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feat/fix(library/algebra/*): add some useful theorems, fix implicit arguments

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Jeremy Avigad 2015-09-20 20:47:30 -04:00
parent 9561e379c7
commit aa8dfba5a5
3 changed files with 66 additions and 56 deletions
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106
library/algebra/ordered_group.lean
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@ -280,119 +280,119 @@ section
theorem le_neg_iff_le_neg : a ≤ -b ↔ b ≤ -a := !neg_neg ▸ !neg_le_neg_iff_le theorem le_neg_iff_le_neg : a ≤ -b ↔ b ≤ -a := !neg_neg ▸ !neg_le_neg_iff_le
theorem le_neg_of_le_neg : a ≤ -b → b ≤ -a := iff.mp !le_neg_iff_le_neg theorem le_neg_of_le_neg {a b : A} : a ≤ -b → b ≤ -a := iff.mp !le_neg_iff_le_neg
theorem neg_le_iff_neg_le : -a ≤ b ↔ -b ≤ a := !neg_neg ▸ !neg_le_neg_iff_le theorem neg_le_iff_neg_le : -a ≤ b ↔ -b ≤ a := !neg_neg ▸ !neg_le_neg_iff_le
theorem neg_le_of_neg_le : -a ≤ b → -b ≤ a := iff.mp !neg_le_iff_neg_le theorem neg_le_of_neg_le {a b : A} : -a ≤ b → -b ≤ a := iff.mp !neg_le_iff_neg_le
theorem lt_neg_iff_lt_neg : a < -b ↔ b < -a := !neg_neg ▸ !neg_lt_neg_iff_lt theorem lt_neg_iff_lt_neg : a < -b ↔ b < -a := !neg_neg ▸ !neg_lt_neg_iff_lt
theorem lt_neg_of_lt_neg : a < -b → b < -a := iff.mp !lt_neg_iff_lt_neg theorem lt_neg_of_lt_neg {a b : A} : a < -b → b < -a := iff.mp !lt_neg_iff_lt_neg
theorem neg_lt_iff_neg_lt : -a < b ↔ -b < a := !neg_neg ▸ !neg_lt_neg_iff_lt theorem neg_lt_iff_neg_lt : -a < b ↔ -b < a := !neg_neg ▸ !neg_lt_neg_iff_lt
theorem neg_lt_of_neg_lt : -a < b → -b < a := iff.mp !neg_lt_iff_neg_lt theorem neg_lt_of_neg_lt {a b : A} : -a < b → -b < a := iff.mp !neg_lt_iff_neg_lt
theorem sub_nonneg_iff_le : 0 ≤ a - b ↔ b ≤ a := !sub_self ▸ !add_le_add_right_iff theorem sub_nonneg_iff_le : 0 ≤ a - b ↔ b ≤ a := !sub_self ▸ !add_le_add_right_iff
theorem sub_nonneg_of_le : b ≤ a → 0 ≤ a - b := iff.mpr !sub_nonneg_iff_le theorem sub_nonneg_of_le {a b : A} : b ≤ a → 0 ≤ a - b := iff.mpr !sub_nonneg_iff_le
theorem le_of_sub_nonneg : 0 ≤ a - b → b ≤ a := iff.mp !sub_nonneg_iff_le theorem le_of_sub_nonneg {a b : A} : 0 ≤ a - b → b ≤ a := iff.mp !sub_nonneg_iff_le
theorem sub_nonpos_iff_le : a - b ≤ 0 ↔ a ≤ b := !sub_self ▸ !add_le_add_right_iff theorem sub_nonpos_iff_le : a - b ≤ 0 ↔ a ≤ b := !sub_self ▸ !add_le_add_right_iff
theorem sub_nonpos_of_le : a ≤ b → a - b ≤ 0 := iff.mpr !sub_nonpos_iff_le theorem sub_nonpos_of_le {a b : A} : a ≤ b → a - b ≤ 0 := iff.mpr !sub_nonpos_iff_le
theorem le_of_sub_nonpos : a - b ≤ 0 → a ≤ b := iff.mp !sub_nonpos_iff_le theorem le_of_sub_nonpos {a b : A} : a - b ≤ 0 → a ≤ b := iff.mp !sub_nonpos_iff_le
theorem sub_pos_iff_lt : 0 < a - b ↔ b < a := !sub_self ▸ !add_lt_add_right_iff theorem sub_pos_iff_lt : 0 < a - b ↔ b < a := !sub_self ▸ !add_lt_add_right_iff
theorem sub_pos_of_lt : b < a → 0 < a - b := iff.mpr !sub_pos_iff_lt theorem sub_pos_of_lt {a b : A} : b < a → 0 < a - b := iff.mpr !sub_pos_iff_lt
theorem lt_of_sub_pos : 0 < a - b → b < a := iff.mp !sub_pos_iff_lt theorem lt_of_sub_pos {a b : A} : 0 < a - b → b < a := iff.mp !sub_pos_iff_lt
theorem sub_neg_iff_lt : a - b < 0 ↔ a < b := !sub_self ▸ !add_lt_add_right_iff theorem sub_neg_iff_lt : a - b < 0 ↔ a < b := !sub_self ▸ !add_lt_add_right_iff
theorem sub_neg_of_lt : a < b → a - b < 0 := iff.mpr !sub_neg_iff_lt theorem sub_neg_of_lt {a b : A} : a < b → a - b < 0 := iff.mpr !sub_neg_iff_lt
theorem lt_of_sub_neg : a - b < 0 → a < b := iff.mp !sub_neg_iff_lt theorem lt_of_sub_neg {a b : A} : a - b < 0 → a < b := iff.mp !sub_neg_iff_lt
theorem add_le_iff_le_neg_add : a + b ≤ c ↔ b ≤ -a + c := theorem add_le_iff_le_neg_add : a + b ≤ c ↔ b ≤ -a + c :=
have H: a + b ≤ c ↔ -a + (a + b) ≤ -a + c, from iff.symm (!add_le_add_left_iff), have H: a + b ≤ c ↔ -a + (a + b) ≤ -a + c, from iff.symm (!add_le_add_left_iff),
!neg_add_cancel_left ▸ H !neg_add_cancel_left ▸ H
theorem add_le_of_le_neg_add : b ≤ -a + c → a + b ≤ c := theorem add_le_of_le_neg_add {a b c : A} : b ≤ -a + c → a + b ≤ c :=
iff.mpr !add_le_iff_le_neg_add iff.mpr !add_le_iff_le_neg_add
theorem le_neg_add_of_add_le : a + b ≤ c → b ≤ -a + c := theorem le_neg_add_of_add_le {a b c : A} : a + b ≤ c → b ≤ -a + c :=
iff.mp !add_le_iff_le_neg_add iff.mp !add_le_iff_le_neg_add
theorem add_le_iff_le_sub_left : a + b ≤ c ↔ b ≤ c - a := theorem add_le_iff_le_sub_left : a + b ≤ c ↔ b ≤ c - a :=
by rewrite [sub_eq_add_neg, {c+_}add.comm]; apply add_le_iff_le_neg_add by rewrite [sub_eq_add_neg, {c+_}add.comm]; apply add_le_iff_le_neg_add
theorem add_le_of_le_sub_left : b ≤ c - a → a + b ≤ c := theorem add_le_of_le_sub_left {a b c : A} : b ≤ c - a → a + b ≤ c :=
iff.mpr !add_le_iff_le_sub_left iff.mpr !add_le_iff_le_sub_left
theorem le_sub_left_of_add_le : a + b ≤ c → b ≤ c - a := theorem le_sub_left_of_add_le {a b c : A} : a + b ≤ c → b ≤ c - a :=
iff.mp !add_le_iff_le_sub_left iff.mp !add_le_iff_le_sub_left
theorem add_le_iff_le_sub_right : a + b ≤ c ↔ a ≤ c - b := theorem add_le_iff_le_sub_right : a + b ≤ c ↔ a ≤ c - b :=
have H: a + b ≤ c ↔ a + b - b ≤ c - b, from iff.symm (!add_le_add_right_iff), have H: a + b ≤ c ↔ a + b - b ≤ c - b, from iff.symm (!add_le_add_right_iff),
!add_neg_cancel_right ▸ H !add_neg_cancel_right ▸ H
theorem add_le_of_le_sub_right : a ≤ c - b → a + b ≤ c := theorem add_le_of_le_sub_right {a b c : A} : a ≤ c - b → a + b ≤ c :=
iff.mpr !add_le_iff_le_sub_right iff.mpr !add_le_iff_le_sub_right
theorem le_sub_right_of_add_le : a + b ≤ c → a ≤ c - b := theorem le_sub_right_of_add_le {a b c : A} : a + b ≤ c → a ≤ c - b :=
iff.mp !add_le_iff_le_sub_right iff.mp !add_le_iff_le_sub_right
theorem le_add_iff_neg_add_le : a ≤ b + c ↔ -b + a ≤ c := theorem le_add_iff_neg_add_le : a ≤ b + c ↔ -b + a ≤ c :=
assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff), assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff),
by rewrite neg_add_cancel_left at H; exact H by rewrite neg_add_cancel_left at H; exact H
theorem le_add_of_neg_add_le : -b + a ≤ c → a ≤ b + c := theorem le_add_of_neg_add_le {a b c : A} : -b + a ≤ c → a ≤ b + c :=
iff.mpr !le_add_iff_neg_add_le iff.mpr !le_add_iff_neg_add_le
theorem neg_add_le_of_le_add : a ≤ b + c → -b + a ≤ c := theorem neg_add_le_of_le_add {a b c : A} : a ≤ b + c → -b + a ≤ c :=
iff.mp !le_add_iff_neg_add_le iff.mp !le_add_iff_neg_add_le
theorem le_add_iff_sub_left_le : a ≤ b + c ↔ a - b ≤ c := theorem le_add_iff_sub_left_le : a ≤ b + c ↔ a - b ≤ c :=
by rewrite [sub_eq_add_neg, {a+_}add.comm]; apply le_add_iff_neg_add_le by rewrite [sub_eq_add_neg, {a+_}add.comm]; apply le_add_iff_neg_add_le
theorem le_add_of_sub_left_le : a - b ≤ c → a ≤ b + c := theorem le_add_of_sub_left_le {a b c : A} : a - b ≤ c → a ≤ b + c :=
iff.mpr !le_add_iff_sub_left_le iff.mpr !le_add_iff_sub_left_le
theorem sub_left_le_of_le_add : a ≤ b + c → a - b ≤ c := theorem sub_left_le_of_le_add {a b c : A} : a ≤ b + c → a - b ≤ c :=
iff.mp !le_add_iff_sub_left_le iff.mp !le_add_iff_sub_left_le
theorem le_add_iff_sub_right_le : a ≤ b + c ↔ a - c ≤ b := theorem le_add_iff_sub_right_le : a ≤ b + c ↔ a - c ≤ b :=
assert H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_add_right_iff), assert H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_add_right_iff),
by rewrite add_neg_cancel_right at H; exact H by rewrite add_neg_cancel_right at H; exact H
theorem le_add_of_sub_right_le : a - c ≤ b → a ≤ b + c := theorem le_add_of_sub_right_le {a b c : A} : a - c ≤ b → a ≤ b + c :=
iff.mpr !le_add_iff_sub_right_le iff.mpr !le_add_iff_sub_right_le
theorem sub_right_le_of_le_add : a ≤ b + c → a - c ≤ b := theorem sub_right_le_of_le_add {a b c : A} : a ≤ b + c → a - c ≤ b :=
iff.mp !le_add_iff_sub_right_le iff.mp !le_add_iff_sub_right_le
theorem le_add_iff_neg_add_le_left : a ≤ b + c ↔ -b + a ≤ c := theorem le_add_iff_neg_add_le_left : a ≤ b + c ↔ -b + a ≤ c :=
assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff), assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff),
by rewrite neg_add_cancel_left at H; exact H by rewrite neg_add_cancel_left at H; exact H
theorem le_add_of_neg_add_le_left : -b + a ≤ c → a ≤ b + c := theorem le_add_of_neg_add_le_left {a b c : A} : -b + a ≤ c → a ≤ b + c :=
iff.mpr !le_add_iff_neg_add_le_left iff.mpr !le_add_iff_neg_add_le_left
theorem neg_add_le_left_of_le_add : a ≤ b + c → -b + a ≤ c := theorem neg_add_le_left_of_le_add {a b c : A} : a ≤ b + c → -b + a ≤ c :=
iff.mp !le_add_iff_neg_add_le_left iff.mp !le_add_iff_neg_add_le_left
theorem le_add_iff_neg_add_le_right : a ≤ b + c ↔ -c + a ≤ b := theorem le_add_iff_neg_add_le_right : a ≤ b + c ↔ -c + a ≤ b :=
by rewrite add.comm; apply le_add_iff_neg_add_le_left by rewrite add.comm; apply le_add_iff_neg_add_le_left
theorem le_add_of_neg_add_le_right : -c + a ≤ b → a ≤ b + c := theorem le_add_of_neg_add_le_right {a b c : A} : -c + a ≤ b → a ≤ b + c :=
iff.mpr !le_add_iff_neg_add_le_right iff.mpr !le_add_iff_neg_add_le_right
theorem neg_add_le_right_of_le_add : a ≤ b + c → -c + a ≤ b := theorem neg_add_le_right_of_le_add {a b c : A} : a ≤ b + c → -c + a ≤ b :=
iff.mp !le_add_iff_neg_add_le_right iff.mp !le_add_iff_neg_add_le_right
theorem le_add_iff_neg_le_sub_left : c ≤ a + b ↔ -a ≤ b - c := theorem le_add_iff_neg_le_sub_left : c ≤ a + b ↔ -a ≤ b - c :=
@ -400,38 +400,38 @@ section
assert H' : -a + c ≤ b ↔ -a ≤ b - c, from !add_le_iff_le_sub_right, assert H' : -a + c ≤ b ↔ -a ≤ b - c, from !add_le_iff_le_sub_right,
iff.trans H H' iff.trans H H'
theorem le_add_of_neg_le_sub_left : -a ≤ b - c → c ≤ a + b := theorem le_add_of_neg_le_sub_left {a b c : A} : -a ≤ b - c → c ≤ a + b :=
iff.mpr !le_add_iff_neg_le_sub_left iff.mpr !le_add_iff_neg_le_sub_left
theorem neg_le_sub_left_of_le_add : c ≤ a + b → -a ≤ b - c := theorem neg_le_sub_left_of_le_add {a b c : A} : c ≤ a + b → -a ≤ b - c :=
iff.mp !le_add_iff_neg_le_sub_left iff.mp !le_add_iff_neg_le_sub_left
theorem le_add_iff_neg_le_sub_right : c ≤ a + b ↔ -b ≤ a - c := theorem le_add_iff_neg_le_sub_right : c ≤ a + b ↔ -b ≤ a - c :=
by rewrite add.comm; apply le_add_iff_neg_le_sub_left by rewrite add.comm; apply le_add_iff_neg_le_sub_left
theorem le_add_of_neg_le_sub_right : -b ≤ a - c → c ≤ a + b := theorem le_add_of_neg_le_sub_right {a b c : A} : -b ≤ a - c → c ≤ a + b :=
iff.mpr !le_add_iff_neg_le_sub_right iff.mpr !le_add_iff_neg_le_sub_right
theorem neg_le_sub_right_of_le_add : c ≤ a + b → -b ≤ a - c := theorem neg_le_sub_right_of_le_add {a b c : A} : c ≤ a + b → -b ≤ a - c :=
iff.mp !le_add_iff_neg_le_sub_right iff.mp !le_add_iff_neg_le_sub_right
theorem add_lt_iff_lt_neg_add_left : a + b < c ↔ b < -a + c := theorem add_lt_iff_lt_neg_add_left : a + b < c ↔ b < -a + c :=
assert H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_add_left_iff), assert H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_add_left_iff),
begin rewrite neg_add_cancel_left at H, exact H end begin rewrite neg_add_cancel_left at H, exact H end
theorem add_lt_of_lt_neg_add_left : b < -a + c → a + b < c := theorem add_lt_of_lt_neg_add_left {a b c : A} : b < -a + c → a + b < c :=
iff.mpr !add_lt_iff_lt_neg_add_left iff.mpr !add_lt_iff_lt_neg_add_left
theorem lt_neg_add_left_of_add_lt : a + b < c → b < -a + c := theorem lt_neg_add_left_of_add_lt {a b c : A} : a + b < c → b < -a + c :=
iff.mp !add_lt_iff_lt_neg_add_left iff.mp !add_lt_iff_lt_neg_add_left
theorem add_lt_iff_lt_neg_add_right : a + b < c ↔ a < -b + c := theorem add_lt_iff_lt_neg_add_right : a + b < c ↔ a < -b + c :=
by rewrite add.comm; apply add_lt_iff_lt_neg_add_left by rewrite add.comm; apply add_lt_iff_lt_neg_add_left
theorem add_lt_of_lt_neg_add_right : a < -b + c → a + b < c := theorem add_lt_of_lt_neg_add_right {a b c : A} : a < -b + c → a + b < c :=
iff.mpr !add_lt_iff_lt_neg_add_right iff.mpr !add_lt_iff_lt_neg_add_right
theorem lt_neg_add_right_of_add_lt : a + b < c → a < -b + c := theorem lt_neg_add_right_of_add_lt {a b c : A} : a + b < c → a < -b + c :=
iff.mp !add_lt_iff_lt_neg_add_right iff.mp !add_lt_iff_lt_neg_add_right
theorem add_lt_iff_lt_sub_left : a + b < c ↔ b < c - a := theorem add_lt_iff_lt_sub_left : a + b < c ↔ b < c - a :=
@ -440,71 +440,71 @@ section
apply add_lt_iff_lt_neg_add_left apply add_lt_iff_lt_neg_add_left
end end
theorem add_lt_of_lt_sub_left : b < c - a → a + b < c := theorem add_lt_of_lt_sub_left {a b c : A} : b < c - a → a + b < c :=
iff.mpr !add_lt_iff_lt_sub_left iff.mpr !add_lt_iff_lt_sub_left
theorem lt_sub_left_of_add_lt : a + b < c → b < c - a := theorem lt_sub_left_of_add_lt {a b c : A} : a + b < c → b < c - a :=
iff.mp !add_lt_iff_lt_sub_left iff.mp !add_lt_iff_lt_sub_left
theorem add_lt_iff_lt_sub_right : a + b < c ↔ a < c - b := theorem add_lt_iff_lt_sub_right : a + b < c ↔ a < c - b :=
assert H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_add_right_iff), assert H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_add_right_iff),
by rewrite add_neg_cancel_right at H; exact H by rewrite add_neg_cancel_right at H; exact H
theorem add_lt_of_lt_sub_right : a < c - b → a + b < c := theorem add_lt_of_lt_sub_right {a b c : A} : a < c - b → a + b < c :=
iff.mpr !add_lt_iff_lt_sub_right iff.mpr !add_lt_iff_lt_sub_right
theorem lt_sub_right_of_add_lt : a + b < c → a < c - b := theorem lt_sub_right_of_add_lt {a b c : A} : a + b < c → a < c - b :=
iff.mp !add_lt_iff_lt_sub_right iff.mp !add_lt_iff_lt_sub_right
theorem lt_add_iff_neg_add_lt_left : a < b + c ↔ -b + a < c := theorem lt_add_iff_neg_add_lt_left : a < b + c ↔ -b + a < c :=
assert H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_add_left_iff), assert H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_add_left_iff),
by rewrite neg_add_cancel_left at H; exact H by rewrite neg_add_cancel_left at H; exact H
theorem lt_add_of_neg_add_lt_left : -b + a < c → a < b + c := theorem lt_add_of_neg_add_lt_left {a b c : A} : -b + a < c → a < b + c :=
iff.mpr !lt_add_iff_neg_add_lt_left iff.mpr !lt_add_iff_neg_add_lt_left
theorem neg_add_lt_left_of_lt_add : a < b + c → -b + a < c := theorem neg_add_lt_left_of_lt_add {a b c : A} : a < b + c → -b + a < c :=
iff.mp !lt_add_iff_neg_add_lt_left iff.mp !lt_add_iff_neg_add_lt_left
theorem lt_add_iff_neg_add_lt_right : a < b + c ↔ -c + a < b := theorem lt_add_iff_neg_add_lt_right : a < b + c ↔ -c + a < b :=
by rewrite add.comm; apply lt_add_iff_neg_add_lt_left by rewrite add.comm; apply lt_add_iff_neg_add_lt_left
theorem lt_add_of_neg_add_lt_right : -c + a < b → a < b + c := theorem lt_add_of_neg_add_lt_right {a b c : A} : -c + a < b → a < b + c :=
iff.mpr !lt_add_iff_neg_add_lt_right iff.mpr !lt_add_iff_neg_add_lt_right
theorem neg_add_lt_right_of_lt_add : a < b + c → -c + a < b := theorem neg_add_lt_right_of_lt_add {a b c : A} : a < b + c → -c + a < b :=
iff.mp !lt_add_iff_neg_add_lt_right iff.mp !lt_add_iff_neg_add_lt_right
theorem lt_add_iff_sub_lt_left : a < b + c ↔ a - b < c := theorem lt_add_iff_sub_lt_left : a < b + c ↔ a - b < c :=
by rewrite [sub_eq_add_neg, {a + _}add.comm]; apply lt_add_iff_neg_add_lt_left by rewrite [sub_eq_add_neg, {a + _}add.comm]; apply lt_add_iff_neg_add_lt_left
theorem lt_add_of_sub_lt_left : a - b < c → a < b + c := theorem lt_add_of_sub_lt_left {a b c : A} : a - b < c → a < b + c :=
iff.mpr !lt_add_iff_sub_lt_left iff.mpr !lt_add_iff_sub_lt_left
theorem sub_lt_left_of_lt_add : a < b + c → a - b < c := theorem sub_lt_left_of_lt_add {a b c : A} : a < b + c → a - b < c :=
iff.mp !lt_add_iff_sub_lt_left iff.mp !lt_add_iff_sub_lt_left
theorem lt_add_iff_sub_lt_right : a < b + c ↔ a - c < b := theorem lt_add_iff_sub_lt_right : a < b + c ↔ a - c < b :=
by rewrite add.comm; apply lt_add_iff_sub_lt_left by rewrite add.comm; apply lt_add_iff_sub_lt_left
theorem lt_add_of_sub_lt_right : a - c < b → a < b + c := theorem lt_add_of_sub_lt_right {a b c : A} : a - c < b → a < b + c :=
iff.mpr !lt_add_iff_sub_lt_right iff.mpr !lt_add_iff_sub_lt_right
theorem sub_lt_right_of_lt_add : a < b + c → a - c < b := theorem sub_lt_right_of_lt_add {a b c : A} : a < b + c → a - c < b :=
iff.mp !lt_add_iff_sub_lt_right iff.mp !lt_add_iff_sub_lt_right
theorem sub_lt_of_sub_lt : a - b < c → a - c < b := theorem sub_lt_of_sub_lt {a b c : A} : a - b < c → a - c < b :=
begin begin
intro H, intro H,
apply sub_lt_left_of_lt_add, apply sub_lt_left_of_lt_add,
apply lt_add_of_sub_lt_right _ _ _ H apply lt_add_of_sub_lt_right H
end end
theorem sub_le_of_sub_le : a - b ≤ c → a - c ≤ b := theorem sub_le_of_sub_le {a b c : A} : a - b ≤ c → a - c ≤ b :=
begin begin
intro H, intro H,
apply sub_left_le_of_le_add, apply sub_left_le_of_le_add,
apply le_add_of_sub_right_le _ _ _ H apply le_add_of_sub_right_le H
end end
-- TODO: the Isabelle library has varations on a + b ≤ b ↔ a ≤ 0 -- TODO: the Isabelle library has varations on a + b ≤ b ↔ a ≤ 0
@ -564,7 +564,7 @@ section
apply le.refl apply le.refl
end end
theorem sub_le_of_nonneg (H : b ≥ 0) : a - b ≤ a := theorem sub_le_of_nonneg {b : A} (H : b ≥ 0) : a - b ≤ a :=
add_le_of_le_of_nonpos (le.refl a) (neg_nonpos_of_nonneg H) add_le_of_le_of_nonpos (le.refl a) (neg_nonpos_of_nonneg H)
theorem sub_lt_of_pos {b : A} (H : b > 0) : a - b < a := theorem sub_lt_of_pos {b : A} (H : b > 0) : a - b < a :=

6
library/algebra/ordered_ring.lean
View file

@ -663,7 +663,7 @@ section
if Hz : 0 ≤ a - b then if Hz : 0 ≤ a - b then
(calc (calc
a ≥ b : (iff.mp !sub_nonneg_iff_le) Hz a ≥ b : (iff.mp !sub_nonneg_iff_le) Hz
... ≥ b - c : sub_le_of_nonneg _ _ (le.trans !abs_nonneg H)) ... ≥ b - c : sub_le_of_nonneg _ (le.trans !abs_nonneg H))
else 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 - 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.mpr !le_add_iff_sub_right_le) Habs, have Habs' : b ≤ c + a, from (iff.mpr !le_add_iff_sub_right_le) Habs,
@ -679,8 +679,8 @@ section
... > b - c : sub_lt_of_pos _ (lt_of_le_of_lt !abs_nonneg H)) ... > b - c : sub_lt_of_pos _ (lt_of_le_of_lt !abs_nonneg H))
else 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 - 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 lt_add_of_sub_lt_right _ _ _ Habs, have Habs' : b < c + a, from lt_add_of_sub_lt_right Habs,
sub_lt_left_of_lt_add _ _ _ Habs') sub_lt_left_of_lt_add Habs')
theorem sub_lt_of_abs_sub_lt_right (H : abs (a - b) < c) : a - c < b := theorem sub_lt_of_abs_sub_lt_right (H : abs (a - b) < c) : a - c < b :=
sub_lt_of_abs_sub_lt_left (!abs_sub ▸ H) sub_lt_of_abs_sub_lt_left (!abs_sub ▸ H)

10
library/algebra/ring.lean
View file

@ -351,6 +351,16 @@ section
-- TODO: do we want the iff versions? -- TODO: do we want the iff versions?
theorem eq_zero_of_mul_eq_self_right {a b : A} (H₁ : b ≠ 1) (H₂ : a * b = a) : a = 0 :=
have b - 1 ≠ 0, from
suppose b - 1 = 0, H₁ (!zero_add ▸ eq_add_of_sub_eq this),
have a * b - a = 0, by rewrite H₂; apply sub_self,
have a * (b - 1) = 0, by+ rewrite [mul_sub_left_distrib, mul_one]; apply this,
show a = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero this) `b - 1 ≠ 0`
theorem eq_zero_of_mul_eq_self_left {a b : A} (H₁ : b ≠ 1) (H₂ : b * a = a) : a = 0 :=
eq_zero_of_mul_eq_self_right H₁ (!mul.comm ▸ H₂)
theorem mul_self_eq_mul_self_iff (a b : A) : a * a = b * b ↔ a = b ∨ a = -b := theorem mul_self_eq_mul_self_iff (a b : A) : a * a = b * b ↔ a = b ∨ a = -b :=
iff.intro iff.intro
(suppose a * a = b * b, (suppose a * a = b * b,