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feat(library): add '[rewrite]' annotation some some theorems

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This commit is contained in:
Leonardo de Moura 2015-07-13 16:39:53 -04:00
parent 1881ad0aef
commit ebe6ec0017
7 changed files with 139 additions and 97 deletions
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36
library/data/equiv.lean
View file

@ -50,22 +50,22 @@ lemma arrow_congr {A₁ B₁ A₂ B₂ : Type} : A₁ ≃ A₂ → B₁ ≃ B₂
section
open unit
lemma arrow_unit_equiv_unit (A : Type) : (A → unit) ≃ unit :=
lemma arrow_unit_equiv_unit [rewrite] (A : Type) : (A → unit) ≃ unit :=
mk (λ f, star) (λ u, (λ f, star))
(λ f, funext (λ x, by cases (f x); reflexivity))
(λ u, by cases u; reflexivity)
lemma unit_arrow_equiv (A : Type) : (unit → A) ≃ A :=
lemma unit_arrow_equiv [rewrite] (A : Type) : (unit → A) ≃ A :=
mk (λ f, f star) (λ a, (λ u, a))
(λ f, funext (λ x, by cases x; reflexivity))
(λ u, rfl)
lemma empty_arrow_equiv_unit (A : Type) : (empty → A) ≃ unit :=
lemma empty_arrow_equiv_unit [rewrite] (A : Type) : (empty → A) ≃ unit :=
mk (λ f, star) (λ u, λ e, empty.rec _ e)
(λ f, funext (λ x, empty.rec _ x))
(λ u, by cases u; reflexivity)
lemma false_arrow_equiv_unit (A : Type) : (false → A) ≃ unit :=
lemma false_arrow_equiv_unit [rewrite] (A : Type) : (false → A) ≃ unit :=
calc (false → A) ≃ (empty → A) : arrow_congr false_equiv_empty !equiv.refl
... ≃ unit : empty_arrow_equiv_unit
end
@ -78,13 +78,13 @@ lemma prod_congr {A₁ B₁ A₂ B₂ : Type} : A₁ ≃ A₂ → B₁ ≃ B₂
(λ p, begin cases p, esimp, rewrite [l₁, l₂] end)
(λ p, begin cases p, esimp, rewrite [r₁, r₂] end)
lemma prod_comm (A B : Type) : (A × B) ≃ (B × A) :=
lemma prod_comm [rewrite] (A B : Type) : (A × B) ≃ (B × A) :=
mk (λ p, match p with (a, b) := (b, a) end)
(λ p, match p with (b, a) := (a, b) end)
(λ p, begin cases p, esimp end)
(λ p, begin cases p, esimp end)
lemma prod_assoc (A B C : Type) : ((A × B) × C) ≃ (A × (B × C)) :=
lemma prod_assoc [rewrite] (A B C : Type) : ((A × B) × C) ≃ (A × (B × C)) :=
mk (λ t, match t with ((a, b), c) := (a, (b, c)) end)
(λ t, match t with (a, (b, c)) := ((a, b), c) end)
(λ t, begin cases t with ab c, cases ab, esimp end)
@ -92,20 +92,20 @@ mk (λ t, match t with ((a, b), c) := (a, (b, c)) end)
section
open unit prod.ops
lemma prod_unit_right (A : Type) : (A × unit) ≃ A :=
lemma prod_unit_right [rewrite] (A : Type) : (A × unit) ≃ A :=
mk (λ p, p.1)
(λ a, (a, star))
(λ p, begin cases p with a u, cases u, esimp end)
(λ a, rfl)
lemma prod_unit_left (A : Type) : (unit × A) ≃ A :=
lemma prod_unit_left [rewrite] (A : Type) : (unit × A) ≃ A :=
calc (unit × A) ≃ (A × unit) : prod_comm
... ≃ A : prod_unit_right
lemma prod_empty_right (A : Type) : (A × empty) ≃ empty :=
lemma prod_empty_right [rewrite] (A : Type) : (A × empty) ≃ empty :=
mk (λ p, empty.rec _ p.2) (λ e, empty.rec _ e) (λ p, empty.rec _ p.2) (λ e, empty.rec _ e)
lemma prod_empty_left (A : Type) : (empty × A) ≃ empty :=
lemma prod_empty_left [rewrite] (A : Type) : (empty × A) ≃ empty :=
calc (empty × A) ≃ (A × empty) : prod_comm
... ≃ empty : prod_empty_right
end
@ -127,25 +127,25 @@ mk (λ b, match b with tt := inl star | ff := inr star end)
(λ b, begin cases b, esimp, esimp end)
(λ s, begin cases s with u u, {cases u, esimp}, {cases u, esimp} end)
lemma sum_comm (A B : Type) : (A + B) ≃ (B + A) :=
lemma sum_comm [rewrite] (A B : Type) : (A + B) ≃ (B + A) :=
mk (λ s, match s with inl a := inr a | inr b := inl b end)
(λ s, match s with inl b := inr b | inr a := inl a end)
(λ s, begin cases s, esimp, esimp end)
(λ s, begin cases s, esimp, esimp end)
lemma sum_assoc (A B C : Type) : ((A + B) + C) ≃ (A + (B + C)) :=
lemma sum_assoc [rewrite] (A B C : Type) : ((A + B) + C) ≃ (A + (B + C)) :=
mk (λ s, match s with inl (inl a) := inl a | inl (inr b) := inr (inl b) | inr c := inr (inr c) end)
(λ s, match s with inl a := inl (inl a) | inr (inl b) := inl (inr b) | inr (inr c) := inr c end)
(λ s, begin cases s with ab c, cases ab, repeat esimp end)
(λ s, begin cases s with a bc, esimp, cases bc, repeat esimp end)
lemma sum_empty_right (A : Type) : (A + empty) ≃ A :=
lemma sum_empty_right [rewrite] (A : Type) : (A + empty) ≃ A :=
mk (λ s, match s with inl a := a | inr e := empty.rec _ e end)
(λ a, inl a)
(λ s, begin cases s with a e, esimp, exact empty.rec _ e end)
(λ a, rfl)
lemma sum_empty_left (A : Type) : (empty + A) ≃ A :=
lemma sum_empty_left [rewrite] (A : Type) : (empty + A) ≃ A :=
calc (empty + A) ≃ (A + empty) : sum_comm
... ≃ A : sum_empty_right
end
@ -191,23 +191,23 @@ mk (λ n, match n with 0 := inr star | succ a := inl a end)
(λ n, begin cases n, repeat esimp end)
(λ s, begin cases s with a u, esimp, {cases u, esimp} end)
lemma nat_sum_unit_equiv_nat : (nat + unit) ≃ nat :=
lemma nat_sum_unit_equiv_nat [rewrite] : (nat + unit) ≃ nat :=
equiv.symm nat_equiv_nat_sum_unit
lemma nat_prod_nat_equiv_nat : (nat × nat) ≃ nat :=
lemma nat_prod_nat_equiv_nat [rewrite] : (nat × nat) ≃ nat :=
mk (λ p, mkpair p.1 p.2)
(λ n, unpair n)
(λ p, begin cases p, apply unpair_mkpair end)
(λ n, mkpair_unpair n)
lemma nat_sum_bool_equiv_nat : (nat + bool) ≃ nat :=
lemma nat_sum_bool_equiv_nat [rewrite] : (nat + bool) ≃ nat :=
calc (nat + bool) ≃ (nat + (unit + unit)) : sum_congr !equiv.refl bool_equiv_unit_sum_unit
... ≃ ((nat + unit) + unit) : sum_assoc
... ≃ (nat + unit) : sum_congr nat_sum_unit_equiv_nat !equiv.refl
... ≃ nat : nat_sum_unit_equiv_nat
open decidable
lemma nat_sum_nat_equiv_nat : (nat + nat) ≃ nat :=
lemma nat_sum_nat_equiv_nat [rewrite] : (nat + nat) ≃ nat :=
mk (λ s, match s with inl n := 2*n | inr n := 2*n+1 end)
(λ n, if even n then inl (n div 2) else inr ((n - 1) div 2))
(λ s, begin

6
library/data/finset/basic.lean
View file

@ -84,7 +84,7 @@ theorem mem_list_of_mem {a : A} {l : nodup_list A} : a ∈ ⟦l⟧ → a ∈ elt
definition singleton (a : A) : finset A :=
to_finset_of_nodup [a] !nodup_singleton
theorem mem_singleton (a : A) : a ∈ singleton a :=
theorem mem_singleton [rewrite] (a : A) : a ∈ singleton a :=
mem_of_mem_list !mem_cons
theorem eq_of_mem_singleton {x a : A} : x ∈ singleton a → x = a :=
@ -116,10 +116,10 @@ to_finset_of_nodup [] nodup_nil
notation `∅` := !empty
theorem not_mem_empty (a : A) : a ∉ ∅ :=
theorem not_mem_empty [rewrite] (a : A) : a ∉ ∅ :=
λ aine : a ∈ ∅, aine
theorem mem_empty_iff (x : A) : x ∈ ∅ ↔ false :=
theorem mem_empty_iff [rewrite] (x : A) : x ∈ ∅ ↔ false :=
iff.mp' !iff_false_iff_not !not_mem_empty
theorem mem_empty_eq (x : A) : x ∈ ∅ = false :=

66
library/data/list/basic.lean
View file

@ -18,7 +18,7 @@ notation `[` l:(foldr `,` (h t, cons h t) nil `]`) := l
variable {T : Type}
lemma cons_ne_nil (a : T) (l : list T) : a::l ≠ [] :=
lemma cons_ne_nil [rewrite] (a : T) (l : list T) : a::l ≠ [] :=
by contradiction
lemma head_eq_of_cons_eq {A : Type} {h₁ h₂ : A} {t₁ t₂ : list A} :
@ -40,17 +40,17 @@ definition append : list T → list T → list T
notation l₁ ++ l₂ := append l₁ l₂
theorem append_nil_left (t : list T) : [] ++ t = t
theorem append_nil_left [rewrite] (t : list T) : [] ++ t = t
theorem append_cons (x : T) (s t : list T) : (x::s) ++ t = x::(s ++ t)
theorem append_cons [rewrite] (x : T) (s t : list T) : (x::s) ++ t = x::(s ++ t)
theorem append_nil_right : ∀ (t : list T), t ++ [] = t
theorem append_nil_right [rewrite] : ∀ (t : list T), t ++ [] = t
| [] := rfl
| (a :: l) := calc
(a :: l) ++ [] = a :: (l ++ []) : rfl
... = a :: l : append_nil_right l
theorem append.assoc : ∀ (s t u : list T), s ++ t ++ u = s ++ (t ++ u)
theorem append.assoc [rewrite] : ∀ (s t u : list T), s ++ t ++ u = s ++ (t ++ u)
| [] t u := rfl
| (a :: l) t u :=
show a :: (l ++ t ++ u) = (a :: l) ++ (t ++ u),
@ -61,11 +61,11 @@ definition length : list T → nat
| [] := 0
| (a :: l) := length l + 1
theorem length_nil : length (@nil T) = 0
theorem length_nil [rewrite] : length (@nil T) = 0
theorem length_cons (x : T) (t : list T) : length (x::t) = length t + 1
theorem length_cons [rewrite] (x : T) (t : list T) : length (x::t) = length t + 1
theorem length_append : ∀ (s t : list T), length (s ++ t) = length s + length t
theorem length_append [rewrite] : ∀ (s t : list T), length (s ++ t) = length s + length t
| [] t := calc
length ([] ++ t) = length t : rfl
... = length [] + length t : zero_add
@ -91,9 +91,9 @@ definition concat : Π (x : T), list T → list T
| a [] := [a]
| a (b :: l) := b :: concat a l
theorem concat_nil (x : T) : concat x [] = [x]
theorem concat_nil [rewrite] (x : T) : concat x [] = [x]
theorem concat_cons (x y : T) (l : list T) : concat x (y::l) = y::(concat x l)
theorem concat_cons [rewrite] (x y : T) (l : list T) : concat x (y::l) = y::(concat x l)
theorem concat_eq_append (a : T) : ∀ (l : list T), concat a l = l ++ [a]
| [] := rfl
@ -101,7 +101,7 @@ theorem concat_eq_append (a : T) : ∀ (l : list T), concat a l = l ++ [a]
show b :: (concat a l) = (b :: l) ++ (a :: []),
by rewrite concat_eq_append
theorem concat_ne_nil (a : T) : ∀ (l : list T), concat a l ≠ [] :=
theorem concat_ne_nil [rewrite] (a : T) : ∀ (l : list T), concat a l ≠ [] :=
by intro l; induction l; repeat contradiction
/- last -/
@ -111,16 +111,16 @@ definition last : Π l : list T, l ≠ [] → T
| [a] h := a
| (a₁::a₂::l) h := last (a₂::l) !cons_ne_nil
lemma last_singleton (a : T) (h : [a] ≠ []) : last [a] h = a :=
lemma last_singleton [rewrite] (a : T) (h : [a] ≠ []) : last [a] h = a :=
rfl
lemma last_cons_cons (a₁ a₂ : T) (l : list T) (h : a₁::a₂::l ≠ []) : last (a₁::a₂::l) h = last (a₂::l) !cons_ne_nil :=
lemma last_cons_cons [rewrite] (a₁ a₂ : T) (l : list T) (h : a₁::a₂::l ≠ []) : last (a₁::a₂::l) h = last (a₂::l) !cons_ne_nil :=
rfl
theorem last_congr {l₁ l₂ : list T} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) : last l₁ h₁ = last l₂ h₂ :=
by subst l₁
theorem last_concat {x : T} : ∀ {l : list T} (h : concat x l ≠ []), last (concat x l) h = x
theorem last_concat [rewrite] {x : T} : ∀ {l : list T} (h : concat x l ≠ []), last (concat x l) h = x
| [] h := rfl
| [a] h := rfl
| (a₁::a₂::l) h :=
@ -139,13 +139,13 @@ definition reverse : list T → list T
| [] := []
| (a :: l) := concat a (reverse l)
theorem reverse_nil : reverse (@nil T) = []
theorem reverse_nil [rewrite] : reverse (@nil T) = []
theorem reverse_cons (x : T) (l : list T) : reverse (x::l) = concat x (reverse l)
theorem reverse_cons [rewrite] (x : T) (l : list T) : reverse (x::l) = concat x (reverse l)
theorem reverse_singleton (x : T) : reverse [x] = [x]
theorem reverse_singleton [rewrite] (x : T) : reverse [x] = [x]
theorem reverse_append : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (reverse s)
theorem reverse_append [rewrite] : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (reverse s)
| [] t2 := calc
reverse ([] ++ t2) = reverse t2 : rfl
... = (reverse t2) ++ [] : append_nil_right
@ -158,7 +158,7 @@ theorem reverse_append : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (
... = reverse t2 ++ concat a2 (reverse s2) : concat_eq_append
... = reverse t2 ++ reverse (a2 :: s2) : rfl
theorem reverse_reverse : ∀ (l : list T), reverse (reverse l) = l
theorem reverse_reverse [rewrite] : ∀ (l : list T), reverse (reverse l) = l
| [] := rfl
| (a :: l) := calc
reverse (reverse (a :: l)) = reverse (concat a (reverse l)) : rfl
@ -178,9 +178,9 @@ definition head [h : inhabited T] : list T → T
| [] := arbitrary T
| (a :: l) := a
theorem head_cons [h : inhabited T] (a : T) (l : list T) : head (a::l) = a
theorem head_cons [rewrite] [h : inhabited T] (a : T) (l : list T) : head (a::l) = a
theorem head_append [h : inhabited T] (t : list T) : ∀ {s : list T}, s ≠ [] → head (s ++ t) = head s
theorem head_append [rewrite] [h : inhabited T] (t : list T) : ∀ {s : list T}, s ≠ [] → head (s ++ t) = head s
| [] H := absurd rfl H
| (a :: s) H :=
show head (a :: (s ++ t)) = head (a :: s),
@ -190,9 +190,9 @@ definition tail : list T → list T
| [] := []
| (a :: l) := l
theorem tail_nil : tail (@nil T) = []
theorem tail_nil [rewrite] : tail (@nil T) = []
theorem tail_cons (a : T) (l : list T) : tail (a::l) = l
theorem tail_cons [rewrite] (a : T) (l : list T) : tail (a::l) = l
theorem cons_head_tail [h : inhabited T] {l : list T} : l ≠ [] → (head l)::(tail l) = l :=
list.cases_on l
@ -208,13 +208,13 @@ definition mem : T → list T → Prop
notation e ∈ s := mem e s
notation e ∉ s := ¬ e ∈ s
theorem mem_nil_iff (x : T) : x ∈ [] ↔ false :=
theorem mem_nil_iff [rewrite] (x : T) : x ∈ [] ↔ false :=
iff.rfl
theorem not_mem_nil (x : T) : x ∉ [] :=
iff.mp !mem_nil_iff
theorem mem_cons (x : T) (l : list T) : x ∈ x :: l :=
theorem mem_cons [rewrite] (x : T) (l : list T) : x ∈ x :: l :=
or.inl rfl
theorem mem_cons_of_mem (y : T) {x : T} {l : list T} : x ∈ l → x ∈ y :: l :=
@ -340,16 +340,16 @@ definition sublist (l₁ l₂ : list T) := ∀ ⦃a : T⦄, a ∈ l₁ → a ∈
infix `⊆` := sublist
theorem nil_sub (l : list T) : [] ⊆ l :=
theorem nil_sub [rewrite] (l : list T) : [] ⊆ l :=
λ b i, false.elim (iff.mp (mem_nil_iff b) i)
theorem sub.refl (l : list T) : l ⊆ l :=
theorem sub.refl [rewrite] (l : list T) : l ⊆ l :=
λ b i, i
theorem sub.trans {l₁ l₂ l₃ : list T} (H₁ : l₁ ⊆ l₂) (H₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ :=
λ b i, H₂ (H₁ i)
theorem sub_cons (a : T) (l : list T) : l ⊆ a::l :=
theorem sub_cons [rewrite] (a : T) (l : list T) : l ⊆ a::l :=
λ b i, or.inr i
theorem sub_of_cons_sub {a : T} {l₁ l₂ : list T} : a::l₁ ⊆ l₂ → l₁ ⊆ l₂ :=
@ -360,10 +360,10 @@ theorem cons_sub_cons {l₁ l₂ : list T} (a : T) (s : l₁ ⊆ l₂) : (a::l
(λ e : b = a, or.inl e)
(λ i : b ∈ l₁, or.inr (s i))
theorem sub_append_left (l₁ l₂ : list T) : l₁ ⊆ l₁++l₂ :=
theorem sub_append_left [rewrite] (l₁ l₂ : list T) : l₁ ⊆ l₁++l₂ :=
λ b i, iff.mp' (mem_append_iff b l₁ l₂) (or.inl i)
theorem sub_append_right (l₁ l₂ : list T) : l₂ ⊆ l₁++l₂ :=
theorem sub_append_right [rewrite] (l₁ l₂ : list T) : l₂ ⊆ l₁++l₂ :=
λ b i, iff.mp' (mem_append_iff b l₁ l₂) (or.inr i)
theorem sub_cons_of_sub (a : T) {l₁ l₂ : list T} : l₁ ⊆ l₂ → l₁ ⊆ (a::l₂) :=
@ -399,7 +399,7 @@ definition find : T → list T → nat
| a [] := 0
| a (b :: l) := if a = b then 0 else succ (find a l)
theorem find_nil (x : T) : find x [] = 0
theorem find_nil [rewrite] (x : T) : find x [] = 0
theorem find_cons (x y : T) (l : list T) : find x (y::l) = if x = y then 0 else succ (find x l)
@ -461,9 +461,9 @@ definition nth : list T → nat → option T
| (a :: l) 0 := some a
| (a :: l) (n+1) := nth l n
theorem nth_zero (a : T) (l : list T) : nth (a :: l) 0 = some a
theorem nth_zero [rewrite] (a : T) (l : list T) : nth (a :: l) 0 = some a
theorem nth_succ (a : T) (l : list T) (n : nat) : nth (a::l) (succ n) = nth l n
theorem nth_succ [rewrite] (a : T) (l : list T) (n : nat) : nth (a::l) (succ n) = nth l n
theorem nth_eq_some : ∀ {l : list T} {n : nat}, n < length l → Σ a : T, nth l n = some a
| [] n h := absurd h !not_lt_zero

37
library/data/nat/basic.lean
View file

@ -49,10 +49,10 @@ by contradiction
-- add_rewrite succ_ne_zero
theorem pred_zero : pred 0 = 0 :=
theorem pred_zero [rewrite] : pred 0 = 0 :=
rfl
theorem pred_succ (n : ℕ) : pred (succ n) = n :=
theorem pred_succ [rewrite] (n : ℕ) : pred (succ n) = n :=
rfl
theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
@ -103,13 +103,13 @@ general m
/- addition -/
theorem add_zero (n : ℕ) : n + 0 = n :=
theorem add_zero [rewrite] (n : ℕ) : n + 0 = n :=
rfl
theorem add_succ (n m : ℕ) : n + succ m = succ (n + m) :=
theorem add_succ [rewrite] (n m : ℕ) : n + succ m = succ (n + m) :=
rfl
theorem zero_add (n : ℕ) : 0 + n = n :=
theorem zero_add [rewrite] (n : ℕ) : 0 + n = n :=
nat.induction_on n
!add_zero
(take m IH, show 0 + succ m = succ m, from
@ -117,7 +117,7 @@ nat.induction_on n
0 + succ m = succ (0 + m) : add_succ
... = succ m : IH)
theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m) :=
theorem succ_add [rewrite] (n m : ℕ) : (succ n) + m = succ (n + m) :=
nat.induction_on m
(!add_zero ▸ !add_zero)
(take k IH, calc
@ -125,7 +125,7 @@ nat.induction_on m
... = succ (succ (n + k)) : IH
... = succ (n + succ k) : add_succ)
theorem add.comm (n m : ℕ) : n + m = m + n :=
theorem add.comm [rewrite] (n m : ℕ) : n + m = m + n :=
nat.induction_on m
(!add_zero ⬝ !zero_add⁻¹)
(take k IH, calc
@ -136,7 +136,7 @@ nat.induction_on m
theorem succ_add_eq_succ_add (n m : ℕ) : succ n + m = n + succ m :=
!succ_add ⬝ !add_succ⁻¹
theorem add.assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
theorem add.assoc [rewrite] (n m k : ℕ) : (n + m) + k = n + (m + k) :=
nat.induction_on k
(!add_zero ▸ !add_zero)
(take l IH,
@ -146,7 +146,7 @@ nat.induction_on k
... = n + succ (m + l) : add_succ
... = n + (m + succ l) : add_succ)
theorem add.left_comm (n m k : ℕ) : n + (m + k) = m + (n + k) :=
theorem add.left_comm [rewrite] (n m k : ℕ) : n + (m + k) = m + (n + k) :=
left_comm add.comm add.assoc n m k
theorem add.right_comm (n m k : ℕ) : n + m + k = n + k + m :=
@ -186,7 +186,7 @@ eq_zero_of_add_eq_zero_right (!add.comm ⬝ H)
theorem eq_zero_and_eq_zero_of_add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
and.intro (eq_zero_of_add_eq_zero_right H) (eq_zero_of_add_eq_zero_left H)
theorem add_one (n : ℕ) : n + 1 = succ n :=
theorem add_one [rewrite] (n : ℕ) : n + 1 = succ n :=
!add_zero ▸ !add_succ
theorem one_add (n : ℕ) : 1 + n = succ n :=
@ -194,20 +194,20 @@ theorem one_add (n : ℕ) : 1 + n = succ n :=
/- multiplication -/
theorem mul_zero (n : ℕ) : n * 0 = 0 :=
theorem mul_zero [rewrite] (n : ℕ) : n * 0 = 0 :=
rfl
theorem mul_succ (n m : ℕ) : n * succ m = n * m + n :=
theorem mul_succ [rewrite] (n m : ℕ) : n * succ m = n * m + n :=
rfl
-- commutativity, distributivity, associativity, identity
theorem zero_mul (n : ℕ) : 0 * n = 0 :=
theorem zero_mul [rewrite] (n : ℕ) : 0 * n = 0 :=
nat.induction_on n
!mul_zero
(take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
theorem succ_mul (n m : ℕ) : (succ n) * m = (n * m) + m :=
theorem succ_mul [rewrite] (n m : ℕ) : (succ n) * m = (n * m) + m :=
nat.induction_on m
(!mul_zero ⬝ !mul_zero⁻¹ ⬝ !add_zero⁻¹)
(take k IH, calc
@ -219,7 +219,7 @@ nat.induction_on m
... = n * k + n + succ k : add.assoc
... = n * succ k + succ k : mul_succ)
theorem mul.comm (n m : ℕ) : n * m = m * n :=
theorem mul.comm [rewrite] (n m : ℕ) : n * m = m * n :=
nat.induction_on m
(!mul_zero ⬝ !zero_mul⁻¹)
(take k IH, calc
@ -250,7 +250,7 @@ calc
... = n * m + k * n : mul.comm
... = n * m + n * k : mul.comm
theorem mul.assoc (n m k : ℕ) : (n * m) * k = n * (m * k) :=
theorem mul.assoc [rewrite] (n m k : ℕ) : (n * m) * k = n * (m * k) :=
nat.induction_on k
(calc
(n * m) * 0 = n * (m * 0) : mul_zero)
@ -261,13 +261,13 @@ nat.induction_on k
... = n * (m * l + m) : mul.left_distrib
... = n * (m * succ l) : mul_succ)
theorem mul_one (n : ℕ) : n * 1 = n :=
theorem mul_one [rewrite] (n : ℕ) : n * 1 = n :=
calc
n * 1 = n * 0 + n : mul_succ
... = 0 + n : mul_zero
... = n : zero_add
theorem one_mul (n : ℕ) : 1 * n = n :=
theorem one_mul [rewrite] (n : ℕ) : 1 * n = n :=
calc
1 * n = n * 1 : mul.comm
... = n : mul_one
@ -314,5 +314,6 @@ section migrate_algebra
notation a ∣ b := dvd a b
migrate from algebra with nat replacing dvd → dvd
end migrate_algebra
end nat

2
library/init/logic.lean
View file

@ -504,7 +504,7 @@ decidable.rec
(λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t e))
H
theorem if_t_t (c : Prop) [H : decidable c] {A : Type} (t : A) : (if c then t else t) = t :=
theorem if_t_t [rewrite] (c : Prop) [H : decidable c] {A : Type} (t : A) : (if c then t else t) = t :=
decidable.rec
(λ Hc : c, eq.refl (@ite c (decidable.inl Hc) A t t))
(λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t t))

53
library/init/nat.lean
View file

@ -62,6 +62,12 @@ namespace nat
theorem pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
theorem le_succ_iff_true [rewrite] (n : ℕ) : n ≤ succ n ↔ true :=
iff_true_intro (le_succ n)
theorem pred_le_iff_true [rewrite] (n : ℕ) : pred n ≤ n ↔ true :=
iff_true_intro (pred_le n)
theorem le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
by induction H2 with n H2 IH;exact H1;exact le.step IH
@ -86,15 +92,24 @@ namespace nat
theorem not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
by induction n with n IH;all_goals intros;cases a;apply IH;exact le_of_succ_le_succ a
theorem succ_le_self_iff_false [rewrite] (n : ℕ) : succ n ≤ n ↔ false :=
iff_false_intro not_succ_le_self
theorem zero_le (n : ℕ) : 0 ≤ n :=
by induction n with n IH;apply le.refl;exact le.step IH
theorem zero_le_iff_true [rewrite] (n : ℕ) : 0 ≤ n ↔ true :=
iff_true_intro (zero_le n)
theorem lt.step {n m : ℕ} (H : n < m) : n < succ m :=
le.step H
theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
by induction n with n IH;apply le.refl;exact le.step IH
theorem zero_lt_succ_iff_true [rewrite] (n : ℕ) : 0 < succ n ↔ true :=
iff_true_intro (zero_lt_succ n)
theorem lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
le.trans (le.step H1) H2
@ -116,9 +131,19 @@ namespace nat
theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
by intro H; cases H
theorem succ_le_zero_iff_false (n : ℕ) : succ n ≤ zero ↔ false :=
iff_false_intro (not_succ_le_zero n)
theorem lt.irrefl (n : ℕ) : ¬n < n := not_succ_le_self
theorem lt_self_iff_false [rewrite] (n : ℕ) : n < n ↔ false :=
iff_false_intro (lt.irrefl n)
theorem self_lt_succ (n : ℕ) : n < succ n := !le.refl
theorem self_lt_succ_iff_true [rewrite] (n : ℕ) : n < succ n ↔ true :=
iff_true_intro (self_lt_succ n)
theorem lt.base (n : ℕ) : n < succ n := !le.refl
theorem le_lt_antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m < n) : false :=
@ -147,15 +172,18 @@ namespace nat
theorem lt.trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
theorem lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
lt.by_cases H1 (λH, H2 (le_of_eq H⁻¹)) (λH, H2 (le_of_lt H))
theorem lt_or_ge (a b : ℕ) : (a < b) ∨ (a ≥ b) :=
lt_ge_by_cases inl inr
definition not_lt_zero (a : ℕ) : ¬ a < zero :=
theorem not_lt_zero (a : ℕ) : ¬ a < zero :=
by intro H; cases H
theorem lt_zero_iff_false [rewrite] (a : ℕ) : a < zero ↔ false :=
iff_false_intro (not_lt_zero a)
-- less-than is well-founded
definition lt.wf [instance] : well_founded lt :=
begin
@ -221,7 +249,7 @@ namespace nat
definition max (a b : ℕ) : ℕ := if a < b then b else a
definition min (a b : ℕ) : ℕ := if a < b then a else b
theorem max_self (a : ℕ) : max a a = a :=
theorem max_self [rewrite] (a : ℕ) : max a a = a :=
eq.rec_on !if_t_t rfl
theorem max_eq_right' {a b : ℕ} (H : a < b) : max a b = b :=
@ -242,6 +270,9 @@ namespace nat
(λ h : a < b, le_of_lt (eq.rec_on (eq_max_right h) h))
(λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
theorem le_max_left_iff_true [rewrite] (a b : ℕ) : a ≤ max a b ↔ true :=
iff_true_intro (le_max_left a b)
theorem le_max_right (a b : ℕ) : b ≤ max a b :=
by_cases
(λ h : a < b, eq.rec_on (eq_max_right h) !le.refl)
@ -251,13 +282,16 @@ namespace nat
have aux : a = max a b, from eq_max_left (lt.asymm h),
eq.rec_on aux (le_of_lt h)))
theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
theorem le_max_right_iff_true [rewrite] (a b : ℕ) : b ≤ max a b ↔ true :=
iff_true_intro (le_max_right a b)
theorem succ_sub_succ_eq_sub [rewrite] (a b : ℕ) : succ a - succ b = a - b :=
by induction b with b IH;reflexivity; apply congr (eq.refl pred) IH
theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
eq.rec_on (succ_sub_succ_eq_sub a b) rfl
theorem zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
theorem zero_sub_eq_zero [rewrite] (a : ℕ) : zero - a = zero :=
nat.rec_on a
rfl
(λ a₁ (ih : zero - a₁ = zero), congr (eq.refl pred) ih)
@ -285,5 +319,12 @@ namespace nat
(le.refl a)
(λ b₁ ih, le.trans !pred_le ih)
lemma sub_lt_succ (a b : ℕ) : a - b < succ a := lt_succ_of_le (sub_le a b)
theorem sub_le_iff_true [rewrite] (a b : ℕ) : a - b ≤ a ↔ true :=
iff_true_intro (sub_le a b)
theorem sub_lt_succ (a b : ℕ) : a - b < succ a :=
lt_succ_of_le (sub_le a b)
theorem sub_lt_succ_iff_true [rewrite] (a b : ℕ) : a - b < succ a ↔ true :=
iff_true_intro (sub_lt_succ a b)
end nat

36
library/logic/connectives.lean
View file

@ -61,10 +61,10 @@ have Hnp : ¬a, from
assume Hp : a, absurd (or.inl Hp) not_em,
absurd (or.inr Hnp) not_em
theorem not_true : ¬ true ↔ false :=
theorem not_true [rewrite] : ¬ true ↔ false :=
iff.intro (assume H, H trivial) (assume H, false.elim H)
theorem not_false_iff : ¬ false ↔ true :=
theorem not_false_iff [rewrite] : ¬ false ↔ true :=
iff.intro (assume H, trivial) (assume H H1, H1)
theorem not_iff_not (H : a ↔ b) : ¬a ↔ ¬b :=
@ -105,19 +105,19 @@ iff.intro
obtain Ha Hb Hc, from H,
and.intro (and.intro Ha Hb) Hc)
theorem and_true (a : Prop) : a ∧ true ↔ a :=
theorem and_true [rewrite] (a : Prop) : a ∧ true ↔ a :=
iff.intro (assume H, and.left H) (assume H, and.intro H trivial)
theorem true_and (a : Prop) : true ∧ a ↔ a :=
theorem true_and [rewrite] (a : Prop) : true ∧ a ↔ a :=
iff.intro (assume H, and.right H) (assume H, and.intro trivial H)
theorem and_false (a : Prop) : a ∧ false ↔ false :=
theorem and_false [rewrite] (a : Prop) : a ∧ false ↔ false :=
iff.intro (assume H, and.right H) (assume H, false.elim H)
theorem false_and (a : Prop) : false ∧ a ↔ false :=
theorem false_and [rewrite] (a : Prop) : false ∧ a ↔ false :=
iff.intro (assume H, and.left H) (assume H, false.elim H)
theorem and_self (a : Prop) : a ∧ a ↔ a :=
theorem and_self [rewrite] (a : Prop) : a ∧ a ↔ a :=
iff.intro (assume H, and.left H) (assume H, and.intro H H)
theorem and_imp_eq (a b c : Prop) : (a ∧ b → c) = (a → b → c) :=
@ -190,18 +190,18 @@ iff.intro
(assume Hb, or.inl (or.inr Hb))
(assume Hc, or.inr Hc)))
theorem or_true (a : Prop) : a ∨ true ↔ true :=
theorem or_true [rewrite] (a : Prop) : a ∨ true ↔ true :=
iff.intro (assume H, trivial) (assume H, or.inr H)
theorem true_or (a : Prop) : true ∨ a ↔ true :=
theorem true_or [rewrite] (a : Prop) : true ∨ a ↔ true :=
iff.intro (assume H, trivial) (assume H, or.inl H)
theorem or_false (a : Prop) : a ∨ false ↔ a :=
theorem or_false [rewrite] (a : Prop) : a ∨ false ↔ a :=
iff.intro
(assume H, or.elim H (assume H1 : a, H1) (assume H1 : false, false.elim H1))
(assume H, or.inl H)
theorem false_or (a : Prop) : false ∨ a ↔ a :=
theorem false_or [rewrite] (a : Prop) : false ∨ a ↔ a :=
iff.intro
(assume H, or.elim H (assume H1 : false, false.elim H1) (assume H1 : a, H1))
(assume H, or.inr H)
@ -249,22 +249,22 @@ propext (!or.comm) ▸ propext (!or.comm) ▸ propext (!or.comm) ▸ !or.distrib
definition iff.def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
!eq.refl
theorem iff_true (a : Prop) : (a ↔ true) ↔ a :=
theorem iff_true [rewrite] (a : Prop) : (a ↔ true) ↔ a :=
iff.intro
(assume H, iff.mp' H trivial)
(assume H, iff.intro (assume H1, trivial) (assume H1, H))
theorem true_iff (a : Prop) : (true ↔ a) ↔ a :=
theorem true_iff [rewrite] (a : Prop) : (true ↔ a) ↔ a :=
iff.intro
(assume H, iff.mp H trivial)
(assume H, iff.intro (assume H1, H) (assume H1, trivial))
theorem iff_false (a : Prop) : (a ↔ false) ↔ ¬ a :=
theorem iff_false [rewrite] (a : Prop) : (a ↔ false) ↔ ¬ a :=
iff.intro
(assume H, and.left H)
(assume H, and.intro H (assume H1, false.elim H1))
theorem false_iff (a : Prop) : (false ↔ a) ↔ ¬ a :=
theorem false_iff [rewrite] (a : Prop) : (false ↔ a) ↔ ¬ a :=
iff.intro
(assume H, and.right H)
(assume H, and.intro (assume H1, false.elim H1) H)
@ -272,7 +272,7 @@ iff.intro
theorem iff_true_of_self (Ha : a) : a ↔ true :=
iff.intro (assume H, trivial) (assume H, Ha)
theorem iff_self (a : Prop) : (a ↔ a) ↔ true :=
theorem iff_self [rewrite] (a : Prop) : (a ↔ a) ↔ true :=
iff_true_of_self !iff.refl
theorem forall_iff_forall {A : Type} {P Q : A → Prop} (H : ∀a, (P a ↔ Q a)) : (∀a, P a) ↔ ∀a, Q a :=
@ -291,10 +291,10 @@ section
variables {A : Type} {c₁ c₂ : Prop}
definition if_true (t e : A) : (if true then t else e) = t :=
definition if_true [rewrite] (t e : A) : (if true then t else e) = t :=
if_pos trivial
definition if_false (t e : A) : (if false then t else e) = e :=
definition if_false [rewrite] (t e : A) : (if false then t else e) = e :=
if_neg not_false
end