michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

refactor(library): rename iff.mp' to iff.mpr

Browse source
This commit is contained in:
Leonardo de Moura 2015-07-18 05:28:53 -04:00
parent c72e661a9e
commit ade60278d0
20 changed files with 72 additions and 72 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

8
library/algebra/order.lean
View file

@ -252,7 +252,7 @@ theorem lt_or_eq_of_le [s : strong_order_pair A] {a b : A} (le_ab : a ≤ b) : a
iff.mp le_iff_lt_or_eq le_ab
theorem le_of_lt_or_eq [s : strong_order_pair A] {a b : A} (lt_or_eq : a < b ∨ a = b) : a ≤ b :=
iff.mp' le_iff_lt_or_eq lt_or_eq
iff.mpr le_iff_lt_or_eq lt_or_eq
private theorem lt_irrefl' [s : strong_order_pair A] (a : A) : ¬ a < a :=
!strong_order_pair.lt_irrefl
@ -268,7 +268,7 @@ iff.intro
or_resolve_left Hor (and.right Hand))
theorem lt_of_le_of_ne [s : strong_order_pair A] {a b : A} : a ≤ b → a ≠ b → a < b :=
take H1 H2, iff.mp' lt_iff_le_and_ne (and.intro H1 H2)
take H1 H2, iff.mpr lt_iff_le_and_ne (and.intro H1 H2)
private theorem ne_of_lt' [s : strong_order_pair A] {a b : A} (H : a < b) : a ≠ b :=
and.right ((iff.mp lt_iff_le_and_ne) H)
@ -282,7 +282,7 @@ have ne_ac : a ≠ c, from
have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
have eq_ab : a = b, from le.antisymm (le_of_lt' _ _ lt_ab) le_ba,
show false, from ne_of_lt' lt_ab eq_ab,
show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
show a < c, from iff.mpr (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
theorem lt_of_le_of_lt' [s : strong_order_pair A] (a b c : A) : a ≤ b → b < c → a < c :=
assume le_ab : a ≤ b,
@ -293,7 +293,7 @@ have ne_ac : a ≠ c, from
have le_cb : c ≤ b, from eq_ac ▸ le_ab,
have eq_bc : b = c, from le.antisymm (le_of_lt' _ _ lt_bc) le_cb,
show false, from ne_of_lt' lt_bc eq_bc,
show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
show a < c, from iff.mpr (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
definition strong_order_pair.to_order_pair [trans-instance] [coercion] [reducible]
[s : strong_order_pair A] : order_pair A :=

10
library/algebra/ordered_field.lean
View file

@ -64,7 +64,7 @@ section linear_ordered_field
(iff.mp (one_le_div_iff_le Hb)) H
theorem one_le_div_of_le (Hb : b > 0) (H : b ≤ a) : 1 ≤ a / b :=
(iff.mp' (one_le_div_iff_le Hb)) H
(iff.mpr (one_le_div_iff_le Hb)) H
theorem one_lt_div_iff_lt (Hb : b > 0) : 1 < a / b ↔ b < a :=
have Hb' : b ≠ 0, from ne.symm (ne_of_lt Hb),
@ -83,7 +83,7 @@ section linear_ordered_field
(iff.mp (one_lt_div_iff_lt Hb)) H
theorem one_lt_div_of_lt (Hb : b > 0) (H : b < a) : 1 < a / b :=
(iff.mp' (one_lt_div_iff_lt Hb)) H
(iff.mpr (one_lt_div_iff_lt Hb)) H
theorem exists_lt : ∃ x, x < a :=
have H : a - 1 < a, from add_lt_of_le_of_neg (le.refl _) zero_gt_neg_one,
@ -296,7 +296,7 @@ theorem nonneg_le_nonneg_of_squares_le (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a
end
theorem div_two : (a + a) / 2 = a :=
symm (iff.mp' (eq_div_iff_mul_eq (ne_of_gt (add_pos zero_lt_one zero_lt_one)))
symm (iff.mpr (eq_div_iff_mul_eq (ne_of_gt (add_pos zero_lt_one zero_lt_one)))
(by rewrite [left_distrib, *mul_one]))
@ -434,7 +434,7 @@ section discrete_linear_ordered_field
rewrite -mul_sub_left_distrib,
apply mul_neg_of_pos_of_neg,
exact Hc,
apply iff.mp' (sub_neg_iff_lt _ _),
apply iff.mpr (sub_neg_iff_lt _ _),
apply div_lt_div_of_lt,
exact Hb, exact H
end
@ -457,7 +457,7 @@ section discrete_linear_ordered_field
... ≥ b - c : sub_le_of_nonneg _ _ (le.trans !abs_nonneg H))
else
(have Habs : b - a ≤ c, by rewrite [abs_of_neg (lt_of_not_ge Hz) at H, neg_sub at H]; apply H,
have Habs' : b ≤ c + a, from (iff.mp' !le_add_iff_sub_right_le) Habs,
have Habs' : b ≤ c + a, from (iff.mpr !le_add_iff_sub_right_le) Habs,
(iff.mp !le_add_iff_sub_left_le) Habs')
theorem ge_sub_of_abs_sub_le_right (H : abs (a - b) ≤ c) : b ≥ a - c :=

6
library/algebra/ordered_group.lean
View file

@ -439,7 +439,7 @@ section
assume H₂ : d ≤ a + c,
have H : d - a ≤ min b c,
from le_min (iff.mp !le_add_iff_sub_left_le H₁) (iff.mp !le_add_iff_sub_left_le H₂),
show d ≤ a + min b c, from iff.mp' !le_add_iff_sub_left_le H))
show d ≤ a + min b c, from iff.mpr !le_add_iff_sub_left_le H))
theorem min_add_add_right : min (a + c) (b + c) = min a b + c :=
by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply min_add_add_left
@ -451,7 +451,7 @@ section
(λ d H₁ H₂,
have H : max b c ≤ d - a,
from max_le (iff.mp !add_le_iff_le_sub_left H₁) (iff.mp !add_le_iff_le_sub_left H₂),
show a + max b c ≤ d, from iff.mp' !add_le_iff_le_sub_left H))
show a + max b c ≤ d, from iff.mpr !add_le_iff_le_sub_left H))
theorem max_add_add_right : max (a + c) (b + c) = max a b + c :=
by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply max_add_add_left
@ -630,7 +630,7 @@ section
(assume H2 : 0 ≤ a + b, aux2 H2)
(assume H2 : a + b ≤ 0,
assert H3 : -a + -b = -(a + b), by rewrite neg_add,
assert H4 : -(a + b) ≥ 0, from iff.mp' (neg_nonneg_iff_nonpos (a+b)) H2,
assert H4 : -(a + b) ≥ 0, from iff.mpr (neg_nonneg_iff_nonpos (a+b)) H2,
have H5 : -a + -b ≥ 0, begin rewrite -H3 at H4, exact H4 end,
calc
abs (a + b) = abs (-a + -b) : by rewrite [-abs_neg, neg_add]

10
library/algebra/ordered_ring.lean
View file

@ -249,7 +249,7 @@ section
include s
theorem mul_le_mul_of_nonpos_left (H : b ≤ a) (Hc : c ≤ 0) : c * a ≤ c * b :=
have Hc' : -c ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos Hc,
have Hc' : -c ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos Hc,
assert H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc',
have H2 : -(c * b) ≤ -(c * a),
begin
@ -259,7 +259,7 @@ section
iff.mp !neg_le_neg_iff_le H2
theorem mul_le_mul_of_nonpos_right (H : b ≤ a) (Hc : c ≤ 0) : a * c ≤ b * c :=
have Hc' : -c ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos Hc,
have Hc' : -c ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos Hc,
assert H1 : b * -c ≤ a * -c, from mul_le_mul_of_nonneg_right H Hc',
have H2 : -(b * c) ≤ -(a * c),
begin
@ -276,7 +276,7 @@ section
end
theorem mul_lt_mul_of_neg_left (H : b < a) (Hc : c < 0) : c * a < c * b :=
have Hc' : -c > 0, from iff.mp' !neg_pos_iff_neg Hc,
have Hc' : -c > 0, from iff.mpr !neg_pos_iff_neg Hc,
assert H1 : -c * b < -c * a, from mul_lt_mul_of_pos_left H Hc',
have H2 : -(c * b) < -(c * a),
begin
@ -286,7 +286,7 @@ section
iff.mp !neg_lt_neg_iff_lt H2
theorem mul_lt_mul_of_neg_right (H : b < a) (Hc : c < 0) : a * c < b * c :=
have Hc' : -c > 0, from iff.mp' !neg_pos_iff_neg Hc,
have Hc' : -c > 0, from iff.mpr !neg_pos_iff_neg Hc,
assert H1 : b * -c < a * -c, from mul_lt_mul_of_pos_right H Hc',
have H2 : -(b * c) < -(a * c),
begin
@ -413,7 +413,7 @@ section
theorem gt_of_mul_lt_mul_neg_left {a b c : A} (H : c * a < c * b) (Hc : c ≤ 0) : a > b :=
have nhc : -c ≥ 0, from neg_nonneg_of_nonpos Hc,
have H2 : -(c * b) < -(c * a), from iff.mp' (neg_lt_neg_iff_lt _ _) H,
have H2 : -(c * b) < -(c * a), from iff.mpr (neg_lt_neg_iff_lt _ _) H,
have H3 : (-c) * b < (-c) * a, from calc
(-c) * b = - (c * b) : neg_mul_eq_neg_mul
... < -(c * a) : H2

2
library/data/finset/basic.lean
View file

@ -123,7 +123,7 @@ theorem not_mem_empty [rewrite] (a : A) : a ∉ ∅ :=
λ aine : a ∈ ∅, aine
theorem mem_empty_iff [rewrite] (x : A) : x ∈ ∅ ↔ false :=
iff.mp' !iff_false_iff_not !not_mem_empty
iff.mpr !iff_false_iff_not !not_mem_empty
theorem mem_empty_eq (x : A) : x ∈ ∅ = false :=
propext !mem_empty_iff

6
library/data/int/div.lean
View file

@ -304,7 +304,7 @@ have H2 : a mod (abs b) ≥ 0, from
calc
-[1+m] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... ≥ 0 : iff.mp' !sub_nonneg_iff_le H3),
... ≥ 0 : iff.mpr !sub_nonneg_iff_le H3),
!mod_abs ▸ H2
theorem mod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) : a mod b < (abs b) :=
@ -419,7 +419,7 @@ by rewrite [↑modulo, !mul_div_mul_of_pos H, mul_sub_left_distrib, mul.left_com
theorem lt_div_add_one_mul_self (a : ℤ) {b : ℤ} (H : b > 0) : a < (a div b + 1) * b :=
have H : a - a div b * b < b, from !mod_lt_of_pos H,
calc
a < a div b * b + b : iff.mp' !lt_add_iff_sub_lt_left H
a < a div b * b + b : iff.mpr !lt_add_iff_sub_lt_left H
... = (a div b + 1) * b : by rewrite [mul.right_distrib, one_mul]
theorem div_le_of_nonneg_of_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≤ a :=
@ -488,7 +488,7 @@ nat.by_cases_zero_pos n
theorem of_nat_dvd_of_nat_of_dvd {m n : ℕ} (H : #nat m ∣ n) : of_nat m ∣ of_nat n :=
nat.dvd.elim H
(take k, assume H1 : #nat n = m * k,
dvd.intro (!iff.mp' !of_nat_eq_of_nat H1⁻¹))
dvd.intro (!iff.mpr !of_nat_eq_of_nat H1⁻¹))
theorem of_nat_dvd_of_nat (m n : ℕ) : of_nat m ∣ of_nat n ↔ (#nat m ∣ n) :=
iff.intro dvd_of_of_nat_dvd_of_nat of_nat_dvd_of_nat_of_dvd

14
library/data/int/order.lean
View file

@ -40,7 +40,7 @@ show nonneg (b - a), from H1⁻¹ ▸ H2
theorem le.elim {a b : ℤ} (H : a ≤ b) : ∃n : ℕ, a + n = b :=
obtain (n : ℕ) (H1 : b - a = n), from nonneg.elim H,
exists.intro n (!add.comm ▸ iff.mp' !add_eq_iff_eq_add_neg (H1⁻¹))
exists.intro n (!add.comm ▸ iff.mpr !add_eq_iff_eq_add_neg (H1⁻¹))
theorem le.total (a b : ℤ) : a ≤ b ∨ b ≤ a :=
or.elim (nonneg_or_nonneg_neg (b - a))
@ -91,7 +91,7 @@ theorem lt_of_of_nat_lt_of_nat {m n : ℕ} (H : of_nat m < of_nat n) : #nat m <
iff.mp !of_nat_lt_of_nat H
theorem of_nat_lt_of_nat_of_lt {m n : ℕ} (H : #nat m < n) : of_nat m < of_nat n :=
iff.mp' !of_nat_lt_of_nat H
iff.mpr !of_nat_lt_of_nat H
/- show that the integers form an ordered additive group -/
@ -186,7 +186,7 @@ le.intro H2
theorem add_lt_add_left {a b : ℤ} (H : a < b) (c : ℤ) : c + a < c + b :=
let H' := le_of_lt H in
(iff.mp' (lt_iff_le_and_ne _ _)) (and.intro (add_le_add_left H' _)
(iff.mpr (lt_iff_le_and_ne _ _)) (and.intro (add_le_add_left H' _)
(take Heq, let Heq' := add_left_cancel Heq in
!lt.irrefl (Heq' ▸ H)))
@ -228,13 +228,13 @@ theorem not_le_of_gt {a b : ℤ} (H : a < b) : ¬ b ≤ a :=
theorem lt_of_lt_of_le {a b c : ℤ} (Hab : a < b) (Hbc : b ≤ c) : a < c :=
let Hab' := le_of_lt Hab in
let Hac := le.trans Hab' Hbc in
(iff.mp' !lt_iff_le_and_ne) (and.intro Hac
(iff.mpr !lt_iff_le_and_ne) (and.intro Hac
(assume Heq, not_le_of_gt (Heq ▸ Hab) Hbc))
theorem lt_of_le_of_lt {a b c : ℤ} (Hab : a ≤ b) (Hbc : b < c) : a < c :=
let Hbc' := le_of_lt Hbc in
let Hac := le.trans Hab Hbc' in
(iff.mp' !lt_iff_le_and_ne) (and.intro Hac
(iff.mpr !lt_iff_le_and_ne) (and.intro Hac
(assume Heq, not_le_of_gt (Heq⁻¹ ▸ Hbc) Hab))
section migrate_algebra
@ -308,7 +308,7 @@ obtain (n : ℕ) (H1 : 0 + of_nat n = a), from le.elim H,
exists.intro n (!zero_add ▸ (H1⁻¹))
theorem exists_eq_neg_of_nat {a : ℤ} (H : a ≤ 0) : ∃n : ℕ, a = -(of_nat n) :=
have H2 : -a ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos H,
have H2 : -a ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos H,
obtain (n : ℕ) (Hn : -a = of_nat n), from exists_eq_of_nat H2,
exists.intro n (eq_neg_of_eq_neg (Hn⁻¹))
@ -317,7 +317,7 @@ obtain (n : ℕ) (Hn : a = of_nat n), from exists_eq_of_nat H,
Hn⁻¹ ▸ congr_arg of_nat (nat_abs_of_nat n)
theorem of_nat_nat_abs_of_nonpos {a : ℤ} (H : a ≤ 0) : of_nat (nat_abs a) = -a :=
have H1 : (-a) ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos H,
have H1 : (-a) ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos H,
calc
of_nat (nat_abs a) = of_nat (nat_abs (-a)) : nat_abs_neg
... = -a : of_nat_nat_abs_of_nonneg H1

4
library/data/list/basic.lean
View file

@ -368,10 +368,10 @@ theorem cons_sub_cons {l₁ l₂ : list T} (a : T) (s : l₁ ⊆ l₂) : (a::l
(λ i : b ∈ l₁, or.inr (s i))
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)
λ b i, iff.mpr (mem_append_iff b l₁ l₂) (or.inl i)
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)
λ b i, iff.mpr (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) :=
λ (s : l₁ ⊆ l₂) (x : T) (i : x ∈ l₁), or.inr (s i)

6
library/data/list/perm.lean
View file

@ -685,7 +685,7 @@ open eq.ops
theorem perm_ext : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → (∀a, a ∈ l₁ ↔ a ∈ l₂) → l₁ ~ l₂
| [] [] d₁ d₂ e := !perm.nil
| [] (a₂::t₂) d₁ d₂ e := absurd (iff.mp' (e a₂) !mem_cons) (not_mem_nil a₂)
| [] (a₂::t₂) d₁ d₂ e := absurd (iff.mpr (e a₂) !mem_cons) (not_mem_nil a₂)
| (a₁::t₁) [] d₁ d₂ e := absurd (iff.mp (e a₁) !mem_cons) (not_mem_nil a₁)
| (a₁::t₁) (a₂::t₂) d₁ d₂ e :=
have a₁inl₂ : a₁ ∈ a₂::t₂, from iff.mp (e a₁) !mem_cons,
@ -711,13 +711,13 @@ theorem perm_ext : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → (∀a
(λ ains₁s₂ : a ∈ s₁ ++ s₂, or.elim (mem_or_mem_of_mem_append ains₁s₂)
(λ ains₁ : a ∈ s₁,
have aina₂t₂ : a ∈ a₂::t₂, from by rewrite [t₂_eq]; exact (mem_append_left _ ains₁),
have aina₁t₁ : a ∈ a₁::t₁, from iff.mp' (e a) aina₂t₂,
have aina₁t₁ : a ∈ a₁::t₁, from iff.mpr (e a) aina₂t₂,
or.elim (eq_or_mem_of_mem_cons aina₁t₁)
(λ aeqa₁ : a = a₁, absurd (aeqa₁ ▸ ains₁) na₁s₁)
(λ aint₁ : a ∈ t₁, aint₁))
(λ ains₂ : a ∈ s₂,
have aina₂t₂ : a ∈ a₂::t₂, from by rewrite [t₂_eq]; exact (mem_append_right _ (mem_cons_of_mem _ ains₂)),
have aina₁t₁ : a ∈ a₁::t₁, from iff.mp' (e a) aina₂t₂,
have aina₁t₁ : a ∈ a₁::t₁, from iff.mpr (e a) aina₂t₂,
or.elim (eq_or_mem_of_mem_cons aina₁t₁)
(λ aeqa₁ : a = a₁, absurd (aeqa₁ ▸ ains₂) na₁s₂)
(λ aint₁ : a ∈ t₁, aint₁))),

4
library/data/nat/parity.lean
View file

@ -45,13 +45,13 @@ lemma odd_iff_not_even : ∀ n, odd n ↔ ¬ even n :=
λ n, !iff.refl
lemma odd_of_not_even {n} : ¬ even n → odd n :=
λ h, iff.mp' !odd_iff_not_even h
λ h, iff.mpr !odd_iff_not_even h
lemma even_of_not_odd {n} : ¬ odd n → even n :=
λ h, not_not_elim (iff.mp (not_iff_not_of_iff !odd_iff_not_even) h)
lemma not_odd_of_even {n} : even n → ¬ odd n :=
λ h, iff.mp' (not_iff_not_of_iff !odd_iff_not_even) (not_not_intro h)
λ h, iff.mpr (not_iff_not_of_iff !odd_iff_not_even) (not_not_intro h)
lemma not_even_of_odd {n} : odd n → ¬ even n :=
λ h, iff.mp !odd_iff_not_even h

8
library/data/pnat.lean
View file

@ -106,16 +106,16 @@ theorem pnat.to_rat_of_nat (n : ℕ+) : rat_of_pnat n = of_nat n~ :=
theorem rat_of_nat_nonneg (n : ℕ) : 0 ≤ of_nat n := trivial
theorem rat_of_pnat_ge_one (n : ℕ+) : rat_of_pnat n ≥ 1 :=
pnat.rec_on n (λ m h, (iff.mp' !of_nat_le_of_nat) (succ_le_of_lt h))
pnat.rec_on n (λ m h, (iff.mpr !of_nat_le_of_nat) (succ_le_of_lt h))
theorem rat_of_pnat_is_pos (n : ℕ+) : rat_of_pnat n > 0 :=
pnat.rec_on n (λ m h, (iff.mp' !of_nat_pos) h)
pnat.rec_on n (λ m h, (iff.mpr !of_nat_pos) h)
theorem of_nat_le_of_nat_of_le {m n : ℕ} (H : m ≤ n) : of_nat m ≤ of_nat n :=
(iff.mp' !of_nat_le_of_nat) H
(iff.mpr !of_nat_le_of_nat) H
theorem of_nat_lt_of_nat_of_lt {m n : ℕ} (H : m < n) : of_nat m < of_nat n :=
(iff.mp' !of_nat_lt_of_nat) H
(iff.mpr !of_nat_lt_of_nat) H
theorem rat_of_pnat_le_of_pnat_le {m n : ℕ+} (H : m ≤ n) : rat_of_pnat m ≤ rat_of_pnat n :=
begin

4
library/data/rat/basic.lean
View file

@ -511,7 +511,7 @@ end migrate_algebra
theorem eq_num_div_denom (a : ℚ) : a = num a / denom a :=
have H : of_int (denom a) ≠ 0, from assume H', ne_of_gt (denom_pos a) (of_int.inj H'),
iff.mp' (eq_div_iff_mul_eq H) (mul_denom a)
iff.mpr (eq_div_iff_mul_eq H) (mul_denom a)
theorem of_nat_div {a b : ℤ} (H : b ∣ a) : of_int (a div b) = of_int a / of_int b :=
decidable.by_cases
@ -523,7 +523,7 @@ decidable.by_cases
int.dvd.elim H
(take c, assume Hc : a = b * c,
by rewrite [Hc, !int.mul_div_cancel_left bnz, mul.comm]),
iff.mp' (eq_div_iff_mul_eq bnz') H')
iff.mpr (eq_div_iff_mul_eq bnz') H')
end rat

16
library/data/rat/order.lean
View file

@ -181,7 +181,7 @@ theorem of_nat_pos (a : ℕ) : (of_nat a > 0) ↔ (#nat a > nat.zero) :=
!of_nat_lt_of_nat
theorem of_nat_nonneg (a : ℕ) : (of_nat a ≥ 0) :=
iff.mp' !of_nat_le_of_nat !nat.zero_le
iff.mpr !of_nat_le_of_nat !nat.zero_le
theorem le.refl (a : ℚ) : a ≤ a :=
by rewrite [↑rat.le, sub_self]; apply nonneg_zero
@ -223,7 +223,7 @@ iff.intro
(assume H : a ≤ b,
decidable.by_cases
(assume H1 : a = b, or.inr H1)
(assume H1 : a ≠ b, or.inl (iff.mp' !lt_iff_le_and_ne (and.intro H H1))))
(assume H1 : a ≠ b, or.inl (iff.mpr !lt_iff_le_and_ne (and.intro H H1))))
(assume H : a < b ∨ a = b,
or.elim H
(assume H1 : a < b, and.left (iff.mp !lt_iff_le_and_ne H1))
@ -251,7 +251,7 @@ have H : pos (a * b), from pos_mul (to_pos H1) (to_pos H2),
definition decidable_lt [instance] : decidable_rel rat.lt :=
take a b, decidable_pos (b - a)
theorem le_of_lt (H : a < b) : a ≤ b := iff.mp' !le_iff_lt_or_eq (or.inl H)
theorem le_of_lt (H : a < b) : a ≤ b := iff.mpr !le_iff_lt_or_eq (or.inl H)
theorem lt_irrefl (a : ℚ) : ¬ a < a :=
take Ha,
@ -266,20 +266,20 @@ theorem not_le_of_gt (H : a < b) : ¬ b ≤ a :=
theorem lt_of_lt_of_le (Hab : a < b) (Hbc : b ≤ c) : a < c :=
let Hab' := le_of_lt Hab in
let Hac := le.trans Hab' Hbc in
(iff.mp' !lt_iff_le_and_ne) (and.intro Hac
(iff.mpr !lt_iff_le_and_ne) (and.intro Hac
(assume Heq, not_le_of_gt (Heq ▸ Hab) Hbc))
theorem lt_of_le_of_lt (Hab : a ≤ b) (Hbc : b < c) : a < c :=
let Hbc' := le_of_lt Hbc in
let Hac := le.trans Hab Hbc' in
(iff.mp' !lt_iff_le_and_ne) (and.intro Hac
(iff.mpr !lt_iff_le_and_ne) (and.intro Hac
(assume Heq, not_le_of_gt (Heq⁻¹ ▸ Hbc) Hab))
theorem zero_lt_one : (0 : ℚ) < 1 := trivial
theorem add_lt_add_left (H : a < b) (c : ℚ) : c + a < c + b :=
let H' := le_of_lt H in
(iff.mp' (lt_iff_le_and_ne _ _)) (and.intro (add_le_add_left H' _)
(iff.mpr (lt_iff_le_and_ne _ _)) (and.intro (add_le_add_left H' _)
(take Heq, let Heq' := add_left_cancel Heq in
!lt_irrefl (Heq' ▸ H)))
@ -334,9 +334,9 @@ definition ubound : ℚ → ℕ := λ a : ℚ, nat.succ (int.nat_abs (num a))
theorem ubound_ge (a : ℚ) : of_nat (ubound a) ≥ a :=
have H : abs a * abs (of_int (denom a)) = abs (of_int (num a)), from !abs_mul ▸ !mul_denom ▸ rfl,
have H'' : 1 ≤ abs (of_int (denom a)), begin
have J : of_int (denom a) > 0, from (iff.mp' !of_int_pos) !denom_pos,
have J : of_int (denom a) > 0, from (iff.mpr !of_int_pos) !denom_pos,
rewrite (abs_of_pos J),
apply iff.mp' !of_int_le_of_int,
apply iff.mpr !of_int_le_of_int,
apply denom_pos
end,
have H' : abs a ≤ abs (of_int (num a)), from

4
library/data/real/basic.lean
View file

@ -31,7 +31,7 @@ theorem find_midpoint {a b : ℚ} (H : a > b) : ∃ c : ℚ, a > b + c ∧ c > 0
have H3 [visible] : (a + a) / (1 + 1) > ((b + b) + (a - b)) / (1 + 1),
from div_lt_div_of_lt_of_pos H2 dec_trivial,
by rewrite [div_two at H3, -div_add_div_same at H3, div_two at H3]; exact H3)
(pos_div_of_pos_of_pos (iff.mp' !sub_pos_iff_lt H) dec_trivial))
(pos_div_of_pos_of_pos (iff.mpr !sub_pos_iff_lt H) dec_trivial))
theorem add_sub_comm (a b c d : ℚ) : a + b - (c + d) = (a - c) + (b - d) := sorry
@ -73,7 +73,7 @@ theorem squeeze_2 {a b : ℚ} (H : ∀ ε : ℚ, ε > 0 → a ≥ b - ε) : a
apply (exists.elim (find_midpoint Hb)),
intro c Hc,
let Hc' := H c (and.right Hc),
apply (rat.not_le_of_gt (and.left Hc)) ((iff.mp' !le_add_iff_sub_right_le) Hc')
apply (rat.not_le_of_gt (and.left Hc)) (iff.mpr !le_add_iff_sub_right_le Hc')
end
theorem rewrite_helper (a b c d : ℚ) : a * b - c * d = a * (b - d) + (a - c) * d :=

16
library/data/real/order.lean
View file

@ -66,9 +66,9 @@ theorem bdd_away_of_pos {s : seq} (Hs : regular s) (H : pos s) :
existsi N,
intro m Hm,
have Habs : abs (s m - s n) ≥ s n - s m, by rewrite abs_sub; apply le_abs_self,
have Habs' : s m + abs (s m - s n) ≥ s n, from (iff.mp' (le_add_iff_sub_left_le _ _ _)) Habs,
have Habs' : s m + abs (s m - s n) ≥ s n, from (iff.mpr (le_add_iff_sub_left_le _ _ _)) Habs,
have HN' : N⁻¹ + N⁻¹ ≤ s n - n⁻¹, begin
apply iff.mp' (le_add_iff_sub_right_le _ _ _),
apply iff.mpr (le_add_iff_sub_right_le _ _ _),
rewrite [sub_neg_eq_add, add.comm, -add.assoc],
apply le_of_lt HN
end,
@ -164,7 +164,7 @@ theorem pos_of_pos_equiv {s t : seq} (Hs : regular s) (Heq : s ≡ t) (Hp : pos
have Hs4 : N⁻¹ ≤ s (2 * 2 * N), from HN _ (!mul_le_mul_left),
apply lt_of_lt_of_le,
rotate 1,
apply iff.mp' (rat.add_le_add_right_iff _ _ _),
apply iff.mpr (rat.add_le_add_right_iff _ _ _),
apply Hs4,
rewrite [*pnat.mul.assoc, pnat.add_halves, -(add_halves N), rat.add_sub_cancel],
apply inv_two_mul_lt_inv
@ -443,7 +443,7 @@ theorem le_and_sep_of_lt {s t : seq} (Hs : regular s) (Ht : regular t) (Lst : s_
... ≥ 0 - n⁻¹: begin
apply rat.sub_le_sub_right,
apply le_of_lt,
apply (iff.mp' (sub_pos_iff_lt _ _)),
apply (iff.mpr (sub_pos_iff_lt _ _)),
apply HN
end
... = -n⁻¹ : by rewrite zero_sub),
@ -464,7 +464,7 @@ theorem lt_of_le_and_sep {s t : seq} (Hs : regular s) (Ht : regular t) (H : s_le
apply exists.elim Hlt,
intro N HN,
let LeN := Le N,
let HN' := (iff.mp' (neg_lt_neg_iff_lt _ _)) HN,
let HN' := (iff.mpr (neg_lt_neg_iff_lt _ _)) HN,
rewrite [↑sadd at HN', ↑sneg at HN', neg_add at HN', neg_neg at HN', add.comm at HN'],
let HN'' := not_le_of_gt HN',
apply absurd LeN HN''
@ -819,11 +819,11 @@ theorem sep_well_defined {s t u v : seq} (Hs : regular s) (Ht : regular t) (Hu :
apply or.elim Hor,
intro Hlt,
apply or.inl,
apply iff.mp' (lt_well_defined Hs Ht Hu Hv Hsu Htv),
apply iff.mpr (lt_well_defined Hs Ht Hu Hv Hsu Htv),
assumption,
intro Hlt,
apply or.inr,
apply iff.mp' (lt_well_defined Ht Hs Hv Hu Htv Hsu),
apply iff.mpr (lt_well_defined Ht Hs Hv Hu Htv Hsu),
assumption
end
@ -920,7 +920,7 @@ theorem const_le_const_of_le {a b : ℚ} (H : a ≤ b) : s_le (const a) (const b
apply rat.neg_nonpos_of_nonneg,
apply rat.le_of_lt,
apply inv_pos,
apply iff.mp' !rat.sub_nonneg_iff_le,
apply iff.mpr !rat.sub_nonneg_iff_le,
apply H
end

4
library/data/set/function.lean
View file

@ -196,8 +196,8 @@ iff.intro
(assume H,
obtain Hmaps Hinj Hsurj, from H,
(and.intro
(iff.mp' !injective_iff_inj_on_univ Hinj)
(iff.mp' !surjective_iff_surj_on_univ Hsurj)))
(iff.mpr !injective_iff_inj_on_univ Hinj)
(iff.mpr !surjective_iff_surj_on_univ Hsurj)))
/- left inverse -/

14
library/init/logic.lean
View file

@ -227,7 +227,7 @@ namespace iff
theorem elim_right (H : a ↔ b) : b → a :=
elim (assume H₁ H₂, H₂) H
definition mp' := @elim_right
definition mpr := @elim_right
theorem refl (a : Prop) : a ↔ a :=
intro (assume H, H) (assume H, H)
@ -597,14 +597,14 @@ decidable.rec
theorem dif_ctx_congr {A : Type} {b c : Prop} [dec_b : decidable b] [dec_c : decidable c]
{x : b → A} {u : c → A} {y : ¬b → A} {v : ¬c → A}
(h_c : b ↔ c)
(h_t : ∀ (h : c), x (iff.mp' h_c h) = u h)
(h_e : ∀ (h : ¬c), y (iff.mp' (not_iff_not_of_iff h_c) h) = v h) :
(h_t : ∀ (h : c), x (iff.mpr h_c h) = u h)
(h_e : ∀ (h : ¬c), y (iff.mpr (not_iff_not_of_iff h_c) h) = v h) :
(@dite b dec_b A x y) = (@dite c dec_c A u v) :=
decidable.rec_on dec_b
(λ hp : b, calc
(if h : b then x h else y h)
= x hp : dif_pos hp
... = x (iff.mp' h_c (iff.mp h_c hp)) : proof_irrel
... = x (iff.mpr h_c (iff.mp h_c hp)) : proof_irrel
... = u (iff.mp h_c hp) : h_t
... = (if h : c then u h else v h) : dif_pos (iff.mp h_c hp))
(λ hn : ¬b,
@ -612,15 +612,15 @@ decidable.rec_on dec_b
calc
(if h : b then x h else y h)
= y hn : dif_neg hn
... = y (iff.mp' h_nc (iff.mp h_nc hn)) : proof_irrel
... = y (iff.mpr h_nc (iff.mp h_nc hn)) : proof_irrel
... = v (iff.mp h_nc hn) : h_e
... = (if h : c then u h else v h) : dif_neg (iff.mp h_nc hn))
theorem dif_ctx_rw_congr {A : Type} {b c : Prop} [dec_b : decidable b]
{x : b → A} {u : c → A} {y : ¬b → A} {v : ¬c → A}
(h_c : b ↔ c)
(h_t : ∀ (h : c), x (iff.mp' h_c h) = u h)
(h_e : ∀ (h : ¬c), y (iff.mp' (not_iff_not_of_iff h_c) h) = v h) :
(h_t : ∀ (h : c), x (iff.mpr h_c h) = u h)
(h_e : ∀ (h : ¬c), y (iff.mpr (not_iff_not_of_iff h_c) h) = v h) :
(@dite b dec_b A x y) = (@dite c (decidable_of_decidable_of_iff dec_b h_c) A u v) :=
@dif_ctx_congr A b c dec_b (decidable_of_decidable_of_iff dec_b h_c) x u y v h_c h_t h_e

2
library/logic/connectives.lean
View file

@ -251,7 +251,7 @@ definition iff.def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
theorem iff_true [rewrite] (a : Prop) : (a ↔ true) ↔ a :=
iff.intro
(assume H, iff.mp' H trivial)
(assume H, iff.mpr H trivial)
(assume H, iff.intro (assume H1, trivial) (assume H1, H))
theorem true_iff [rewrite] (a : Prop) : (true ↔ a) ↔ a :=

4
library/logic/quantifiers.lean
View file

@ -106,12 +106,12 @@ section
theorem congr_forall : (∀x, p₁ x) ↔ (∀x, p₂ x) :=
iff.intro
(assume H', take x, iff.mp (H x) (H' x))
(assume H', take x, iff.mp' (H x) (H' x))
(assume H', take x, iff.mpr (H x) (H' x))
theorem congr_exists : (∃x, p₁ x) ↔ (∃x, p₂ x) :=
iff.intro
(assume H', exists.elim H' (λ x H₁, exists.intro x (iff.mp (H x) H₁)))
(assume H', exists.elim H' (λ x H₁, exists.intro x (iff.mp' (H x) H₁)))
(assume H', exists.elim H' (λ x H₁, exists.intro x (iff.mpr (H x) H₁)))
include H
theorem congr_exists_unique : (∃!x, p₁ x) ↔ (∃!x, p₂ x) :=

2
library/theories/number_theory/primes.lean
View file

@ -71,7 +71,7 @@ lt_of_succ_le (le.trans h₂ h₃)
definition sub_dvd_of_not_prime {n : nat} : n ≥ 2 → ¬ prime n → {m | m ∣ n ∧ m ≠ 1 ∧ m ≠ n} :=
assume h₁ h₂,
have h₃ : ¬ prime_ext n, from iff.mp' (not_iff_not_of_iff !prime_ext_iff_prime) h₂,
have h₃ : ¬ prime_ext n, from iff.mpr (not_iff_not_of_iff !prime_ext_iff_prime) h₂,
have h₄ : ¬ n ≥ 2 ∨ ¬ (∀ m, m ≤ n → m ∣ n → m = 1 ∨ m = n), from iff.mp !not_and_iff_not_or_not h₃,
have h₅ : ¬ (∀ m, m ≤ n → m ∣ n → m = 1 ∨ m = n), from or_resolve_right h₄ (not_not_intro h₁),
have h₆ : ¬ (∀ m, m < succ n → m ∣ n → m = 1 ∨ m = n), from