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refactor(library/data/nat): declare lt and le asap using inductive definitions, and make key theorems transparent for definitional package

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We also define key theorems that will be used to generate the
automatically generated a well-founded subterm relation for inductive
datatypes.
We also prove decidability and wf theorems asap.
This commit is contained in:
Leonardo de Moura 2014-11-22 00:15:51 -08:00
parent 7c54dbce10
commit 064ecd3e3d
15 changed files with 426 additions and 350 deletions
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4
library/data/fin.lean
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@ -74,8 +74,8 @@ namespace fin
to_nat (of_nat_core (succ p₁)) = succ (to_nat (of_nat_core p₁)) : rfl
... = succ p₁ : ih)
private lemma of_nat_eq {p : nat} {n : nat} (H : p < n) : n - succ p + succ p = n :=
add_sub_ge_left (eq.subst (lt_def p n) H)
private lemma of_nat_eq {p n : nat} (H : p < n) : n - succ p + succ p = n :=
add_sub_ge_left (lt_imp_le_succ H)
definition of_nat (p : nat) (n : nat) : p < n → fin n :=
λ H : p < n,

2
library/data/int/order.lean
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@ -235,7 +235,7 @@ theorem lt_of_nat (n m : ℕ) : (of_nat n < of_nat m) ↔ (n < m) :=
calc
(of_nat n + 1 ≤ of_nat m) ↔ (of_nat (succ n) ≤ of_nat m) : by simp
... ↔ (succ n ≤ m) : le_of_nat (succ n) m
... ↔ (n < m) : iff.symm (eq_to_iff (nat.lt_def n m))
... ↔ (n < m) : iff.symm (nat.lt_def n m)
theorem gt_of_nat (n m : ℕ) : (of_nat n > of_nat m) ↔ (n > m) :=
lt_of_nat m n

94
library/data/nat/basic.lean
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@ -1,44 +1,18 @@
--- 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
--- Author: Floris van Doorn, Leonardo de Moura
-- data.nat.basic
-- ==============
--
-- Basic operations on the natural numbers.
import logic data.num tools.tactic algebra.binary tools.helper_tactics
import logic.inhabited
import .decl data.num algebra.binary
open tactic binary eq.ops
open decidable
open relation -- for subst_iff
open helper_tactics
-- Definition of the type
-- ----------------------
inductive nat : Type :=
zero : nat,
succ : nat → nat
open eq.ops binary
namespace nat
notation `ℕ` := nat
theorem rec_zero {P : ℕ → Type} (x : P zero) (f : ∀m, P m → P (succ m)) : nat.rec x f zero = x
theorem rec_succ {P : ℕ → Type} (x : P zero) (f : ∀m, P m → P (succ m)) (n : ℕ) :
nat.rec x f (succ n) = f n (nat.rec x f n)
protected definition is_inhabited [instance] : inhabited nat :=
inhabited.mk zero
-- Coercion from num
-- -----------------
definition add (x y : ℕ) : ℕ :=
nat.rec x (λ n r, succ r) y
notation a + b := add a b
definition of_num [coercion] [reducible] (n : num) : ℕ :=
num.rec zero
(λ n, pos_num.rec (succ zero) (λ n r, r + r + (succ zero)) (λ n r, r + r) n) n
definition addl (x y : ℕ) : ℕ :=
nat.rec y (λ n r, succ r) x
@ -70,10 +44,6 @@ nat.induction_on x
... = succ (succ x₁ ⊕ y₁) : {ih₂}
... = succ x₁ ⊕ succ y₁ : addl.succ_right))
definition of_num [coercion] [reducible] (n : num) : ℕ :=
num.rec zero
(λ n, pos_num.rec (succ zero) (λ n r, r + r + (succ zero)) (λ n r, r + r) n) n
-- Successor and predecessor
-- -------------------------
@ -82,11 +52,11 @@ assume H, no_confusion H
-- add_rewrite succ_ne_zero
definition pred (n : ℕ) := nat.rec 0 (fun m x, m) n
theorem pred.zero : pred 0 = 0 :=
rfl
theorem pred.zero : pred 0 = 0
theorem pred.succ (n : ℕ) : pred (succ n) = n
theorem pred.succ (n : ℕ) : pred (succ n) = n :=
rfl
irreducible pred
@ -103,11 +73,6 @@ or.imp_or (zero_or_succ_pred n) (assume H, H)
theorem case {P : ℕ → Prop} (n : 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 : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
or.elim (zero_or_succ_pred n)
(take H3 : n = 0, H1 H3)
(take H3 : n = succ (pred n), H2 (pred n) H3)
theorem succ.inj {n m : ℕ} (H : succ n = succ m) : n = m :=
no_confusion H (λe, e)
@ -118,25 +83,11 @@ induction_on n
absurd H ne)
(take k IH H, IH (succ.inj H))
protected definition has_decidable_eq [instance] : decidable_eq ℕ :=
take n m : ℕ,
have general : ∀n, decidable (n = m), from
rec_on m
(take n,
rec_on n
(inl rfl)
(λ m iH, inr !succ_ne_zero))
(λ (m' : ℕ) (iH1 : ∀n, decidable (n = m')),
take n, rec_on n
(inr (ne.symm !succ_ne_zero))
(λ (n' : ℕ) (iH2 : decidable (n' = succ m')),
decidable.by_cases
(assume Heq : n' = m', inl (congr_arg 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 discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
or.elim (zero_or_succ_pred n)
(take H3 : n = 0, H1 H3)
(take H3 : n = succ (pred n), H2 (pred n) H3)
theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
(H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a :=
@ -163,9 +114,11 @@ general m
-- Addition
-- --------
theorem add.zero_right (n : ℕ) : n + 0 = n
theorem add.zero_right (n : ℕ) : n + 0 = n :=
rfl
theorem add.succ_right (n m : ℕ) : n + succ m = succ (n + m)
theorem add.succ_right (n m : ℕ) : n + succ m = succ (n + m) :=
rfl
irreducible add
@ -267,12 +220,11 @@ nat.rec H1 (take n IH, !add.one ▸ (H2 n IH)) a
-- Multiplication
-- --------------
definition mul (n m : ℕ) := nat.rec 0 (fun m x, x + n) m
notation a * b := mul a b
theorem mul.zero_right (n : ℕ) : n * 0 = 0 :=
rfl
theorem mul.zero_right (n : ℕ) : n * 0 = 0
theorem mul.succ_right (n m : ℕ) : n * succ m = n * m + n
theorem mul.succ_right (n m : ℕ) : n * succ m = n * m + n :=
rfl
irreducible mul

277
library/data/nat/decl.lean Normal file
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@ -0,0 +1,277 @@
--- 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, Leonardo de Moura
import logic.eq logic.heq logic.wf logic.decidable logic.if data.prod
open eq.ops decidable
inductive nat :=
zero : nat,
succ : nat → nat
namespace nat
notation `ℕ` := nat
inductive lt (a : nat) : nat → Prop :=
base : lt a (succ a),
step : Π {b}, lt a b → lt a (succ b)
notation a < b := lt a b
inductive le (a : nat) : nat → Prop :=
refl : le a a,
of_lt : ∀ {b}, lt a b → le a b
notation a ≤ b := le a b
definition pred (a : nat) : nat :=
cases_on a zero (λ a₁, a₁)
protected definition is_inhabited [instance] : inhabited nat :=
inhabited.mk zero
protected definition has_decidable_eq [instance] : decidable_eq nat :=
λn m : nat,
have general : ∀n, decidable (n = m), from
rec_on m
(λ n, cases_on n
(inl rfl)
(λ m, inr (λ (e : succ m = zero), no_confusion e)))
(λ (m' : nat) (ih : ∀n, decidable (n = m')) (n : nat), cases_on n
(inr (λ h, no_confusion h))
(λ (n' : nat),
decidable.rec_on (ih n')
(assume Heq : n' = m', inl (eq.rec_on Heq rfl))
(assume Hne : n' ≠ m',
have H1 : succ n' ≠ succ m', from
assume Heq, no_confusion Heq (λ e : n' = m', Hne e),
inr H1))),
general n
-- less-than is well-founded
definition lt.wf [instance] : well_founded lt :=
well_founded.intro (λn, rec_on n
(acc.intro zero (λ (y : nat) (hlt : y < zero),
have aux : ∀ {n₁}, y < n₁ → zero = n₁ → acc lt y, from
λ n₁ hlt, lt.cases_on hlt
(λ heq, no_confusion heq)
(λ b hlt heq, no_confusion heq),
aux hlt rfl))
(λ (n : nat) (ih : acc lt n),
acc.intro (succ n) (λ (m : nat) (hlt : m < succ n),
have aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m, from
λ n₁ hlt, lt.cases_on hlt
(λ (heq : succ n = succ m),
nat.no_confusion heq (λ (e : n = m),
eq.rec_on e ih))
(λ b (hlt : m < b) (heq : succ n = succ b),
nat.no_confusion heq (λ (e : n = b),
acc.inv (eq.rec_on e ih) hlt)),
aux hlt rfl)))
definition not_lt_zero (a : nat) : ¬ a < zero :=
have aux : ∀ {b}, a < b → b = zero → false, from
λ b H, lt.cases_on H
(λ heq, nat.no_confusion heq)
(λ b h₁ heq, nat.no_confusion heq),
λ H, aux H rfl
definition zero_lt_succ (a : nat) : zero < succ a :=
rec_on a
(lt.base zero)
(λ a (hlt : zero < succ a), lt.step hlt)
definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c :=
have aux : ∀ {d}, d < c → b = d → a < b → a < c, from
(λ d H, lt.rec_on H
(λ h₁ h₂, lt.step (eq.rec_on h₁ h₂))
(λ b hl ih h₁ h₂, lt.step (ih h₁ h₂))),
aux H₂ rfl H₁
definition lt.imp_succ {a b : nat} (H : a < b) : succ a < succ b :=
lt.rec_on H
(lt.base (succ a))
(λ b hlt ih, lt.trans ih (lt.base (succ b)))
definition lt.cancel_succ_left {a b : nat} (H : succ a < b) : a < b :=
have aux : ∀ {a₁}, a₁ < b → succ a = a₁ → a < b, from
λ a₁ H, lt.rec_on H
(λ e₁, eq.rec_on e₁ (lt.step (lt.base a)))
(λ d hlt ih e₁, lt.step (ih e₁)),
aux H rfl
definition lt.cancel_succ_left_right {a b : nat} (H : succ a < succ b) : a < b :=
have aux : pred (succ a) < pred (succ b), from
lt.rec_on H
(lt.base a)
(λ (b : nat) (hlt : succ a < b) ih,
show pred (succ a) < pred (succ b), from
lt.cancel_succ_left hlt),
aux
definition lt.is_decidable_rel [instance] : decidable_rel lt :=
λ a b, rec_on b
(λ (a : nat), inr (not_lt_zero a))
(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), cases_on a
(inl !zero_lt_succ)
(λ a, decidable.rec_on (ih a)
(λ h_pos : a < b₁, inl (lt.imp_succ h_pos))
(λ h_neg : ¬ a < b₁,
have aux : ¬ succ a < succ b₁, from
λ h : succ a < succ b₁, h_neg (lt.cancel_succ_left_right h),
inr aux)))
a
definition le_def_right {a b : nat} (H : a ≤ b) : a = b ∨ a < b :=
le.cases_on H
(or.inl rfl)
(λ b hlt, or.inr hlt)
definition le_def_left {a b : nat} (H : a = b ∨ a < b) : a ≤ b :=
or.rec_on H
(λ hl, eq.rec_on hl !le.refl)
(λ hr, le.of_lt hr)
definition le.is_decidable_rel [instance] : decidable_rel le :=
λ a b, decidable_iff_equiv _ (iff.intro le_def_left le_def_right)
definition lt.irrefl (a : nat) : ¬ a < a :=
rec_on a
!not_lt_zero
(λ (a : nat) (ih : ¬ a < a) (h : succ a < succ a),
ih (lt.cancel_succ_left_right h))
definition lt.asymm {a b : nat} (H : a < b) : ¬ b < a :=
lt.rec_on H
(λ h : succ a < a, !lt.irrefl (lt.cancel_succ_left h))
(λ b hlt (ih : ¬ b < a) (h : succ b < a), ih (lt.cancel_succ_left h))
definition lt.trichotomy (a b : nat) : a < b ∨ a = b ∨ b < a :=
rec_on b
(λa, cases_on a
(or.inr (or.inl rfl))
(λ a₁, or.inr (or.inr !zero_lt_succ)))
(λ b₁ (ih : ∀a, a < b₁ ∨ a = b₁ ∨ b₁ < a) (a : nat), cases_on a
(or.inl !zero_lt_succ)
(λ a, or.rec_on (ih a)
(λ h : a < b₁, or.inl (lt.imp_succ h))
(λ h, or.rec_on h
(λ h : a = b₁, or.inr (or.inl (eq.rec_on h rfl)))
(λ h : b₁ < a, or.inr (or.inr (lt.imp_succ h))))))
a
definition not_lt {a b : nat} (hnlt : ¬ a < b) : a = b ∨ b < a :=
or.rec_on (lt.trichotomy a b)
(λ hlt, absurd hlt hnlt)
(λ h, h)
definition le_imp_lt_succ {a b : nat} (h : a ≤ b) : a < succ b :=
le.rec_on h
(lt.base a)
(λ b h, lt.step h)
definition le_succ_imp_lt {a b : nat} (h : succ a ≤ b) : a < b :=
le.rec_on h
(lt.base a)
(λ b (h : succ a < b), lt.cancel_succ_left_right (lt.step h))
definition le.step {a b : nat} (h : a ≤ b) : a ≤ succ b :=
le.rec_on h
(le.of_lt (lt.base a))
(λ b (h : a < b), le.of_lt (lt.step h))
definition lt_imp_le_succ {a b : nat} (h : a < b) : succ a ≤ b :=
lt.rec_on h
(le.refl (succ a))
(λ b hlt (ih : succ a ≤ b), le.step ih)
definition le.trans {a b c : nat} (h₁ : a ≤ b) : b ≤ c → a ≤ c :=
le.rec_on h₁
(λ h, h)
(λ b (h₁ : a < b) (h₂ : b ≤ c), le.rec_on h₂
(le.of_lt h₁)
(λ c (h₂ : b < c), le.of_lt (lt.trans h₁ h₂)))
definition le_lt.trans {a b c : nat} (h₁ : a ≤ b) : b < c → a < c :=
le.rec_on h₁
(λ h, h)
(λ b (h₁ : a < b) (h₂ : b < c), lt.trans h₁ h₂)
definition lt_le.trans {a b c : nat} (h₁ : a < b) (h₂ : b ≤ c) : a < c :=
le.rec_on h₂
h₁
(λ c (h₂ : b < c), lt.trans h₁ h₂)
definition lt_eq.trans {a b c : nat} (h₁ : a < b) (h₂ : b = c) : a < c :=
eq.rec_on h₂ h₁
definition le_eq.trans {a b c : nat} (h₁ : a ≤ b) (h₂ : b = c) : a ≤ c :=
eq.rec_on h₂ h₁
definition eq_lt.trans {a b c : nat} (h₁ : a = b) (h₂ : b < c) : a < c :=
eq.rec_on (eq.rec_on h₁ rfl) h₂
definition eq_le.trans {a b c : nat} (h₁ : a = b) (h₂ : b ≤ c) : a ≤ c :=
eq.rec_on (eq.rec_on h₁ rfl) h₂
calc_trans lt.trans
calc_trans le.trans
calc_trans le_lt.trans
calc_trans lt_le.trans
calc_trans lt_eq.trans
calc_trans le_eq.trans
calc_trans eq_lt.trans
calc_trans eq_le.trans
definition max (a b : nat) : nat :=
if a < b then b else a
definition min (a b : nat) : nat :=
if a < b then a else b
definition max_a_a (a : nat) : a = max a a :=
eq.rec_on !if_t_t rfl
definition max.eq_right {a b : nat} (H : a < b) : b = max a b :=
eq.rec_on (if_pos H) rfl
definition max.eq_left {a b : nat} (H : ¬ a < b) : a = max a b :=
eq.rec_on (if_neg H) rfl
definition max.left (a b : nat) : a ≤ max a b :=
by_cases
(λ h : a < b, le.of_lt (eq.rec_on (max.eq_right h) h))
(λ h : ¬ a < b, eq.rec_on (max.eq_left h) !le.refl)
definition max.right (a b : nat) : b ≤ max a b :=
by_cases
(λ h : a < b, eq.rec_on (max.eq_right h) !le.refl)
(λ h : ¬ a < b, or.rec_on (not_lt h)
(λ heq, eq.rec_on heq (eq.rec_on (max_a_a a) !le.refl))
(λ h : b < a,
have aux : a = max a b, from max.eq_left (lt.asymm h),
eq.rec_on aux (le.of_lt h)))
definition gt a b := lt b a
notation a > b := gt a b
definition ge a b := le b a
notation a ≥ b := ge a b
definition add (a b : nat) : nat :=
rec_on b a (λ b₁ r, succ r)
notation a + b := add a b
definition sub (a b : nat) : nat :=
rec_on b a (λ b₁ r, pred r)
notation a - b := sub a b
definition mul (a b : nat) : nat :=
rec_on b zero (λ b₁ r, r + a)
notation a * b := mul a b
end nat

2
library/data/nat/default.lean
View file

@ -2,4 +2,4 @@
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Jeremy Avigad
import .basic .order .sub .div .wf
import .basic .order .sub .div

10
library/data/nat/div.lean
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@ -11,7 +11,7 @@
import logic .sub algebra.relation data.prod
import tools.fake_simplifier
open nat relation relation.iff_ops prod
open relation relation.iff_ops prod
open fake_simplifier decidable
open eq.ops
@ -87,7 +87,7 @@ case_strong_induction_on m
assume Hzx : measure z < measure x,
calc
f' m z = restrict default measure f m z : IH m !le_refl z
... = f z : restrict_lt_eq _ _ _ _ _ (lt_le_trans Hzx (lt_succ_imp_le H1)),
... = f z : restrict_lt_eq _ _ _ _ _ (lt_le.trans Hzx (lt_succ_imp_le H1)),
have H2 : f' (succ m) x = rec_val x f, from
calc
f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl
@ -168,7 +168,7 @@ show lhs = rhs, from
have H2b : ¬ x < y, from assume H, H1 (or.inr H),
have ypos : y > 0, from ne_zero_imp_pos H2a,
have xgey : x ≥ y, from not_lt_imp_ge H2b,
have H4 : x - y < x, from sub_lt (lt_le_trans ypos xgey) ypos,
have H4 : x - y < x, from sub_lt (lt_le.trans ypos xgey) ypos,
calc
lhs = succ (g1 (x - y)) : if_neg H1
... = succ (g2 (x - y)) : {H _ H4}
@ -244,7 +244,7 @@ show lhs = rhs, from
have H2b : ¬ x < y, from assume H, H1 (or.inr H),
have ypos : y > 0, from ne_zero_imp_pos H2a,
have xgey : x ≥ y, from not_lt_imp_ge H2b,
have H4 : x - y < x, from sub_lt (lt_le_trans ypos xgey) ypos,
have H4 : x - y < x, from sub_lt (lt_le.trans ypos xgey) ypos,
calc
lhs = g1 (x - y) : if_neg H1
... = g2 (x - y) : H _ H4
@ -382,7 +382,7 @@ theorem quotient_unique {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y)
(H3 : q1 * y + r1 = q2 * y + r2) : q1 = q2 :=
have H4 : q1 * y + r2 = q2 * y + r2, from (remainder_unique H H1 H2 H3) ▸ H3,
have H5 : q1 * y = q2 * y, from add.cancel_right H4,
have H6 : y > 0, from le_lt_trans !zero_le H1,
have H6 : y > 0, from le_lt.trans !zero_le H1,
show q1 = q2, from mul_cancel_right H6 H5
theorem div_mul_mul {z x y : ℕ} (zpos : z > 0) : (z * x) div (z * y) = x div y :=

243
library/data/nat/order.lean
View file

@ -1,41 +1,52 @@
--- 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
--- Author: Floris van Doorn, Leonardo de Moura
-- data.nat.order
-- ==============
--
-- The ordering on the natural numbers
import .basic logic.decidable
import tools.fake_simplifier
import .basic
open nat eq.ops tactic
open fake_simplifier
open eq.ops
namespace nat
-- Less than or equal
-- ------------------
definition le (n m : ℕ) : Prop := exists k : nat, n + k = m
theorem le.succ_right {n m : ℕ} (h : n ≤ m) : n ≤ succ m :=
le.rec_on h
(le.of_lt (lt.base n))
(λ b (h : n < b), le.of_lt (lt.step h))
notation a <= b := le a b
notation a ≤ b := le a b
theorem le.add_right (n k : ℕ) : n ≤ n + k :=
induction_on k
(calc n ≤ n : le.refl n
... = n + zero : add.zero_right)
(λ k (ih : n ≤ n + k), calc
n ≤ succ (n + k) : le.succ_right ih
... = n + succ k : add.succ_right)
theorem le_intro {n m k : ℕ} (H : n + k = m) : n ≤ m :=
exists_intro k H
theorem le_intro {n m k : ℕ} (h : n + k = m) : n ≤ m :=
h ▸ le.add_right n k
theorem le_elim {n m : ℕ} (H : n ≤ m) : ∃k, n + k = m :=
H
irreducible le
theorem le_elim {n m : ℕ} (h : n ≤ m) : ∃k, n + k = m :=
le.rec_on h
(exists_intro 0 rfl)
(λ m (h : n < m), lt.rec_on h
(exists_intro 1 rfl)
(λ b hlt (ih : ∃ (k : ℕ), n + k = b),
obtain (k : ℕ) (h : n + k = b), from ih,
exists_intro (succ k) (calc
n + succ k = succ (n + k) : add.succ_right
... = succ b : h)))
-- ### partial order (totality is part of less than)
theorem le_refl (n : ℕ) : n ≤ n :=
le_intro !add.zero_right
le.refl n
theorem zero_le (n : ℕ) : 0 ≤ n :=
le_intro !add.zero_left
@ -51,13 +62,7 @@ not_intro
absurd H2 !succ_ne_zero)
theorem le_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
obtain (l1 : ℕ) (Hl1 : n + l1 = m), from le_elim H1,
obtain (l2 : ℕ) (Hl2 : m + l2 = k), from le_elim H2,
le_intro
(calc
n + (l1 + l2) = n + l1 + l2 : !add.assoc⁻¹
... = m + l2 : {Hl1}
... = k : Hl2)
le.trans H1 H2
theorem le_antisym {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
obtain (k : ℕ) (Hk : n + k = m), from (le_elim H1),
@ -218,7 +223,7 @@ theorem mul_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k * n ≤ k * m :=
obtain (l : ℕ) (Hl : n + l = m), from (le_elim H),
have H2 : k * n + k * l = k * m, from
calc
k * n + k * l = k * (n + l) : by simp
k * n + k * l = k * (n + l) : mul.distr_left
... = k * m : {Hl},
le_intro H2
@ -228,52 +233,14 @@ theorem mul_le_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n * k ≤ m * k :=
theorem mul_le {n m k l : ℕ} (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
definition le_decidable [instance] (n m : ℕ) : decidable (n ≤ m) :=
have general : ∀n, decidable (n ≤ m), from
rec_on m
(take n,
rec_on n
(decidable.inl !le_refl)
(take m iH, decidable.inr !not_succ_zero_le))
(take (m' : ℕ) (iH1 : ∀n, decidable (n ≤ m')) (n : ℕ),
rec_on n
(decidable.inl !zero_le)
(take (n' : ℕ) (iH2 : decidable (n' ≤ succ m')),
decidable.by_cases
(assume Hp : n' ≤ m', decidable.inl (succ_le Hp))
(assume Hn : ¬ n' ≤ m',
have H : ¬ succ n' ≤ succ m', from
assume Hle : succ n' ≤ succ m',
absurd (succ_le_cancel Hle) Hn,
decidable.inr H))),
general n
-- Less than, Greater than, Greater than or equal
-- ----------------------------------------------
-- ge and gt will be transparent, so we don't need to reprove theorems for le and lt for them
definition lt (n m : ℕ) := succ n ≤ m
notation a < b := lt a b
definition ge (n m : ℕ) := m ≤ n
notation a >= b := ge a b
notation a ≥ b := ge a b
definition gt (n m : ℕ) := m < n
notation a > b := gt a b
theorem lt_def (n m : ℕ) : (n < m) = (succ n ≤ m) := rfl
-- add_rewrite gt_def ge_def --it might be possible to remove this in Lean 0.2
theorem lt_intro {n m k : ℕ} (H : succ n + k = m) : n < m :=
le_intro H
le_succ_imp_lt (le_intro H)
theorem lt_elim {n m : ℕ} (H : n < m) : ∃ k, succ n + k = m :=
le_elim H
le_elim (lt_imp_le_succ H)
theorem lt_add_succ (n m : ℕ) : n < n + succ m :=
lt_intro !add.move_succ
@ -281,16 +248,18 @@ lt_intro !add.move_succ
-- ### basic facts
theorem lt_imp_ne {n m : ℕ} (H : n < m) : n ≠ m :=
and.elim_right (succ_le_imp_le_and_ne H)
λ heq : n = m, absurd H (heq ▸ !lt.irrefl)
theorem lt_irrefl (n : ℕ) : ¬ n < n :=
not_intro (assume H : n < n, absurd rfl (lt_imp_ne H))
theorem succ_pos (n : ℕ) : 0 < succ n :=
succ_le !zero_le
theorem lt_def (n m : ℕ) : n < m ↔ succ n ≤ m :=
iff.intro
(λ h, lt_imp_le_succ h)
(λ h, le_succ_imp_lt h)
theorem not_lt_zero (n : ℕ) : ¬ n < 0 :=
!not_succ_zero_le
theorem succ_pos (n : ℕ) : 0 < succ n :=
!zero_lt_succ
theorem lt_imp_eq_succ {n m : ℕ} (H : n < m) : exists k, m = succ k :=
discriminate
@ -299,90 +268,53 @@ discriminate
-- ### interaction with le
theorem lt_imp_le_succ {n m : ℕ} (H : n < m) : succ n ≤ m :=
H
theorem le_succ_imp_lt {n m : ℕ} (H : succ n ≤ m) : n < m :=
H
theorem self_lt_succ (n : ℕ) : n < succ n :=
!le_refl
lt.base n
theorem lt_imp_le {n m : ℕ} (H : n < m) : n ≤ m :=
and.elim_left (succ_le_imp_le_and_ne H)
le.of_lt H
theorem le_imp_lt_or_eq {n m : ℕ} (H : n ≤ m) : n < m ∨ n = m :=
le_imp_succ_le_or_eq H
or.swap (le_def_right H)
theorem le_ne_imp_lt {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : n < m :=
le_ne_imp_succ_le H1 H2
theorem le_imp_lt_succ {n m : ℕ} (H : n ≤ m) : n < succ m :=
succ_le H
or.resolve_left (le_imp_lt_or_eq H1) H2
theorem lt_succ_imp_le {n m : ℕ} (H : n < succ m) : n ≤ m :=
succ_le_cancel H
-- ### transitivity, antisymmmetry
theorem lt_le_trans {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
le_trans H1 H2
theorem le_lt_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
le_trans (succ_le H1) H2
theorem lt_trans {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
lt_le_trans H1 (lt_imp_le H2)
theorem eq_le_trans {n m k : ℕ} (H1 : n = m) (H2 : m ≤ k) : n ≤ k :=
H1⁻¹ ▸ H2
theorem eq_lt_trans {n m k : ℕ} (H1 : n = m) (H2 : m < k) : n < k :=
H1⁻¹ ▸ H2
theorem le_eq_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m = k) : n ≤ k :=
H2 ▸ H1
theorem lt_eq_trans {n m k : ℕ} (H1 : n < m) (H2 : m = k) : n < k :=
H2 ▸ H1
calc_trans le_trans
calc_trans lt_trans
calc_trans lt_le_trans
calc_trans le_lt_trans
calc_trans eq_le_trans
calc_trans eq_lt_trans
calc_trans le_eq_trans
calc_trans lt_eq_trans
succ_le_cancel (lt_imp_le_succ H)
theorem le_imp_not_gt {n m : ℕ} (H : n ≤ m) : ¬ n > m :=
not_intro (assume H2 : m < n, absurd (le_lt_trans H H2) !lt_irrefl)
le.rec_on H
!lt.irrefl
(λ m (h : n < m), lt.asymm h)
theorem lt_imp_not_ge {n m : ℕ} (H : n < m) : ¬ n ≥ m :=
not_intro (assume H2 : m ≤ n, absurd (lt_le_trans H H2) !lt_irrefl)
not_intro (assume H2 : m ≤ n, absurd (lt_le.trans H H2) !lt_irrefl)
theorem lt_antisym {n m : ℕ} (H : n < m) : ¬ m < n :=
le_imp_not_gt (lt_imp_le H)
lt.asymm H
-- le_imp_not_gt (lt_imp_le H)
-- ### interaction with addition
theorem add_lt_left {n m : ℕ} (H : n < m) (k : ℕ) : k + n < k + m :=
!add.succ_right ▸ add_le_left H k
le_succ_imp_lt (!add.succ_right ▸ add_le_left (lt_imp_le_succ H) k)
theorem add_lt_right {n m : ℕ} (H : n < m) (k : ℕ) : n + k < m + k :=
!add.comm ▸ !add.comm ▸ add_lt_left H k
theorem add_le_lt {n m k l : ℕ} (H1 : n ≤ k) (H2 : m < l) : n + m < k + l :=
le_lt_trans (add_le_right H1 m) (add_lt_left H2 k)
le_lt.trans (add_le_right H1 m) (add_lt_left H2 k)
theorem add_lt_le {n m k l : ℕ} (H1 : n < k) (H2 : m ≤ l) : n + m < k + l :=
lt_le_trans (add_lt_right H1 m) (add_le_left H2 k)
lt_le.trans (add_lt_right H1 m) (add_le_left H2 k)
theorem add_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n + m < k + l :=
add_lt_le H1 (lt_imp_le H2)
theorem add_lt_cancel_left {n m k : ℕ} (H : k + n < k + m) : n < m :=
add_le_cancel_left (!add.succ_right⁻¹ ▸ H)
le_succ_imp_lt (add_le_cancel_left (!add.succ_right⁻¹ ▸ (lt_imp_le_succ H)))
theorem add_lt_cancel_right {n m k : ℕ} (H : n + k < m + k) : n < m :=
add_lt_cancel_left (!add.comm ▸ !add.comm ▸ H)
@ -395,43 +327,27 @@ theorem succ_lt {n m : ℕ} (H : n < m) : succ n < succ m :=
theorem succ_lt_cancel {n m : ℕ} (H : succ n < succ m) : n < m :=
add_lt_cancel_right (!add.one⁻¹ ▸ !add.one⁻¹ ▸ H)
theorem lt_imp_lt_succ {n m : ℕ} (H : n < m) : n < succ m
:= lt_trans H !self_lt_succ
theorem lt_imp_lt_succ {n m : ℕ} (H : n < m) : n < succ m :=
lt.step H
-- ### totality of lt and le
theorem le_or_gt {n m : ℕ} : n ≤ m ∨ n > m :=
induction_on n
(or.inl !zero_le)
(take (k : ℕ),
assume IH : k ≤ m ∨ m < k,
or.elim IH
(assume H : k ≤ m,
obtain (l : ℕ) (Hl : k + l = m), from le_elim H,
discriminate
(assume H2 : l = 0,
have H3 : m = k,
from calc
m = k + l : Hl⁻¹
... = k + 0 : {H2}
... = k : !add.zero_right,
have H4 : m < succ k, from H3 ▸ !self_lt_succ,
or.inr H4)
(take l2 : ℕ,
assume H2 : l = succ l2,
have H3 : succ k + l2 = m,
from calc
succ k + l2 = k + succ l2 : !add.move_succ
... = k + l : {H2⁻¹}
... = m : Hl,
or.inl (le_intro H3)))
(assume H : m < k, or.inr (lt_imp_lt_succ H)))
or.rec_on (lt.trichotomy n m)
(λ h : n < m, or.inl (le.of_lt h))
(λ h : n = m ∨ m < n, or.rec_on h
(λ h : n = m, eq.rec_on h (or.inl !le.refl))
(λ h : m < n, or.inr h))
theorem trichotomy_alt (n m : ℕ) : (n < m ∨ n = m) ∨ n > m :=
or.imp_or_left le_or_gt (assume H : n ≤ m, le_imp_lt_or_eq H)
or.rec_on (lt.trichotomy n m)
(λ h, or.inl (or.inl h))
(λ h, or.rec_on h
(λ h, or.inl (or.inr h))
(λ h, or.inr h))
theorem trichotomy (n m : ℕ) : n < m ∨ n = m ∨ n > m :=
iff.elim_left or.assoc !trichotomy_alt
lt.trichotomy n m
theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n :=
or.imp_or_right le_or_gt (assume H : m < n, lt_imp_le H)
@ -442,13 +358,6 @@ or.resolve_left le_or_gt H
theorem not_le_imp_gt {n m : ℕ} (H : ¬ n ≤ m) : n > m :=
or.resolve_right le_or_gt H
-- The following three theorems are automatically proved using the instance le_decidable
definition lt_decidable [instance] (n m : ℕ) : decidable (n < m) := _
definition gt_decidable [instance] (n m : ℕ) : decidable (n > m) := _
definition ge_decidable [instance] (n m : ℕ) : decidable (n ≥ m) := _
-- Note: interaction with multiplication under "positivity"
-- ### misc
protected theorem strong_induction_on {P : nat → Prop} (n : ℕ) (H : ∀n, (∀m, m < n → P m) → P n) : P n :=
@ -543,22 +452,22 @@ mul_pos_imp_pos_left (!mul.comm ▸ H)
theorem mul_lt_left {n m k : ℕ} (Hk : k > 0) (H : n < m) : k * n < k * m :=
have H2 : k * n < k * n + k, from add_pos_right Hk,
have H3 : k * n + k ≤ k * m, from !mul.succ_right ▸ mul_le_left H k,
lt_le_trans H2 H3
have H3 : k * n + k ≤ k * m, from !mul.succ_right ▸ mul_le_left (lt_imp_le_succ H) k,
lt_le.trans H2 H3
theorem mul_lt_right {n m k : ℕ} (Hk : k > 0) (H : n < m) : n * k < m * k :=
!mul.comm ▸ !mul.comm ▸ mul_lt_left Hk H
theorem mul_le_lt {n m k l : ℕ} (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)
le_lt.trans (mul_le_right H1 m) (mul_lt_left Hk H2)
theorem mul_lt_le {n m k l : ℕ} (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)
le_lt.trans (mul_le_left H2 n) (mul_lt_right Hl H1)
theorem mul_lt {n m k l : ℕ} (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 H1) H2,
le_lt_trans H3 H4
have H4 : k * m < k * l, from mul_lt_left (le_lt.trans !zero_le H1) H2,
le_lt.trans H3 H4
theorem mul_lt_cancel_left {n m k : ℕ} (H : k * n < k * m) : n < m :=
or.elim le_or_gt
@ -571,7 +480,7 @@ theorem mul_lt_cancel_right {n m k : ℕ} (H : n * k < m * k) : n < m :=
mul_lt_cancel_left (!mul.comm ▸ !mul.comm ▸ H)
theorem mul_le_cancel_left {n m k : ℕ} (Hk : k > 0) (H : k * n ≤ k * m) : n ≤ m :=
have H2 : k * n < k * m + k, from le_lt_trans H (add_pos_right Hk),
have H2 : k * n < k * m + k, from le_lt.trans H (add_pos_right Hk),
have H3 : k * n < k * succ m, from !mul.succ_right⁻¹ ▸ H2,
have H4 : n < succ m, from mul_lt_cancel_left H3,
show n ≤ m, from lt_succ_imp_le H4
@ -602,9 +511,9 @@ have H3 : n > 0, from mul_pos_imp_pos_left H2,
have H4 : m > 0, from mul_pos_imp_pos_right H2,
or.elim le_or_gt
(assume H5 : n ≤ 1,
show n = 1, from le_antisym H5 H3)
show n = 1, from le_antisym H5 (lt_imp_le_succ H3))
(assume H5 : n > 1,
have H6 : n * m ≥ 2 * 1, from mul_le H5 H4,
have H6 : n * m ≥ 2 * 1, from mul_le (lt_imp_le_succ H5) (lt_imp_le_succ H4),
have H7 : 1 ≥ 2, from !mul.one_right ▸ H ▸ H6,
absurd !self_lt_succ (le_imp_not_gt H7))

54
library/data/nat/sub.lean
View file

@ -7,11 +7,10 @@
--
-- Subtraction on the natural numbers, as well as min, max, and distance.
import data.nat.order
import .order
import tools.fake_simplifier
open nat eq.ops tactic
open helper_tactics
open eq.ops
open fake_simplifier
namespace nat
@ -19,12 +18,11 @@ namespace nat
-- subtraction
-- -----------
definition sub (n m : ℕ) : nat := rec n (fun m x, pred x) m
notation a - b := sub a b
theorem sub_zero_right (n : ℕ) : n - 0 = n :=
rfl
theorem sub_zero_right (n : ℕ) : n - 0 = n
theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m)
theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m) :=
rfl
irreducible sub
@ -300,7 +298,7 @@ have xsuby_eq : x - y = x' - y', from calc
... = 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
show x - y < x, from xeq⁻¹ ▸ xsuby_eq⁻¹ ▸ le_lt.trans H1 H2
theorem sub_le_right {n m : ℕ} (H : n ≤ m) (k : nat) : n - k ≤ m - k :=
obtain (l : ℕ) (Hl : n + l = m), from le_elim H,
@ -309,7 +307,7 @@ or.elim !le_total
(assume H2 : k ≤ n,
have H3 : n - k + l = m - k, from
calc
n - k + l = l + (n - k) : by simp
n - k + l = l + (n - k) : add.comm
... = l + n - k : (add_sub_assoc H2 l)⁻¹
... = n + l - k : {!add.comm}
... = m - k : {Hl},
@ -324,7 +322,9 @@ sub_split
have H3 : n ≤ k, from le_trans H (le_intro Hm),
have H4 : m' + l + n = k - n + n, from
calc
m' + l + n = n + l + m' : by simp
m' + l + n = n + (m' + l) : add.comm
... = n + (l + m') : add.comm
... = n + l + m' : add.assoc
... = m + m' : {Hl}
... = k : Hm
... = k - n + n : (add_sub_ge_left H3)⁻¹,
@ -352,8 +352,9 @@ sub_split
assume Hkm : k + km = m,
have H : k + (mn + km) = n, from
calc
k + (mn + km) = k + km + mn : by simp
... = m + mn : {Hkm}
k + (mn + km) = k + (km + mn): add.comm
... = k + km + mn : add.assoc
... = m + mn : Hkm
... = n : Hmn,
have H2 : n - k = mn + km, from sub_intro H,
H2 ▸ !le_refl))
@ -362,24 +363,9 @@ sub_split
-- add_rewrite sub_self mul_sub_distr_left mul_sub_distr_right
-- Max, min, iteration, and absolute difference
-- absolute difference
-- --------------------------------------------
definition max (n m : ℕ) : ℕ := n + (m - n)
definition min (n m : ℕ) : ℕ := m - (m - n)
theorem max_le {n m : ℕ} (H : n ≤ m) : max n m = m :=
add_sub_le H
theorem max_ge {n m : ℕ} (H : n ≥ m) : max n m = n :=
add_sub_ge H
theorem left_le_max (n m : ℕ) : n ≤ max n m :=
!le_add_sub_left
theorem right_le_max (n m : ℕ) : m ≤ max n m :=
!le_add_sub_right
-- ### absolute difference
-- This section is still incomplete
@ -391,8 +377,9 @@ theorem dist_comm (n m : ℕ) : dist n m = dist m n :=
theorem dist_self (n : ℕ) : dist n n = 0 :=
calc
(n - n) + (n - n) = 0 + 0 : by simp
... = 0 : by simp
(n - n) + (n - n) = 0 + (n - n) : sub_self
... = 0 + 0 : sub_self
... = 0 : rfl
theorem dist_eq_zero {n m : ℕ} (H : dist n m = 0) : n = m :=
have H2 : n - m = 0, from add.eq_zero_left H,
@ -445,9 +432,10 @@ calc
theorem dist_sub_move_add {n m : ℕ} (H : n ≥ m) (k : ℕ) : dist (n - m) k = dist n (k + m) :=
have H2 : n - m + (k + m) = k + n, from
calc
n - m + (k + m) = n - m + m + k : by simp
n - m + (k + m) = n - m + (m + k) : add.comm
... = n - m + m + k : add.assoc
... = n + k : {add_sub_ge_left H}
... = k + n : by simp,
... = k + n : add.comm,
dist_eq_intro H2
theorem dist_sub_move_add' {k m : ℕ} (H : k ≥ m) (n : ℕ) : dist n (k - m) = dist (n + m) k :=

52
library/data/nat/wf.lean
View file

@ -1,52 +0,0 @@
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import data.nat.order logic.wf
open nat eq.ops
namespace nat
inductive pred_rel : nat → nat → Prop :=
intro : Π (n : nat), pred_rel n (succ n)
definition not_pred_rel_zero (n : nat) : ¬ pred_rel n zero :=
have aux : ∀{m}, pred_rel n m → succ n = m, from
λm H, pred_rel.rec_on H (take n, rfl),
assume H : pred_rel n zero,
absurd (aux H) !succ_ne_zero
definition pred_rel_succ {a b : nat} (H : pred_rel a (succ b)) : b = a :=
have aux : pred (succ b) = a, from
pred_rel.rec_on H (λn, rfl),
aux
-- Predecessor relation is well-founded
definition pred_rel.wf : well_founded pred_rel :=
well_founded.intro
(λn, induction_on n
(acc.intro zero (λy (H : pred_rel y zero), absurd H (not_pred_rel_zero y)))
(λa (iH : acc pred_rel a),
acc.intro (succ a) (λy (H : pred_rel y (succ a)),
have aux : a = y, from pred_rel_succ H,
eq.rec_on aux iH)))
-- Less-than relation is well-founded
definition lt.wf [instance] : well_founded lt :=
well_founded.intro
(λn, induction_on n
(acc.intro zero (λ (y : nat) (H : y < 0),
absurd H !not_lt_zero))
(λ (n : nat) (iH : acc lt n),
acc.intro (succ n) (λ (m : nat) (H : m < succ n),
have H₁ : m < n ∨ m = n, from le_imp_lt_or_eq (succ_le_cancel (lt_imp_le_succ H)),
or.elim H₁
(assume Hlt : m < n, acc.inv iH Hlt)
(assume Heq : m = n, Heq⁻¹ ▸ iH))))
definition measure {A : Type} (f : A → nat) : A → A → Prop :=
inv_image lt f
definition measure.wf {A : Type} (f : A → nat) : well_founded (measure f) :=
inv_image.wf f lt.wf
end nat

2
tests/lean/notation.lean
View file

@ -1,4 +1,4 @@
import logic data.nat.basic
import logic data.num data.nat.basic
open num
constant b : num
check b + b + b

2
tests/lean/run/div2.lean
View file

@ -78,7 +78,7 @@ case_strong_induction_on m
assume Hzx : measure z < measure x,
calc
f' m z = restrict default measure f m z : IH m !le_refl z
... = f z : !restrict_lt_eq (lt_le_trans Hzx (lt_succ_imp_le H1))
... = f z : !restrict_lt_eq (lt_le.trans Hzx (lt_succ_imp_le H1))
∎,
have H2 : f' (succ m) x = rec_val x f,
proof

2
tests/lean/run/div_wf.lean
View file

@ -11,7 +11,7 @@ notation `dif` c `then` t:45 `else` e:45 := dite c t e
-- Auxiliary lemma used to justify recursive call
private lemma lt_aux {x y : nat} (H : 0 < y ∧ y ≤ x) : x - y < x :=
have y0 : 0 < y, from and.elim_left H,
have x0 : 0 < x, from lt_le_trans y0 (and.elim_right H),
have x0 : 0 < x, from lt_le.trans y0 (and.elim_right H),
sub_lt x0 y0
definition wdiv.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=

12
tests/lean/run/fib_wrec.lean
View file

@ -7,8 +7,8 @@ nat.cases_on n
(λ (n₁ : nat), nat.cases_on n₁
(λ (f : Π (m : nat), m < (succ zero) → nat), succ zero)
(λ (n₂ : nat) (f : Π (m : nat), m < (succ (succ n₂)) → nat),
have l₁ : succ n₂ < succ (succ n₂), from self_lt_succ (succ n₂),
have l₂ : n₂ < succ (succ n₂), from lt_trans (self_lt_succ n₂) l₁,
have l₁ : succ n₂ < succ (succ n₂), from lt.base (succ n₂),
have l₂ : n₂ < succ (succ n₂), from lt.trans (lt.base n₂) l₁,
f (succ n₂) l₁ + f n₂ l₂))
definition fib (n : nat) :=
@ -23,6 +23,8 @@ well_founded.fix_eq fib.F 1
theorem fib.succ_succ_eq (n : nat) : fib (succ (succ n)) = fib (succ n) + fib n :=
well_founded.fix_eq fib.F (succ (succ n))
eval fib 5 -- ignores opaque annotations
eval fib 6
eval [strict] fib 5 -- take opaque/irreducible annotations into account
example : fib 5 = 8 :=
rfl
example : fib 6 = 13 :=
rfl

6
tests/lean/run/tree_height.lean
View file

@ -19,13 +19,13 @@ definition height_lt.wf (A : Type) : well_founded (@height_lt A) :=
inv_image.wf height lt.wf
theorem height_lt.node_left {A : Type} (t₁ t₂ : tree A) : height_lt t₁ (node t₁ t₂) :=
le_imp_lt_succ (left_le_max (height t₁) (height t₂))
le_imp_lt_succ (max.left (height t₁) (height t₂))
theorem height_lt.node_right {A : Type} (t₁ t₂ : tree A) : height_lt t₂ (node t₁ t₂) :=
le_imp_lt_succ (right_le_max (height t₁) (height t₂))
le_imp_lt_succ (max.right (height t₁) (height t₂))
theorem height_lt.trans {A : Type} : transitive (@height_lt A) :=
inv_image.trans lt height @lt_trans
inv_image.trans lt height @lt.trans
example : height_lt (leaf 2) (node (leaf 1) (leaf 2)) :=
!height_lt.node_right

4
tests/lean/whnf.lean.expected.out
View file

@ -1,5 +1,5 @@
whnf.lean:2:0: warning: imported file uses 'sorry'
succ (nat.rec 2 (λ (n r : ℕ), succ r) zero)
succ (nat.rec 2 (λ (b₁ r : ℕ), succ r) zero)
succ (succ (succ zero))
succ (nat.rec a (λ (n r : ℕ), succ r) zero)
succ (nat.rec a (λ (b₁ r : ℕ), succ r) zero)
succ a