lean2/library/standard/nat.lean

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-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Floris van Doorn
----------------------------------------------------------------------------------------------------
import logic num tactic decidable binary
using tactic num binary eq_proofs
using decidable (hiding induction_on rec_on)
namespace nat
inductive nat : Type :=
| zero : nat
| succ : nat → nat
notation ``:max := nat
abbreviation plus (x y : ) :
:= nat_rec x (λ n r, succ r) y
definition to_nat [coercion] [inline] (n : num) :
:= num_rec zero (λ n, pos_num_rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n
namespace helper_tactics
definition apply_refl := apply @refl
tactic_hint apply_refl
end
using helper_tactics
theorem nat_rec_zero {P : → Type} (x : P 0) (f : ∀m, P m → P (succ m)) : nat_rec x f 0 = x
theorem nat_rec_succ {P : → Type} (x : P 0) (f : ∀m, P m → P (succ m)) (n : ) : nat_rec x f (succ n) = f n (nat_rec x f n)
theorem induction_on {P : → Prop} (a : ) (H1 : P 0) (H2 : ∀ (n : ) (IH : P n), P (succ n)) : P a
:= nat_rec H1 H2 a
definition rec_on {P : → Type} (n : ) (H1 : P 0) (H2 : ∀m, P m → P (succ m)) : P n
:= nat_rec H1 H2 n
-------------------------------------------------- succ pred
theorem succ_ne_zero (n : ) : succ n ≠ 0
:= assume H : succ n = 0,
have H2 : true = false, from
let f [inline] := (nat_rec false (fun a b, true)) in
calc true = f (succ n) : _
... = f 0 : {H}
... = false : _,
absurd H2 true_ne_false
definition pred (n : ) := nat_rec 0 (fun m x, m) n
theorem pred_zero : pred 0 = 0
theorem pred_succ (n : ) : pred (succ n) = n
theorem zero_or_succ (n : ) : n = 0 n = succ (pred n)
:= induction_on n
(or_inl (refl 0))
(take m IH, or_inr
(show succ m = succ (pred (succ m)), from congr2 succ (pred_succ m⁻¹)))
theorem zero_or_succ2 (n : ) : n = 0 ∃k, n = succ k
:= or_imp_or (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
theorem case {P : → Prop} (n : ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
:= induction_on n H1 (take m IH, H2 m)
theorem discriminate {B : Prop} {n : } (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B
:= or_elim (zero_or_succ n)
(take H3 : n = 0, H1 H3)
(take H3 : n = succ (pred n), H2 (pred n) H3)
theorem succ_inj {n m : } (H : succ n = succ m) : n = m
:= calc
n = pred (succ n) : pred_succ n⁻¹
... = pred (succ m) : {H}
... = m : pred_succ m
theorem succ_ne_self (n : ) : succ n ≠ n
:= induction_on n
(take H : 1 = 0,
have ne : 1 ≠ 0, from succ_ne_zero 0,
absurd H ne)
(take k IH H, IH (succ_inj H))
theorem decidable_eq [instance] (n m : ) : decidable (n = m)
:= have general : ∀n, decidable (n = m), from
rec_on m
(take n,
rec_on n
(inl (refl 0))
(λ m iH, inr (succ_ne_zero m)))
(λ (m' : ) (iH1 : ∀n, decidable (n = m')),
take n, rec_on n
(inr (ne_symm (succ_ne_zero m')))
(λ (n' : ) (iH2 : decidable (n' = succ m')),
have d1 : decidable (n' = m'), from iH1 n',
decidable.rec_on d1
(assume Heq : n' = m', inl (congr2 succ Heq))
(assume Hne : n' ≠ m',
have H1 : succ n' ≠ succ m', from
assume Heq, absurd (succ_inj Heq) Hne,
inr H1))),
general n
theorem two_step_induction_on {P : → Prop} (a : ) (H1 : P 0) (H2 : P 1)
(H3 : ∀ (n : ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a
:= have stronger : P a ∧ P (succ a), from
induction_on a
(and_intro H1 H2)
(take k IH,
have IH1 : P k, from and_elim_left IH,
have IH2 : P (succ k), from and_elim_right IH,
and_intro IH2 (H3 k IH1 IH2)),
and_elim_left stronger
theorem sub_induction {P : → Prop} (n m : ) (H1 : ∀m, P 0 m)
(H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m
:= have general : ∀m, P n m, from induction_on n
(take m : , H1 m)
(take k : ,
assume IH : ∀m, P k m,
take m : ,
discriminate
(assume Hm : m = 0,
Hm⁻¹ ▸ (H2 k))
(take l : ,
assume Hm : m = succ l,
Hm⁻¹ ▸ (H3 k l (IH l)))),
general m
-------------------------------------------------- add
definition add (x y : ) : := plus x y
infixl `+`:65 := add
theorem add_zero_right (n : ) : n + 0 = n
theorem add_succ_right (n m : ) : n + succ m = succ (n + m)
opaque_hint (hiding add)
---------- comm, assoc
theorem add_zero_left (n : ) : 0 + n = n
:= induction_on n
(add_zero_right 0)
(take m IH, show 0 + succ m = succ m, from
calc
0 + succ m = succ (0 + m) : add_succ_right _ _
... = succ m : {IH})
theorem add_succ_left (n m : ) : (succ n) + m = succ (n + m)
:= induction_on m
(calc
succ n + 0 = succ n : add_zero_right (succ n)
... = succ (n + 0) : {symm (add_zero_right n)})
(take k IH,
calc
succ n + succ k = succ (succ n + k) : add_succ_right _ _
... = succ (succ (n + k)) : {IH}
... = succ (n + succ k) : {symm (add_succ_right _ _)})
theorem add_comm (n m : ) : n + m = m + n
:= induction_on m
(trans (add_zero_right _) (symm (add_zero_left _)))
(take k IH,
calc
n + succ k = succ (n+k) : add_succ_right _ _
... = succ (k + n) : {IH}
... = succ k + n : symm (add_succ_left _ _))
theorem add_move_succ (n m : ) : succ n + m = n + succ m
:= calc
succ n + m = succ (n + m) : add_succ_left n m
... = n +succ m : symm (add_succ_right n m)
theorem add_comm_succ (n m : ) : n + succ m = m + succ n
:= calc
n + succ m = succ n + m : symm (add_move_succ n m)
... = m + succ n : add_comm (succ n) m
theorem add_assoc (n m k : ) : (n + m) + k = n + (m + k)
:= induction_on k
(calc
(n + m) + 0 = n + m : add_zero_right _
... = n + (m + 0) : {symm (add_zero_right m)})
(take l IH,
calc
(n + m) + succ l = succ ((n + m) + l) : add_succ_right _ _
... = succ (n + (m + l)) : {IH}
... = n + succ (m + l) : symm (add_succ_right _ _)
... = n + (m + succ l) : {symm (add_succ_right _ _)})
theorem add_left_comm (n m k : ) : n + (m + k) = m + (n + k)
:= left_comm add_comm add_assoc n m k
theorem add_right_comm (n m k : ) : n + m + k = n + k + m
:= right_comm add_comm add_assoc n m k
---------- inversion
theorem add_cancel_left {n m k : } : n + m = n + k → m = k
:=
induction_on n
(take H : 0 + m = 0 + k,
calc
m = 0 + m : symm (add_zero_left m)
... = 0 + k : H
... = k : add_zero_left k)
(take (n : ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
have H2 : succ (n + m) = succ (n + k),
from calc
succ (n + m) = succ n + m : symm (add_succ_left n m)
... = succ n + k : H
... = succ (n + k) : add_succ_left n k,
have H3 : n + m = n + k, from succ_inj H2,
IH H3)
--rename to and_cancel_right
theorem add_cancel_right {n m k : } (H : n + m = k + m) : n = k
:=
have H2 : m + n = m + k,
from calc
m + n = n + m : add_comm m n
... = k + m : H
... = m + k : add_comm k m,
add_cancel_left H2
theorem add_eq_zero_left {n m : } : n + m = 0 → n = 0
:=
induction_on n
(take (H : 0 + m = 0), refl 0)
(take k IH,
assume (H : succ k + m = 0),
absurd_elim (succ k = 0)
(show succ (k + m) = 0, from
calc
succ (k + m) = succ k + m : symm (add_succ_left k m)
... = 0 : H)
(succ_ne_zero (k + m)))
theorem add_eq_zero_right {n m : } (H : n + m = 0) : m = 0
:= add_eq_zero_left (trans (add_comm m n) H)
theorem add_eq_zero {n m : } (H : n + m = 0) : n = 0 ∧ m = 0
:= and_intro (add_eq_zero_left H) (add_eq_zero_right H)
-- add_eq_self below
---------- misc
theorem add_one (n:) : n + 1 = succ n
:=
calc
n + 1 = succ (n + 0) : add_succ_right _ _
... = succ n : {add_zero_right _}
theorem add_one_left (n:) : 1 + n = succ n
:=
calc
1 + n = succ (0 + n) : add_succ_left _ _
... = succ n : {add_zero_left _}
--the following theorem has a terrible name, but since the name is not a substring or superstring of another name, it is at least easy to globally replace it
theorem induction_plus_one {P : → Prop} (a : ) (H1 : P 0)
(H2 : ∀ (n : ) (IH : P n), P (n + 1)) : P a
:= nat_rec H1 (take n IH, (add_one n) ▸ (H2 n IH)) a
-------------------------------------------------- mul
definition mul (n m : ) := nat_rec 0 (fun m x, x + n) m
infixl `*`:75 := mul
theorem mul_zero_right (n:) : n * 0 = 0
theorem mul_succ_right (n m:) : n * succ m = n * m + n
opaque_hint (hiding mul)
---------- comm, distr, assoc, identity
theorem mul_zero_left (n:) : 0 * n = 0
:= induction_on n
(mul_zero_right 0)
(take m IH,
calc
0 * succ m = 0 * m + 0 : mul_succ_right _ _
... = 0 * m : add_zero_right _
... = 0 : IH)
theorem mul_succ_left (n m:) : (succ n) * m = (n * m) + m
:= induction_on m
(calc
succ n * 0 = 0 : mul_zero_right _
... = n * 0 : symm (mul_zero_right _)
... = n * 0 + 0 : symm (add_zero_right _))
(take k IH,
calc
succ n * succ k = (succ n * k) + succ n : mul_succ_right _ _
... = (n * k) + k + succ n : { IH }
... = (n * k) + (k + succ n) : add_assoc _ _ _
... = (n * k) + (n + succ k) : {add_comm_succ _ _}
... = (n * k) + n + succ k : symm (add_assoc _ _ _)
... = (n * succ k) + succ k : {symm (mul_succ_right n k)})
theorem mul_comm (n m:) : n * m = m * n
:= induction_on m
(trans (mul_zero_right _) (symm (mul_zero_left _)))
(take k IH,
calc
n * succ k = n * k + n : mul_succ_right _ _
... = k * n + n : {IH}
... = (succ k) * n : symm (mul_succ_left _ _))
theorem mul_add_distr_left (n m k : ) : (n + m) * k = n * k + m * k
:= induction_on k
(calc
(n + m) * 0 = 0 : mul_zero_right _
... = 0 + 0 : symm (add_zero_right _)
... = n * 0 + 0 : {symm (mul_zero_right _)}
... = n * 0 + m * 0 : {symm (mul_zero_right _)})
(take l IH, calc
(n + m) * succ l = (n + m) * l + (n + m) : mul_succ_right _ _
... = n * l + m * l + (n + m) : {IH}
... = n * l + m * l + n + m : symm (add_assoc _ _ _)
... = n * l + n + m * l + m : {add_right_comm _ _ _}
... = n * l + n + (m * l + m) : add_assoc _ _ _
... = n * succ l + (m * l + m) : {symm (mul_succ_right _ _)}
... = n * succ l + m * succ l : {symm (mul_succ_right _ _)})
theorem mul_add_distr_right (n m k : ) : n * (m + k) = n * m + n * k
:= calc
n * (m + k) = (m + k) * n : mul_comm _ _
... = m * n + k * n : mul_add_distr_left _ _ _
... = n * m + k * n : {mul_comm _ _}
... = n * m + n * k : {mul_comm _ _}
theorem mul_assoc (n m k:) : (n * m) * k = n * (m * k)
:= induction_on k
(calc
(n * m) * 0 = 0 : mul_zero_right _
... = n * 0 : symm (mul_zero_right _)
... = n * (m * 0) : {symm (mul_zero_right _)})
(take l IH,
calc
(n * m) * succ l = (n * m) * l + n * m : mul_succ_right _ _
... = n * (m * l) + n * m : {IH}
... = n * (m * l + m) : symm (mul_add_distr_right _ _ _)
... = n * (m * succ l) : {symm (mul_succ_right _ _)})
theorem mul_comm_left (n m k : ) : n * (m * k) = m * (n * k)
:= left_comm mul_comm mul_assoc n m k
theorem mul_comm_right (n m k : ) : n * m * k = n * k * m
:= right_comm mul_comm mul_assoc n m k
theorem mul_one_right (n : ) : n * 1 = n
:= calc
n * 1 = n * 0 + n : mul_succ_right n 0
... = 0 + n : {mul_zero_right n}
... = n : add_zero_left n
theorem mul_one_left (n : ) : 1 * n = n
:= calc
1 * n = n * 1 : mul_comm _ _
... = n : mul_one_right n
---------- inversion
theorem mul_eq_zero {n m : } (H : n * m = 0) : n = 0 m = 0
:=
discriminate
(take Hn : n = 0, or_inl Hn)
(take (k : ),
assume (Hk : n = succ k),
discriminate
(take (Hm : m = 0), or_inr Hm)
(take (l : ),
assume (Hl : m = succ l),
have Heq : succ (k * succ l + l) = n * m, from
symm (calc
n * m = n * succ l : { Hl }
... = succ k * succ l : { Hk }
... = k * succ l + succ l : mul_succ_left _ _
... = succ (k * succ l + l) : add_succ_right _ _),
absurd_elim _ (trans Heq H) (succ_ne_zero _)))
-- see more under "positivity" below
-------------------------------------------------- le
definition le (n m:) : Prop := ∃k, n + k = m
infix `<=`:50 := le
infix `≤`:50 := le
theorem le_intro {n m k : } (H : n + k = m) : n ≤ m
:= exists_intro k H
theorem le_elim {n m : } (H : n ≤ m) : ∃ k, n + k = m
:= H
opaque_hint (hiding le)
---------- partial order (totality is part of lt)
theorem le_intro2 (n m : ) : n ≤ n + m
:= le_intro (refl (n + m))
theorem le_refl (n : ) : n ≤ n
:= le_intro (add_zero_right n)
theorem zero_le (n : ) : 0 ≤ n
:= le_intro (add_zero_left n)
theorem le_zero {n : } (H : n ≤ 0) : n = 0
:=
obtain (k : ) (Hk : n + k = 0), from le_elim H,
add_eq_zero_left Hk
theorem not_succ_zero_le (n : ) : ¬ succ n ≤ 0
:= assume H : succ n ≤ 0,
have H2 : succ n = 0, from le_zero H,
absurd H2 (succ_ne_zero n)
theorem le_zero_inv {n : } (H : n ≤ 0) : n = 0
:= obtain (k : ) (Hk : n + k = 0), from le_elim H,
add_eq_zero_left Hk
theorem le_trans {n m k : } (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k
:= obtain (l1 : ) (Hl1 : n + l1 = m), from le_elim H1,
obtain (l2 : ) (Hl2 : m + l2 = k), from le_elim H2,
le_intro
(calc
n + (l1 + l2) = n + l1 + l2 : symm (add_assoc n l1 l2)
... = m + l2 : { Hl1 }
... = k : Hl2)
theorem le_antisym {n m : } (H1 : n ≤ m) (H2 : m ≤ n) : n = m
:= obtain (k : ) (Hk : n + k = m), from (le_elim H1),
obtain (l : ) (Hl : m + l = n), from (le_elim H2),
have L1 : k + l = 0, from
add_cancel_left
(calc
n + (k + l) = n + k + l : { symm (add_assoc n k l) }
... = m + l : { Hk }
... = n : Hl
... = n + 0 : symm (add_zero_right n)),
have L2 : k = 0, from add_eq_zero_left L1,
calc
n = n + 0 : symm (add_zero_right n)
... = n + k : { symm L2 }
... = m : Hk
---------- interaction with add
theorem add_le_left {n m : } (H : n ≤ m) (k : ) : k + n ≤ k + m
:= obtain (l : ) (Hl : n + l = m), from (le_elim H),
le_intro
(calc
k + n + l = k + (n + l) : add_assoc k n l
... = k + m : { Hl })
theorem add_le_right {n m : } (H : n ≤ m) (k : ) : n + k ≤ m + k
:= (add_comm k m) ▸ (add_comm k n) ▸ (add_le_left H k)
theorem add_le {n m k l : } (H1 : n ≤ k) (H2 : m ≤ l) : n + m ≤ k + l
:= le_trans (add_le_right H1 m) (add_le_left H2 k)
theorem add_le_left_inv {n m k : } (H : k + n ≤ k + m) : n ≤ m
:=
obtain (l : ) (Hl : k + n + l = k + m), from (le_elim H),
le_intro (add_cancel_left
(calc
k + (n + l) = k + n + l : symm (add_assoc k n l)
... = k + m : Hl))
theorem add_le_right_inv {n m k : } (H : n + k ≤ m + k) : n ≤ m
:= add_le_left_inv (add_comm m k ▸ add_comm n k ▸ H)
---------- interaction with succ and pred
theorem succ_le {n m : } (H : n ≤ m) : succ n ≤ succ m
:= add_one m ▸ add_one n ▸ add_le_right H 1
theorem succ_le_cancel {n m : } (H : succ n ≤ succ m) : n ≤ m
:= add_le_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
theorem self_le_succ (n : ) : n ≤ succ n
:= le_intro (add_one n)
theorem le_imp_le_succ {n m : } (H : n ≤ m) : n ≤ succ m
:= le_trans H (self_le_succ m)
theorem succ_le_left_or {n m : } (H : n ≤ m) : succ n ≤ m n = m
:= obtain (k : ) (Hk : n + k = m), from (le_elim H),
discriminate
(assume H3 : k = 0,
have Heq : n = m,
from calc
n = n + 0 : (add_zero_right n)⁻¹
... = n + k : {H3⁻¹}
... = m : Hk,
or_inr Heq)
(take l:,
assume H3 : k = succ l,
have Hlt : succ n ≤ m, from
(le_intro
(calc
succ n + l = n + succ l : add_move_succ n l
... = n + k : {H3⁻¹}
... = m : Hk)),
or_inl Hlt)
theorem succ_le_left {n m : } (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
:= resolve_left (succ_le_left_or H1) H2
theorem succ_le_right_inv {n m : } (H : n ≤ succ m) : n ≤ m n = succ m
:= or_imp_or (succ_le_left_or H)
(take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
(take H2 : n = succ m, H2)
theorem succ_le_left_inv {n m : } (H : succ n ≤ m) : n ≤ m ∧ n ≠ m
:= obtain (k : ) (H2 : succ n + k = m), from (le_elim H),
and_intro
(have H3 : n + succ k = m,
from calc
n + succ k = succ n + k : symm (add_move_succ n k)
... = m : H2,
show n ≤ m, from le_intro H3)
(assume H3 : n = m,
have H4 : succ n ≤ n, from subst (symm H3) H,
have H5 : succ n = n, from le_antisym H4 (self_le_succ n),
show false, from absurd H5 (succ_ne_self n))
theorem le_pred_self (n : ) : pred n ≤ n
:= case n
(subst (symm pred_zero) (le_refl 0))
(take k : , subst (symm (pred_succ k)) (self_le_succ k))
theorem pred_le {n m : } (H : n ≤ m) : pred n ≤ pred m
:= discriminate
(take Hn : n = 0,
have H2 : pred n = 0,
from calc
pred n = pred 0 : {Hn}
... = 0 : pred_zero,
subst (symm H2) (zero_le (pred m)))
(take k : ,
assume Hn : n = succ k,
obtain (l : ) (Hl : n + l = m), from le_elim H,
have H2 : pred n + l = pred m,
from calc
pred n + l = pred (succ k) + l : {Hn}
... = k + l : {pred_succ k}
... = pred (succ (k + l)) : symm (pred_succ (k + l))
... = pred (succ k + l) : {symm (add_succ_left k l)}
... = pred (n + l) : {symm Hn}
... = pred m : {Hl},
le_intro H2)
theorem pred_le_left_inv {n m : } (H : pred n ≤ m) : n ≤ m n = succ m
:= discriminate
(take Hn : n = 0,
or_inl (subst (symm Hn) (zero_le m)))
(take k : ,
assume Hn : n = succ k,
have H2 : pred n = k,
from calc
pred n = pred (succ k) : {Hn}
... = k : pred_succ k,
have H3 : k ≤ m, from subst H2 H,
have H4 : succ k ≤ m k = m, from succ_le_left_or H3,
show n ≤ m n = succ m, from
or_imp_or H4
(take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
(take H5 : k = m, show n = succ m, from subst H5 Hn))
-- ### interaction with successor and predecessor
theorem le_imp_succ_le_or_eq {n m : } (H : n ≤ m) : succ n ≤ m n = m
:=
obtain (k : ) (Hk : n + k = m), from (le_elim H),
discriminate
(assume H3 : k = 0,
have Heq : n = m,
from calc
n = n + 0 : symm (add_zero_right n)
... = n + k : {symm H3}
... = m : Hk,
or_inr Heq)
(take l : nat,
assume H3 : k = succ l,
have Hlt : succ n ≤ m, from
(le_intro
(calc
succ n + l = n + succ l : add_move_succ n l
... = n + k : {symm H3}
... = m : Hk)),
or_inl Hlt)
theorem le_ne_imp_succ_le {n m : } (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
:= resolve_left (le_imp_succ_le_or_eq H1) H2
theorem le_succ_imp_le_or_eq {n m : } (H : n ≤ succ m) : n ≤ m n = succ m
:= imp_or_left (le_imp_succ_le_or_eq H)
(take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
theorem succ_le_imp_le_and_ne {n m : } (H : succ n ≤ m) : n ≤ m ∧ n ≠ m
:=
and_intro
(le_trans (self_le_succ n) H)
(assume H2 : n = m,
have H3 : succ n ≤ n, from subst (symm H2) H,
have H4 : succ n = n, from le_antisym H3 (self_le_succ n),
show false, from absurd H4 (succ_ne_self n))
theorem pred_le_self (n : ) : pred n ≤ n
:=
case n
(subst (symm pred_zero) (le_refl 0))
(take k : nat, subst (symm (pred_succ k)) (self_le_succ k))
theorem pred_le_imp_le_or_eq {n m : } (H : pred n ≤ m) : n ≤ m n = succ m
:=
discriminate
(take Hn : n = 0,
or_inl (subst (symm Hn) (zero_le m)))
(take k : nat,
assume Hn : n = succ k,
have H2 : pred n = k,
from calc
pred n = pred (succ k) : {Hn}
... = k : pred_succ k,
have H3 : k ≤ m, from subst H2 H,
have H4 : succ k ≤ m k = m, from le_imp_succ_le_or_eq H3,
show n ≤ m n = succ m, from
or_imp_or H4
(take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
(take H5 : k = m, show n = succ m, from subst H5 Hn))
---------- interaction with mul
theorem mul_le_left {n m : } (H : n ≤ m) (k : ) : k * n ≤ k * m
:=
obtain (l : ) (Hl : n + l = m), from (le_elim H),
induction_on k
(have H2 : 0 * n = 0 * m,
from calc
0 * n = 0 : mul_zero_left n
... = 0 * m : symm (mul_zero_left m),
show 0 * n ≤ 0 * m, from subst H2 (le_refl (0 * n)))
(take (l : ),
assume IH : l * n ≤ l * m,
have H2 : l * n + n ≤ l * m + m, from add_le IH H,
have H3 : succ l * n ≤ l * m + m, from subst (symm (mul_succ_left l n)) H2,
show succ l * n ≤ succ l * m, from subst (symm (mul_succ_left l m)) H3)
theorem mul_le_right {n m : } (H : n ≤ m) (k : ) : n * k ≤ m * k
:= mul_comm k m ▸ mul_comm k n ▸ (mul_le_left H k)
theorem mul_le {n m k l : } (H1 : n ≤ k) (H2 : m ≤ l) : n * m ≤ k * l
:= le_trans (mul_le_right H1 m) (mul_le_left H2 k)
-- mul_le_[left|right]_inv below
-------------------------------------------------- lt
definition lt (n m : ) := succ n ≤ m
infix `<`:50 := lt
theorem lt_intro {n m k : } (H : succ n + k = m) : n < m
:= le_intro H
theorem lt_elim {n m : } (H : n < m) : ∃ k, succ n + k = m
:= le_elim H
theorem lt_intro2 (n m : ) : n < n + succ m
:= lt_intro (add_move_succ n m)
-------------------------------------------------- ge, gt
definition ge (n m : ) := m ≤ n
infix `>=`:50 := ge
infix `≥`:50 := ge
definition gt (n m : ) := m < n
infix `>`:50 := gt
---------- basic facts
theorem lt_ne {n m : } (H : n < m) : n ≠ m
:= and_elim_right (succ_le_left_inv H)
theorem lt_irrefl (n : ) : ¬ n < n
:= assume H : n < n, absurd (refl n) (lt_ne H)
theorem lt_zero (n : ) : 0 < succ n
:= succ_le (zero_le n)
theorem lt_zero_inv (n : ) : ¬ n < 0
:= assume H : n < 0,
have H2 : succ n = 0, from le_zero_inv H,
absurd H2 (succ_ne_zero n)
theorem lt_positive {n m : } (H : n < m) : ∃k, m = succ k
:= discriminate
(take (Hm : m = 0), absurd_elim _ (subst Hm H) (lt_zero_inv n))
(take (l : ) (Hm : m = succ l), exists_intro l Hm)
---------- interaction with le
theorem lt_imp_le_succ {n m : } (H : n < m) : succ n ≤ m
:= H
theorem le_succ_imp_lt {n m : } (H : succ n ≤ m) : n < m
:= H
theorem self_lt_succ (n : ) : n < succ n
:= le_refl (succ n)
theorem lt_imp_le {n m : } (H : n < m) : n ≤ m
:= and_elim_left (succ_le_imp_le_and_ne H)
theorem le_imp_lt_or_eq {n m : } (H : n ≤ m) : n < m n = m
:= le_imp_succ_le_or_eq H
theorem le_ne_imp_lt {n m : } (H1 : n ≤ m) (H2 : n ≠ m) : n < m
:= le_ne_imp_succ_le H1 H2
theorem le_imp_lt_succ {n m : } (H : n ≤ m) : n < succ m
:= succ_le H
theorem lt_succ_imp_le {n m : } (H : n < succ m) : n ≤ m
:= succ_le_cancel H
---------- trans, antisym
theorem lt_le_trans {n m k : } (H1 : n < m) (H2 : m ≤ k) : n < k
:= le_trans H1 H2
theorem le_lt_trans {n m k : } (H1 : n ≤ m) (H2 : m < k) : n < k
:= le_trans (succ_le H1) H2
theorem lt_trans {n m k : } (H1 : n < m) (H2 : m < k) : n < k
:= lt_le_trans H1 (lt_imp_le H2)
theorem le_imp_not_gt {n m : } (H : n ≤ m) : ¬ n > m
:= assume H2 : m < n, absurd (le_lt_trans H H2) (lt_irrefl n)
theorem lt_imp_not_ge {n m : } (H : n < m) : ¬ n ≥ m
:= assume H2 : m ≤ n, absurd (lt_le_trans H H2) (lt_irrefl n)
theorem lt_antisym {n m : } (H : n < m) : ¬ m < n
:= le_imp_not_gt (lt_imp_le H)
---------- interaction with add
theorem add_lt_left {n m : } (H : n < m) (k : ) : k + n < k + m
:= add_succ_right k n ▸ add_le_left H k
theorem add_lt_right {n m : } (H : n < m) (k : ) : n + k < m + k
:= add_comm k m ▸ add_comm k n ▸ add_lt_left H k
theorem add_le_lt {n m k l : } (H1 : n ≤ k) (H2 : m < l) : n + m < k + l
:= le_lt_trans (add_le_right H1 m) (add_lt_left H2 k)
theorem add_lt_le {n m k l : } (H1 : n < k) (H2 : m ≤ l) : n + m < k + l
:= lt_le_trans (add_lt_right H1 m) (add_le_left H2 k)
theorem add_lt {n m k l : } (H1 : n < k) (H2 : m < l) : n + m < k + l
:= add_lt_le H1 (lt_imp_le H2)
theorem add_lt_left_inv {n m k : } (H : k + n < k + m) : n < m
:= add_le_left_inv (add_succ_right k n⁻¹ ▸ H)
theorem add_lt_right_inv {n m k : } (H : n + k < m + k) : n < m
:= add_lt_left_inv (add_comm m k ▸ add_comm n k ▸ H)
---------- interaction with succ (see also the interaction with le)
theorem succ_lt {n m : } (H : n < m) : succ n < succ m
:= add_one m ▸ add_one n ▸ add_lt_right H 1
theorem succ_lt_inv {n m : } (H : succ n < succ m) : n < m
:= add_lt_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
theorem lt_self_succ (n : ) : n < succ n
:= le_refl (succ n)
theorem succ_lt_right {n m : } (H : n < m) : n < succ m
:= lt_trans H (lt_self_succ m)
---------- totality of lt and le
theorem le_or_lt (n m : ) : n ≤ m m < n
:= induction_on n
(or_inl (zero_le m))
(take (k : ),
assume IH : k ≤ m m < k,
or_elim IH
(assume H : k ≤ m,
obtain (l : ) (Hl : k + l = m), from le_elim H,
discriminate
(assume H2 : l = 0,
have H3 : m = k,
from calc
m = k + l : symm Hl
... = k + 0 : {H2}
... = k : add_zero_right k,
have H4 : m < succ k, from subst H3 (lt_self_succ m),
or_inr H4)
(take l2 : ,
assume H2 : l = succ l2,
have H3 : succ k + l2 = m,
from calc
succ k + l2 = k + succ l2 : add_move_succ k l2
... = k + l : {symm H2}
... = m : Hl,
or_inl (le_intro H3)))
(assume H : m < k, or_inr (succ_lt_right H)))
theorem trichotomy_alt (n m : ) : (n < m n = m) m < n
:= or_imp_or (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H)
theorem trichotomy (n m : ) : n < m n = m m < n
:= iff_elim_left (or_assoc _ _ _) (trichotomy_alt n m)
theorem le_total (n m : ) : n ≤ m m ≤ n
:= or_imp_or (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H)
-- interaction with mul under "positivity"
theorem strong_induction_on {P : → Prop} (n : ) (IH : ∀n, (∀m, m < n → P m) → P n) : P n
:= have stronger : ∀k, k ≤ n → P k, from
induction_on n
(take (k : ),
assume H : k ≤ 0,
have H2 : k = 0, from le_zero_inv H,
have H3 : ∀m, m < k → P m, from
(take m : ,
assume H4 : m < k,
have H5 : m < 0, from subst H2 H4,
absurd_elim _ H5 (lt_zero_inv m)),
show P k, from IH k H3)
(take l : ,
assume IHl : ∀k, k ≤ l → P k,
take k : ,
assume H : k ≤ succ l,
or_elim (succ_le_right_inv H)
(assume H2 : k ≤ l, show P k, from IHl k H2)
(assume H2 : k = succ l,
have H3 : ∀m, m < k → P m, from
(take m : ,
assume H4 : m < k,
have H5 : m ≤ l, from lt_succ_imp_le (subst H2 H4),
show P m, from IHl m H5),
show P k, from IH k H3)),
stronger n (le_refl n)
theorem case_strong_induction_on {P : → Prop} (a : ) (H0 : P 0) (Hind : ∀(n : ), (∀m, m ≤ n → P m) → P (succ n)) : P a
:= strong_induction_on a
(take n, case n
(assume H : (∀m, m < 0 → P m), H0)
(take n, assume H : (∀m, m < succ n → P m),
Hind n (take m, assume H1 : m ≤ n, H m (le_imp_lt_succ H1))))
theorem add_eq_self {n m : } (H : n + m = n) : m = 0
:= discriminate
(take Hm : m = 0, Hm)
(take k : ,
assume Hm : m = succ k,
have H2 : succ n + k = n,
from calc
succ n + k = n + succ k : add_move_succ n k
... = n + m : {symm Hm}
... = n : H,
have H3 : n < n, from lt_intro H2,
have H4 : n ≠ n, from lt_ne H3,
absurd_elim _ (refl n) H4)
-------------------------------------------------- positivity
-- we use " _ > 0" as canonical way of denoting that a number is positive
---------- basic
theorem zero_or_positive (n : ) : n = 0 n > 0
:= or_imp_or (or_swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) (take H : n > 0, H)
theorem succ_positive {n m : } (H : n = succ m) : n > 0
:= subst (symm H) (lt_zero m)
theorem ne_zero_positive {n : } (H : n ≠ 0) : n > 0
:= or_elim (zero_or_positive n) (take H2 : n = 0, absurd_elim _ H2 H) (take H2 : n > 0, H2)
theorem pos_imp_eq_succ {n : } (H : n > 0) : ∃l, n = succ l
:= discriminate
(take H2, absurd_elim _ (subst H2 H) (lt_irrefl 0))
(take l Hl, exists_intro l Hl)
theorem add_positive_right (n : ) {k : } (H : k > 0) : n + k > n
:= obtain (l : ) (Hl : k = succ l), from pos_imp_eq_succ H,
subst (symm Hl) (lt_intro2 n l)
theorem add_positive_left (n : ) {k : } (H : k > 0) : k + n > n
:= subst (add_comm n k) (add_positive_right n H)
-- Positivity
-- ---------
--
-- Writing "t > 0" is the preferred way to assert that a natural number is positive.
-- ### basic
-- See also succ_pos.
theorem succ_pos (n : ) : 0 < succ n
:= succ_le (zero_le n)
theorem case_zero_pos {P : → Prop} (y : ) (H0 : P 0) (H1 : ∀y, y > 0 → P y) : P y
:= case y H0 (take y', H1 _ (succ_pos _))
theorem zero_or_pos (n : ) : n = 0 n > 0
:= imp_or_left (or_swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H)
theorem succ_imp_pos {n m : } (H : n = succ m) : n > 0
:= subst (symm H) (succ_pos m)
theorem ne_zero_pos {n : } (H : n ≠ 0) : n > 0
:= or_elim (zero_or_pos n) (take H2 : n = 0, absurd_elim _ H2 H) (take H2 : n > 0, H2)
theorem add_pos_right (n : ) {k : } (H : k > 0) : n + k > n
:= subst (add_zero_right n) (add_lt_left H n)
theorem add_pos_left (n : ) {k : } (H : k > 0) : k + n > n
:= subst (add_comm n k) (add_pos_right n H)
---------- mul
theorem mul_positive {n m : } (Hn : n > 0) (Hm : m > 0) : n * m > 0
:= obtain (k : ) (Hk : n = succ k), from pos_imp_eq_succ Hn,
obtain (l : ) (Hl : m = succ l), from pos_imp_eq_succ Hm,
succ_positive (calc
n * m = succ k * m : {Hk}
... = succ k * succ l : {Hl}
... = succ k * l + succ k : mul_succ_right (succ k) l
... = succ (succ k * l + k) : add_succ_right _ _)
theorem mul_positive_inv_left {n m : } (H : n * m > 0) : n > 0
:= discriminate
(assume H2 : n = 0,
have H3 : n * m = 0,
from calc
n * m = 0 * m : {H2}
... = 0 : mul_zero_left m,
have H4 : 0 > 0, from subst H3 H,
absurd_elim _ H4 (lt_irrefl 0))
(take l : ,
assume Hl : n = succ l,
subst (symm Hl) (lt_zero l))
theorem mul_positive_inv_right {n m : } (H : n * m > 0) : m > 0
:= mul_positive_inv_left (subst (mul_comm n m) H)
theorem mul_left_inj {n m k : } (Hn : n > 0) (H : n * m = n * k) : m = k
:=
have general : ∀m, n * m = n * k → m = k, from
induction_on k
(take m:,
assume H : n * m = n * 0,
have H2 : n * m = 0,
from calc
n * m = n * 0 : H
... = 0 : mul_zero_right n,
have H3 : n = 0 m = 0, from mul_eq_zero H2,
resolve_right H3 (ne_symm (lt_ne Hn)))
(take (l : ),
assume (IH : ∀ m, n * m = n * l → m = l),
take (m : ),
assume (H : n * m = n * succ l),
have H2 : n * succ l > 0, from mul_positive Hn (lt_zero l),
have H3 : m > 0, from mul_positive_inv_right (subst (symm H) H2),
obtain (l2:) (Hm : m = succ l2), from pos_imp_eq_succ H3,
have H4 : n * l2 + n = n * l + n,
from calc
n * l2 + n = n * succ l2 : symm (mul_succ_right n l2)
... = n * m : {symm Hm}
... = n * succ l : H
... = n * l + n : mul_succ_right n l,
have H5 : n * l2 = n * l, from add_cancel_right H4,
calc
m = succ l2 : Hm
... = succ l : {IH l2 H5}),
general m H
theorem mul_right_inj {n m k : } (Hm : m > 0) (H : n * m = k * m) : n = k
:= mul_left_inj Hm (subst (mul_comm k m) (subst (mul_comm n m) H))
-- mul_eq_one below
---------- interaction of mul with le and lt
theorem mul_lt_left {n m k : } (Hk : k > 0) (H : n < m) : k * n < k * m
:=
have H2 : k * n < k * n + k, from add_positive_right (k * n) Hk,
have H3 : k * n + k ≤ k * m, from subst (mul_succ_right k n) (mul_le_left H k),
lt_le_trans H2 H3
theorem mul_lt_right {n m k : } (Hk : k > 0) (H : n < m) : n * k < m * k
:= subst (mul_comm k m) (subst (mul_comm k n) (mul_lt_left Hk H))
theorem mul_le_lt {n m k l : } (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) : n * m < k * l
:= le_lt_trans (mul_le_right H1 m) (mul_lt_left Hk H2)
theorem mul_lt_le {n m k l : } (Hl : l > 0) (H1 : n < k) (H2 : m ≤ l) : n * m < k * l
:= le_lt_trans (mul_le_left H2 n) (mul_lt_right Hl H1)
theorem mul_lt {n m k l : } (H1 : n < k) (H2 : m < l) : n * m < k * l
:=
have H3 : n * m ≤ k * m, from mul_le_right (lt_imp_le H1) m,
have H4 : k * m < k * l, from mul_lt_left (le_lt_trans (zero_le n) H1) H2,
le_lt_trans H3 H4
theorem mul_lt_left_inv {n m k : } (H : k * n < k * m) : n < m
:=
have general : ∀ m, k * n < k * m → n < m, from
induction_on n
(take m : ,
assume H2 : k * 0 < k * m,
have H3 : 0 < k * m, from mul_zero_right k ▸ H2,
show 0 < m, from mul_positive_inv_right H3)
(take l : ,
assume IH : ∀ m, k * l < k * m → l < m,
take m : ,
assume H2 : k * succ l < k * m,
have H3 : 0 < k * m, from le_lt_trans (zero_le _) H2,
have H4 : 0 < m, from mul_positive_inv_right H3,
obtain (l2 : ) (Hl2 : m = succ l2), from pos_imp_eq_succ H4,
have H5 : k * l + k < k * m, from mul_succ_right k l ▸ H2,
have H6 : k * l + k < k * succ l2, from Hl2 ▸ H5,
have H7 : k * l + k < k * l2 + k, from mul_succ_right k l2 ▸ H6,
have H8 : k * l < k * l2, from add_lt_right_inv H7,
have H9 : l < l2, from IH l2 H8,
have H10 : succ l < succ l2, from succ_lt H9,
show succ l < m, from Hl2⁻¹ ▸ H10),
general m H
theorem mul_lt_right_inv {n m k : } (H : n * k < m * k) : n < m
:= mul_lt_left_inv (mul_comm m k ▸ mul_comm n k ▸ H)
theorem mul_le_left_inv {n m k : } (H : succ k * n ≤ succ k * m) : n ≤ m
:=
have H2 : succ k * n < succ k * m + succ k, from le_lt_trans H (lt_intro2 _ _),
have H3 : succ k * n < succ k * succ m, from subst (symm (mul_succ_right (succ k) m)) H2,
have H4 : n < succ m, from mul_lt_left_inv H3,
show n ≤ m, from lt_succ_imp_le H4
theorem mul_le_right_inv {n m k : } (H : n * succ m ≤ k * succ m) : n ≤ k
:= mul_le_left_inv (subst (mul_comm k (succ m)) (subst (mul_comm n (succ m)) H))
theorem mul_eq_one_left {n m : } (H : n * m = 1) : n = 1
:=
have H2 : n * m > 0, from subst (symm H) (lt_zero 0),
have H3 : n > 0, from mul_positive_inv_left H2,
have H4 : m > 0, from mul_positive_inv_right H2,
or_elim (le_or_lt n 1)
(assume H5 : n ≤ 1,
show n = 1, from le_antisym H5 H3)
(assume H5 : n > 1,
have H6 : n * m ≥ 2 * 1, from mul_le H5 H4,
have H7 : 1 ≥ 2, from subst (mul_one_right 2) (subst H H6),
absurd_elim _ (self_lt_succ 1) (le_imp_not_gt H7))
theorem mul_eq_one_right {n m : } (H : n * m = 1) : m = 1
:= mul_eq_one_left (subst (mul_comm n m) H)
theorem mul_eq_one {n m : } (H : n * m = 1) : n = 1 ∧ m = 1
:= and_intro (mul_eq_one_left H) (mul_eq_one_right H)
opaque_hint (hiding lt)
-------------------------------------------------- sub
definition sub (n m : ) : := nat_rec n (fun m x, pred x) m
infixl `-`:65 := sub
theorem sub_zero_right (n : ) : n - 0 = n
theorem sub_succ_right (n m : ) : n - succ m = pred (n - m)
opaque_hint (hiding sub)
theorem sub_zero_left (n : ) : 0 - n = 0
:= induction_on n (sub_zero_right 0)
(take k : ,
assume IH : 0 - k = 0,
calc
0 - succ k = pred (0 - k) : sub_succ_right 0 k
... = pred 0 : {IH}
... = 0 : pred_zero)
theorem sub_succ_succ (n m : ) : succ n - succ m = n - m
:= induction_on m
(calc
succ n - 1 = pred (succ n - 0) : sub_succ_right (succ n) 0
... = pred (succ n) : {sub_zero_right (succ n)}
... = n : pred_succ n
... = n - 0 : symm (sub_zero_right n))
(take k : ,
assume IH : succ n - succ k = n - k,
calc
succ n - succ (succ k) = pred (succ n - succ k) : sub_succ_right (succ n) (succ k)
... = pred (n - k) : {IH}
... = n - succ k : symm (sub_succ_right n k))
theorem sub_one (n : ) : n - 1 = pred n
:= calc
n - 1 = pred (n - 0) : sub_succ_right n 0
... = pred n : {sub_zero_right n}
theorem sub_self (n : ) : n - n = 0
:= induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH)
theorem sub_add_add_right (n m k : ) : (n + k) - (m + k) = n - m
:= induction_on k
(calc
(n + 0) - (m + 0) = n - (m + 0) : {add_zero_right _}
... = n - m : {add_zero_right _})
(take l : ,
assume IH : (n + l) - (m + l) = n - m,
calc
(n + succ l) - (m + succ l) = succ (n + l) - (m + succ l) : {add_succ_right _ _}
... = succ (n + l) - succ (m + l) : {add_succ_right _ _}
... = (n + l) - (m + l) : sub_succ_succ _ _
... = n - m : IH)
theorem sub_add_add_left (n m k : ) : (k + n) - (k + m) = n - m
:= subst (add_comm m k) (subst (add_comm n k) (sub_add_add_right n m k))
theorem sub_add_left (n m : ) : n + m - m = n
:= induction_on m
(subst (symm (add_zero_right n)) (sub_zero_right n))
(take k : ,
assume IH : n + k - k = n,
calc
n + succ k - succ k = succ (n + k) - succ k : {add_succ_right n k}
... = n + k - k : sub_succ_succ _ _
... = n : IH)
theorem sub_sub (n m k : ) : n - m - k = n - (m + k)
:= induction_on k
(calc
n - m - 0 = n - m : sub_zero_right _
... = n - (m + 0) : {symm (add_zero_right m)})
(take l : ,
assume IH : n - m - l = n - (m + l),
calc
n - m - succ l = pred (n - m - l) : sub_succ_right (n - m) l
... = pred (n - (m + l)) : {IH}
... = n - succ (m + l) : symm (sub_succ_right n (m + l))
... = n - (m + succ l) : {symm (add_succ_right m l)})
theorem succ_sub_sub (n m k : ) : succ n - m - succ k = n - m - k
:= calc
succ n - m - succ k = succ n - (m + succ k) : sub_sub _ _ _
... = succ n - succ (m + k) : {add_succ_right m k}
... = n - (m + k) : sub_succ_succ _ _
... = n - m - k : symm (sub_sub n m k)
theorem sub_add_right_eq_zero (n m : ) : n - (n + m) = 0
:= calc
n - (n + m) = n - n - m : symm (sub_sub n n m)
... = 0 - m : {sub_self n}
... = 0 : sub_zero_left m
theorem sub_comm (m n k : ) : m - n - k = m - k - n
:= calc
m - n - k = m - (n + k) : sub_sub m n k
... = m - (k + n) : {add_comm n k}
... = m - k - n : symm (sub_sub m k n)
theorem succ_sub_one (n : ) : succ n - 1 = n
:= sub_succ_succ n 0 ⬝ sub_zero_right n
---------- mul
theorem mul_pred_left (n m : ) : pred n * m = n * m - m
:= induction_on n
(calc
pred 0 * m = 0 * m : {pred_zero}
... = 0 : mul_zero_left _
... = 0 - m : symm (sub_zero_left m)
... = 0 * m - m : {symm (mul_zero_left m)})
(take k : ,
assume IH : pred k * m = k * m - m,
calc
pred (succ k) * m = k * m : {pred_succ k}
... = k * m + m - m : symm (sub_add_left _ _)
... = succ k * m - m : {symm (mul_succ_left k m)})
theorem mul_pred_right (n m : ) : n * pred m = n * m - n
:= calc n * pred m = pred m * n : mul_comm _ _
... = m * n - n : mul_pred_left m n
... = n * m - n : {mul_comm m n}
theorem mul_sub_distr_left (n m k : ) : (n - m) * k = n * k - m * k
:= induction_on m
(calc
(n - 0) * k = n * k : {sub_zero_right n}
... = n * k - 0 : symm (sub_zero_right _)
... = n * k - 0 * k : {symm (mul_zero_left _)})
(take l : ,
assume IH : (n - l) * k = n * k - l * k,
calc
(n - succ l) * k = pred (n - l) * k : {sub_succ_right n l}
... = (n - l) * k - k : mul_pred_left _ _
... = n * k - l * k - k : {IH}
... = n * k - (l * k + k) : sub_sub _ _ _
... = n * k - (succ l * k) : {symm (mul_succ_left l k)})
theorem mul_sub_distr_right (n m k : ) : n * (m - k) = n * m - n * k
:= calc
n * (m - k) = (m - k) * n : mul_comm _ _
... = m * n - k * n : mul_sub_distr_left _ _ _
... = n * m - k * n : {mul_comm _ _}
... = n * m - n * k : {mul_comm _ _}
-------------------------------------------------- max, min, iteration, maybe: sub, div
theorem succ_sub {m n : } : m ≥ n → succ m - n = succ (m - n)
:= sub_induction n m
(take k,
assume H : 0 ≤ k,
calc
succ k - 0 = succ k : sub_zero_right (succ k)
... = succ (k - 0) : {symm (sub_zero_right k)})
(take k,
assume H : succ k ≤ 0,
absurd_elim _ H (not_succ_zero_le k))
(take k l,
assume IH : k ≤ l → succ l - k = succ (l - k),
take H : succ k ≤ succ l,
calc
succ (succ l) - succ k = succ l - k : sub_succ_succ (succ l) k
... = succ (l - k) : IH (succ_le_cancel H)
... = succ (succ l - succ k) : {symm (sub_succ_succ l k)})
theorem le_imp_sub_eq_zero {n m : } (H : n ≤ m) : n - m = 0
:= obtain (k : ) (Hk : n + k = m), from le_elim H, subst Hk (sub_add_right_eq_zero n k)
theorem add_sub_le {n m : } : n ≤ m → n + (m - n) = m
:= sub_induction n m
(take k,
assume H : 0 ≤ k,
calc
0 + (k - 0) = k - 0 : add_zero_left (k - 0)
... = k : sub_zero_right k)
(take k, assume H : succ k ≤ 0, absurd_elim _ H (not_succ_zero_le k))
(take k l,
assume IH : k ≤ l → k + (l - k) = l,
take H : succ k ≤ succ l,
calc
succ k + (succ l - succ k) = succ k + (l - k) : {sub_succ_succ l k}
... = succ (k + (l - k)) : add_succ_left k (l - k)
... = succ l : {IH (succ_le_cancel H)})
theorem add_sub_ge_left {n m : } : n ≥ m → n - m + m = n
:= subst (add_comm m (n - m)) add_sub_le
theorem add_sub_ge {n m : } (H : n ≥ m) : n + (m - n) = n
:= calc
n + (m - n) = n + 0 : {le_imp_sub_eq_zero H}
... = n : add_zero_right n
theorem add_sub_le_left {n m : } : n ≤ m → n - m + m = m
:= subst (add_comm m (n - m)) add_sub_ge
theorem le_add_sub_left (n m : ) : n ≤ n + (m - n)
:= or_elim (le_total n m)
(assume H : n ≤ m, subst (symm (add_sub_le H)) H)
(assume H : m ≤ n, subst (symm (add_sub_ge H)) (le_refl n))
theorem le_add_sub_right (n m : ) : m ≤ n + (m - n)
:= or_elim (le_total n m)
(assume H : n ≤ m, subst (symm (add_sub_le H)) (le_refl m))
(assume H : m ≤ n, subst (symm (add_sub_ge H)) H)
theorem sub_split {P : → Prop} {n m : } (H1 : n ≤ m → P 0) (H2 : ∀k, m + k = n -> P k)
: P (n - m)
:= or_elim (le_total n m)
(assume H3 : n ≤ m, subst (symm (le_imp_sub_eq_zero H3)) (H1 H3))
(assume H3 : m ≤ n, H2 (n - m) (add_sub_le H3))
theorem sub_le_self (n m : ) : n - m ≤ n
:=
sub_split
(assume H : n ≤ m, zero_le n)
(take k : , assume H : m + k = n, le_intro (subst (add_comm m k) H))
theorem le_elim_sub (n m : ) (H : n ≤ m) : ∃k, m - k = n
:=
obtain (k : ) (Hk : n + k = m), from le_elim H,
exists_intro k
(calc
m - k = n + k - k : {symm Hk}
... = n : sub_add_left n k)
theorem add_sub_assoc {m k : } (H : k ≤ m) (n : ) : n + m - k = n + (m - k)
:= have l1 : k ≤ m → n + m - k = n + (m - k), from
sub_induction k m
(take m : ,
assume H : 0 ≤ m,
calc
n + m - 0 = n + m : sub_zero_right (n + m)
... = n + (m - 0) : {symm (sub_zero_right m)})
(take k : , assume H : succ k ≤ 0, absurd_elim _ H (not_succ_zero_le k))
(take k m,
assume IH : k ≤ m → n + m - k = n + (m - k),
take H : succ k ≤ succ m,
calc
n + succ m - succ k = succ (n + m) - succ k : {add_succ_right n m}
... = n + m - k : sub_succ_succ (n + m) k
... = n + (m - k) : IH (succ_le_cancel H)
... = n + (succ m - succ k) : {symm (sub_succ_succ m k)}),
l1 H
theorem sub_eq_zero_imp_le {n m : } : n - m = 0 → n ≤ m
:= sub_split
(assume H1 : n ≤ m, assume H2 : 0 = 0, H1)
(take k : ,
assume H1 : m + k = n,
assume H2 : k = 0,
have H3 : n = m, from subst (add_zero_right m) (subst H2 (symm H1)),
subst H3 (le_refl n))
theorem sub_sub_split {P : → Prop} {n m : } (H1 : ∀k, n = m + k -> P k 0)
(H2 : ∀k, m = n + k → P 0 k) : P (n - m) (m - n)
:= or_elim (le_total n m)
(assume H3 : n ≤ m,
le_imp_sub_eq_zero H3⁻¹ ▸ (H2 (m - n) (add_sub_le H3⁻¹)))
(assume H3 : m ≤ n,
le_imp_sub_eq_zero H3⁻¹ ▸ (H1 (n - m) (add_sub_le H3⁻¹)))
theorem sub_intro {n m k : } (H : n + m = k) : k - n = m
:= have H2 : k - n + n = m + n, from
calc
k - n + n = k : add_sub_ge_left (le_intro H)
... = n + m : symm H
... = m + n : add_comm n m,
add_cancel_right H2
theorem sub_lt {x y : } (xpos : x > 0) (ypos : y > 0) : x - y < x
:= obtain (x' : ) (xeq : x = succ x'), from pos_imp_eq_succ xpos,
obtain (y' : ) (yeq : y = succ y'), from pos_imp_eq_succ ypos,
have xsuby_eq : x - y = x' - y', from
calc
x - y = succ x' - y : {xeq}
... = succ x' - succ y' : {yeq}
... = x' - y' : sub_succ_succ _ _,
have H1 : x' - y' ≤ x', from sub_le_self _ _,
have H2 : x' < succ x', from self_lt_succ _,
show x - y < x, from xeq⁻¹ ▸ xsuby_eq⁻¹ ▸ le_lt_trans H1 H2
-- Max, min, iteration, and absolute difference
-- --------------------------------------------
definition max (n m : ) : := n + (m - n)
definition min (n m : ) : := m - (m - n)
theorem max_le {n m : } (H : n ≤ m) : n + (m - n) = m := add_sub_le H
theorem max_ge {n m : } (H : n ≥ m) : n + (m - n) = n := add_sub_ge H
theorem left_le_max (n m : ) : n ≤ n + (m - n) := le_add_sub_left n m
theorem right_le_max (n m : ) : m ≤ max n m := le_add_sub_right n m
-- ### absolute difference
-- This section is still incomplete
definition dist (n m : ) := (n - m) + (m - n)
theorem dist_comm (n m : ) : dist n m = dist m n
:= add_comm (n - m) (m - n)
theorem dist_eq_zero {n m : } (H : dist n m = 0) : n = m
:=
have H2 : n - m = 0, from add_eq_zero_left H,
have H3 : n ≤ m, from sub_eq_zero_imp_le H2,
have H4 : m - n = 0, from add_eq_zero_right H,
have H5 : m ≤ n, from sub_eq_zero_imp_le H4,
le_antisym H3 H5
theorem dist_le {n m : } (H : n ≤ m) : dist n m = m - n
:= calc
dist n m = (n - m) + (m - n) : refl _
... = 0 + (m - n) : {le_imp_sub_eq_zero H}
... = m - n : add_zero_left (m - n)
theorem dist_ge {n m : } (H : n ≥ m) : dist n m = n - m
:= subst (dist_comm m n) (dist_le H)
theorem dist_zero_right (n : ) : dist n 0 = n
:= trans (dist_ge (zero_le n)) (sub_zero_right n)
theorem dist_zero_left (n : ) : dist 0 n = n
:= trans (dist_le (zero_le n)) (sub_zero_right n)
theorem dist_intro {n m k : } (H : n + m = k) : dist k n = m
:= calc
dist k n = k - n : dist_ge (le_intro H)
... = m : sub_intro H
theorem dist_add_right (n k m : ) : dist (n + k) (m + k) = dist n m
:=
calc
dist (n + k) (m + k) = ((n+k) - (m+k)) + ((m+k)-(n+k)) : refl _
... = (n - m) + ((m + k) - (n + k)) : {sub_add_add_right _ _ _}
... = (n - m) + (m - n) : {sub_add_add_right _ _ _}
theorem dist_add_left (k n m : ) : dist (k + n) (k + m) = dist n m
:= subst (add_comm m k) (subst (add_comm n k) (dist_add_right n k m))
theorem dist_ge_add_right {n m : } (H : n ≥ m) : dist n m + m = n
:= calc
dist n m + m = n - m + m : {dist_ge H}
... = n : add_sub_ge_left H
theorem dist_eq_intro {n m k l : } (H : n + m = k + l) : dist n k = dist l m
:= calc
dist n k = dist (n + m) (k + m) : symm (dist_add_right n m k)
... = dist (k + l) (k + m) : {H}
... = dist l m : dist_add_left k l m
end --namespace nat