = succ (n + (m + l)) : {IH}
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... = n + succ (m + l) : symm (add_succ_right _ _)
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... = n + (m + succ l) : {symm (add_succ_right _ _)})
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theorem add_left_comm (n m k : ℕ) : n + (m + k) = m + (n + k) :=
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left_comm add_comm add_assoc n m k
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theorem add_right_comm (n m k : ℕ) : n + m + k = n + k + m :=
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right_comm add_comm add_assoc n m k
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-- add_rewrite add_zero_left add_zero_right
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-- add_rewrite add_succ_left add_succ_right
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-- add_rewrite add_comm add_assoc add_left_comm
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-- ### cancelation
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theorem add_cancel_left {n m k : ℕ} : n + m = n + k → m = k :=
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induction_on n
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(take H : 0 + m = 0 + k,
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calc
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m = 0 + m : symm (add_zero_left m)
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... = 0 + k : H
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... = k : add_zero_left k)
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(take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
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have H2 : succ (n + m) = succ (n + k),
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from calc
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succ (n + m) = succ n + m : symm (add_succ_left n m)
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... = succ n + k : H
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... = succ (n + k) : add_succ_left n k,
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have H3 : n + m = n + k, from succ_inj H2,
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IH H3)
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theorem add_cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k :=
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have H2 : m + n = m + k,
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from calc
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m + n = n + m : add_comm m n
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... = k + m : H
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... = m + k : add_comm k m,
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add_cancel_left H2
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theorem add_eq_zero_left {n m : ℕ} : n + m = 0 → n = 0 :=
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induction_on n
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(take (H : 0 + m = 0), refl 0)
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(take k IH,
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assume (H : succ k + m = 0),
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absurd
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(show succ (k + m) = 0, from
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calc
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succ (k + m) = succ k + m : symm (add_succ_left k m)
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... = 0 : H)
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(succ_ne_zero (k + m)))
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theorem add_eq_zero_right {n m : ℕ} (H : n + m = 0) : m = 0 :=
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add_eq_zero_left (trans (add_comm m n) H)
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theorem add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
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and_intro (add_eq_zero_left H) (add_eq_zero_right H)
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-- ### misc
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theorem add_one (n : ℕ) : n + 1 = succ n :=
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calc
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n + 1 = succ (n + 0) : add_succ_right _ _
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... = succ n : {add_zero_right _}
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theorem add_one_left (n : ℕ) : 1 + n = succ n :=
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calc
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1 + n = succ (0 + n) : add_succ_left _ _
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... = succ n : {add_zero_left _}
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-- TODO: rename? remove?
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theorem induction_plus_one {P : nat → Prop} (a : ℕ) (H1 : P 0)
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(H2 : ∀ (n : ℕ) (IH : P n), P (n + 1)) : P a :=
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nat_rec H1 (take n IH, (add_one n) ▸ (H2 n IH)) a
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-- Multiplication
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-- --------------
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definition mul (n m : ℕ) := nat_rec 0 (fun m x, x + n) m
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infixl `*` := mul
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theorem mul_zero_right (n:ℕ) : n * 0 = 0
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theorem mul_succ_right (n m:ℕ) : n * succ m = n * m + n
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opaque_hint (hiding mul)
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-- ### commutativity, distributivity, associativity, identity
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theorem mul_zero_left (n:ℕ) : 0 * n = 0 :=
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induction_on n
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(mul_zero_right 0)
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(take m IH,
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calc
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0 * succ m = 0 * m + 0 : mul_succ_right _ _
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... = 0 * m : add_zero_right _
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... = 0 : IH)
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theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m :=
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induction_on m
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(calc
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succ n * 0 = 0 : mul_zero_right _
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... = n * 0 : symm (mul_zero_right _)
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... = n * 0 + 0 : symm (add_zero_right _))
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(take k IH,
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calc
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succ n * succ k = (succ n * k) + succ n : mul_succ_right _ _
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... = (n * k) + k + succ n : { IH }
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... = (n * k) + (k + succ n) : add_assoc _ _ _
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... = (n * k) + (n + succ k) : {add_comm_succ _ _}
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... = (n * k) + n + succ k : symm (add_assoc _ _ _)
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... = (n * succ k) + succ k : {symm (mul_succ_right n k)})
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theorem mul_comm (n m:ℕ) : n * m = m * n :=
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induction_on m
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(trans (mul_zero_right _) (symm (mul_zero_left _)))
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(take k IH,
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calc
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n * succ k = n * k + n : mul_succ_right _ _
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... = k * n + n : {IH}
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... = (succ k) * n : symm (mul_succ_left _ _))
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theorem mul_distr_right (n m k : ℕ) : (n + m) * k = n * k + m * k :=
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induction_on k
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(calc
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(n + m) * 0 = 0 : mul_zero_right _
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... = 0 + 0 : symm (add_zero_right _)
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... = n * 0 + 0 : {symm (mul_zero_right _)}
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... = n * 0 + m * 0 : {symm (mul_zero_right _)})
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(take l IH, calc
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(n + m) * succ l = (n + m) * l + (n + m) : mul_succ_right _ _
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... = n * l + m * l + (n + m) : {IH}
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... = n * l + m * l + n + m : symm (add_assoc _ _ _)
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... = n * l + n + m * l + m : {add_right_comm _ _ _}
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... = n * l + n + (m * l + m) : add_assoc _ _ _
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... = n * succ l + (m * l + m) : {symm (mul_succ_right _ _)}
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... = n * succ l + m * succ l : {symm (mul_succ_right _ _)})
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theorem mul_distr_left (n m k : ℕ) : n * (m + k) = n * m + n * k :=
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calc
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n * (m + k) = (m + k) * n : mul_comm _ _
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... = m * n + k * n : mul_distr_right _ _ _
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... = n * m + k * n : {mul_comm _ _}
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... = n * m + n * k : {mul_comm _ _}
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theorem mul_assoc (n m k:ℕ) : (n * m) * k = n * (m * k) :=
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induction_on k
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(calc
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(n * m) * 0 = 0 : mul_zero_right _
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... = n * 0 : symm (mul_zero_right _)
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... = n * (m * 0) : {symm (mul_zero_right _)})
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(take l IH,
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calc
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(n * m) * succ l = (n * m) * l + n * m : mul_succ_right _ _
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... = n * (m * l) + n * m : {IH}
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... = n * (m * l + m) : symm (mul_distr_left _ _ _)
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... = n * (m * succ l) : {symm (mul_succ_right _ _)})
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theorem mul_left_comm (n m k : ℕ) : n * (m * k) = m * (n * k) :=
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left_comm mul_comm mul_assoc n m k
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theorem mul_right_comm (n m k : ℕ) : n * m * k = n * k * m :=
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right_comm mul_comm mul_assoc n m k
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theorem mul_one_right (n : ℕ) : n * 1 = n :=
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calc
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n * 1 = n * 0 + n : mul_succ_right n 0
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... = 0 + n : {mul_zero_right n}
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... = n : add_zero_left n
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theorem mul_one_left (n : ℕ) : 1 * n = n :=
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calc
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1 * n = n * 1 : mul_comm _ _
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... = n : mul_one_right n
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theorem mul_eq_zero {n m : ℕ} (H : n * m = 0) : n = 0 ∨ m = 0 :=
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discriminate
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(take Hn : n = 0, or_intro_left _ Hn)
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(take (k : ℕ),
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assume (Hk : n = succ k),
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discriminate
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(take (Hm : m = 0), or_intro_right _ Hm)
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(take (l : ℕ),
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assume (Hl : m = succ l),
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have Heq : succ (k * succ l + l) = n * m, from
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symm (calc
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n * m = n * succ l : {Hl}
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... = succ k * succ l : {Hk}
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... = k * succ l + succ l : mul_succ_left _ _
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... = succ (k * succ l + l) : add_succ_right _ _),
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absurd (trans Heq H) (succ_ne_zero _)))
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---other inversion theorems appear below
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-- add_rewrite mul_zero_left mul_zero_right mul_one_right mul_one_left
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-- add_rewrite mul_succ_left mul_succ_right
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-- add_rewrite mul_comm mul_assoc mul_left_comm
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-- add_rewrite mul_distr_right mul_distr_left
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end nat
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