feat/fix(library/algebra/*): add some useful theorems, fix implicit arguments
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3 changed files with 66 additions and 56 deletions
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@ -280,119 +280,119 @@ section
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theorem le_neg_iff_le_neg : a ≤ -b ↔ b ≤ -a := !neg_neg ▸ !neg_le_neg_iff_le
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theorem le_neg_of_le_neg : a ≤ -b → b ≤ -a := iff.mp !le_neg_iff_le_neg
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theorem le_neg_of_le_neg {a b : A} : a ≤ -b → b ≤ -a := iff.mp !le_neg_iff_le_neg
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theorem neg_le_iff_neg_le : -a ≤ b ↔ -b ≤ a := !neg_neg ▸ !neg_le_neg_iff_le
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theorem neg_le_of_neg_le : -a ≤ b → -b ≤ a := iff.mp !neg_le_iff_neg_le
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theorem neg_le_of_neg_le {a b : A} : -a ≤ b → -b ≤ a := iff.mp !neg_le_iff_neg_le
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theorem lt_neg_iff_lt_neg : a < -b ↔ b < -a := !neg_neg ▸ !neg_lt_neg_iff_lt
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theorem lt_neg_of_lt_neg : a < -b → b < -a := iff.mp !lt_neg_iff_lt_neg
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theorem lt_neg_of_lt_neg {a b : A} : a < -b → b < -a := iff.mp !lt_neg_iff_lt_neg
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theorem neg_lt_iff_neg_lt : -a < b ↔ -b < a := !neg_neg ▸ !neg_lt_neg_iff_lt
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theorem neg_lt_of_neg_lt : -a < b → -b < a := iff.mp !neg_lt_iff_neg_lt
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theorem neg_lt_of_neg_lt {a b : A} : -a < b → -b < a := iff.mp !neg_lt_iff_neg_lt
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theorem sub_nonneg_iff_le : 0 ≤ a - b ↔ b ≤ a := !sub_self ▸ !add_le_add_right_iff
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theorem sub_nonneg_of_le : b ≤ a → 0 ≤ a - b := iff.mpr !sub_nonneg_iff_le
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theorem sub_nonneg_of_le {a b : A} : b ≤ a → 0 ≤ a - b := iff.mpr !sub_nonneg_iff_le
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theorem le_of_sub_nonneg : 0 ≤ a - b → b ≤ a := iff.mp !sub_nonneg_iff_le
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theorem le_of_sub_nonneg {a b : A} : 0 ≤ a - b → b ≤ a := iff.mp !sub_nonneg_iff_le
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theorem sub_nonpos_iff_le : a - b ≤ 0 ↔ a ≤ b := !sub_self ▸ !add_le_add_right_iff
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theorem sub_nonpos_of_le : a ≤ b → a - b ≤ 0 := iff.mpr !sub_nonpos_iff_le
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theorem sub_nonpos_of_le {a b : A} : a ≤ b → a - b ≤ 0 := iff.mpr !sub_nonpos_iff_le
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theorem le_of_sub_nonpos : a - b ≤ 0 → a ≤ b := iff.mp !sub_nonpos_iff_le
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theorem le_of_sub_nonpos {a b : A} : a - b ≤ 0 → a ≤ b := iff.mp !sub_nonpos_iff_le
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theorem sub_pos_iff_lt : 0 < a - b ↔ b < a := !sub_self ▸ !add_lt_add_right_iff
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theorem sub_pos_of_lt : b < a → 0 < a - b := iff.mpr !sub_pos_iff_lt
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theorem sub_pos_of_lt {a b : A} : b < a → 0 < a - b := iff.mpr !sub_pos_iff_lt
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theorem lt_of_sub_pos : 0 < a - b → b < a := iff.mp !sub_pos_iff_lt
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theorem lt_of_sub_pos {a b : A} : 0 < a - b → b < a := iff.mp !sub_pos_iff_lt
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theorem sub_neg_iff_lt : a - b < 0 ↔ a < b := !sub_self ▸ !add_lt_add_right_iff
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theorem sub_neg_of_lt : a < b → a - b < 0 := iff.mpr !sub_neg_iff_lt
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theorem sub_neg_of_lt {a b : A} : a < b → a - b < 0 := iff.mpr !sub_neg_iff_lt
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theorem lt_of_sub_neg : a - b < 0 → a < b := iff.mp !sub_neg_iff_lt
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theorem lt_of_sub_neg {a b : A} : a - b < 0 → a < b := iff.mp !sub_neg_iff_lt
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theorem add_le_iff_le_neg_add : a + b ≤ c ↔ b ≤ -a + c :=
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have H: a + b ≤ c ↔ -a + (a + b) ≤ -a + c, from iff.symm (!add_le_add_left_iff),
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!neg_add_cancel_left ▸ H
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theorem add_le_of_le_neg_add : b ≤ -a + c → a + b ≤ c :=
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theorem add_le_of_le_neg_add {a b c : A} : b ≤ -a + c → a + b ≤ c :=
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iff.mpr !add_le_iff_le_neg_add
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theorem le_neg_add_of_add_le : a + b ≤ c → b ≤ -a + c :=
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theorem le_neg_add_of_add_le {a b c : A} : a + b ≤ c → b ≤ -a + c :=
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iff.mp !add_le_iff_le_neg_add
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theorem add_le_iff_le_sub_left : a + b ≤ c ↔ b ≤ c - a :=
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by rewrite [sub_eq_add_neg, {c+_}add.comm]; apply add_le_iff_le_neg_add
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theorem add_le_of_le_sub_left : b ≤ c - a → a + b ≤ c :=
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theorem add_le_of_le_sub_left {a b c : A} : b ≤ c - a → a + b ≤ c :=
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iff.mpr !add_le_iff_le_sub_left
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theorem le_sub_left_of_add_le : a + b ≤ c → b ≤ c - a :=
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theorem le_sub_left_of_add_le {a b c : A} : a + b ≤ c → b ≤ c - a :=
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iff.mp !add_le_iff_le_sub_left
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theorem add_le_iff_le_sub_right : a + b ≤ c ↔ a ≤ c - b :=
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have H: a + b ≤ c ↔ a + b - b ≤ c - b, from iff.symm (!add_le_add_right_iff),
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!add_neg_cancel_right ▸ H
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theorem add_le_of_le_sub_right : a ≤ c - b → a + b ≤ c :=
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theorem add_le_of_le_sub_right {a b c : A} : a ≤ c - b → a + b ≤ c :=
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iff.mpr !add_le_iff_le_sub_right
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theorem le_sub_right_of_add_le : a + b ≤ c → a ≤ c - b :=
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theorem le_sub_right_of_add_le {a b c : A} : a + b ≤ c → a ≤ c - b :=
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iff.mp !add_le_iff_le_sub_right
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theorem le_add_iff_neg_add_le : a ≤ b + c ↔ -b + a ≤ c :=
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assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff),
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by rewrite neg_add_cancel_left at H; exact H
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theorem le_add_of_neg_add_le : -b + a ≤ c → a ≤ b + c :=
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theorem le_add_of_neg_add_le {a b c : A} : -b + a ≤ c → a ≤ b + c :=
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iff.mpr !le_add_iff_neg_add_le
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theorem neg_add_le_of_le_add : a ≤ b + c → -b + a ≤ c :=
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theorem neg_add_le_of_le_add {a b c : A} : a ≤ b + c → -b + a ≤ c :=
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iff.mp !le_add_iff_neg_add_le
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theorem le_add_iff_sub_left_le : a ≤ b + c ↔ a - b ≤ c :=
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by rewrite [sub_eq_add_neg, {a+_}add.comm]; apply le_add_iff_neg_add_le
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theorem le_add_of_sub_left_le : a - b ≤ c → a ≤ b + c :=
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theorem le_add_of_sub_left_le {a b c : A} : a - b ≤ c → a ≤ b + c :=
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iff.mpr !le_add_iff_sub_left_le
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theorem sub_left_le_of_le_add : a ≤ b + c → a - b ≤ c :=
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theorem sub_left_le_of_le_add {a b c : A} : a ≤ b + c → a - b ≤ c :=
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iff.mp !le_add_iff_sub_left_le
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theorem le_add_iff_sub_right_le : a ≤ b + c ↔ a - c ≤ b :=
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assert H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_add_right_iff),
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by rewrite add_neg_cancel_right at H; exact H
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theorem le_add_of_sub_right_le : a - c ≤ b → a ≤ b + c :=
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theorem le_add_of_sub_right_le {a b c : A} : a - c ≤ b → a ≤ b + c :=
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iff.mpr !le_add_iff_sub_right_le
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theorem sub_right_le_of_le_add : a ≤ b + c → a - c ≤ b :=
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theorem sub_right_le_of_le_add {a b c : A} : a ≤ b + c → a - c ≤ b :=
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iff.mp !le_add_iff_sub_right_le
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theorem le_add_iff_neg_add_le_left : a ≤ b + c ↔ -b + a ≤ c :=
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assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff),
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by rewrite neg_add_cancel_left at H; exact H
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theorem le_add_of_neg_add_le_left : -b + a ≤ c → a ≤ b + c :=
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theorem le_add_of_neg_add_le_left {a b c : A} : -b + a ≤ c → a ≤ b + c :=
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iff.mpr !le_add_iff_neg_add_le_left
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theorem neg_add_le_left_of_le_add : a ≤ b + c → -b + a ≤ c :=
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theorem neg_add_le_left_of_le_add {a b c : A} : a ≤ b + c → -b + a ≤ c :=
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iff.mp !le_add_iff_neg_add_le_left
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theorem le_add_iff_neg_add_le_right : a ≤ b + c ↔ -c + a ≤ b :=
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by rewrite add.comm; apply le_add_iff_neg_add_le_left
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theorem le_add_of_neg_add_le_right : -c + a ≤ b → a ≤ b + c :=
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theorem le_add_of_neg_add_le_right {a b c : A} : -c + a ≤ b → a ≤ b + c :=
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iff.mpr !le_add_iff_neg_add_le_right
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theorem neg_add_le_right_of_le_add : a ≤ b + c → -c + a ≤ b :=
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theorem neg_add_le_right_of_le_add {a b c : A} : a ≤ b + c → -c + a ≤ b :=
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iff.mp !le_add_iff_neg_add_le_right
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theorem le_add_iff_neg_le_sub_left : c ≤ a + b ↔ -a ≤ b - c :=
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@ -400,38 +400,38 @@ section
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assert H' : -a + c ≤ b ↔ -a ≤ b - c, from !add_le_iff_le_sub_right,
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iff.trans H H'
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theorem le_add_of_neg_le_sub_left : -a ≤ b - c → c ≤ a + b :=
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theorem le_add_of_neg_le_sub_left {a b c : A} : -a ≤ b - c → c ≤ a + b :=
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iff.mpr !le_add_iff_neg_le_sub_left
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theorem neg_le_sub_left_of_le_add : c ≤ a + b → -a ≤ b - c :=
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theorem neg_le_sub_left_of_le_add {a b c : A} : c ≤ a + b → -a ≤ b - c :=
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iff.mp !le_add_iff_neg_le_sub_left
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theorem le_add_iff_neg_le_sub_right : c ≤ a + b ↔ -b ≤ a - c :=
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by rewrite add.comm; apply le_add_iff_neg_le_sub_left
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theorem le_add_of_neg_le_sub_right : -b ≤ a - c → c ≤ a + b :=
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theorem le_add_of_neg_le_sub_right {a b c : A} : -b ≤ a - c → c ≤ a + b :=
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iff.mpr !le_add_iff_neg_le_sub_right
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theorem neg_le_sub_right_of_le_add : c ≤ a + b → -b ≤ a - c :=
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theorem neg_le_sub_right_of_le_add {a b c : A} : c ≤ a + b → -b ≤ a - c :=
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iff.mp !le_add_iff_neg_le_sub_right
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theorem add_lt_iff_lt_neg_add_left : a + b < c ↔ b < -a + c :=
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assert H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_add_left_iff),
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begin rewrite neg_add_cancel_left at H, exact H end
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theorem add_lt_of_lt_neg_add_left : b < -a + c → a + b < c :=
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theorem add_lt_of_lt_neg_add_left {a b c : A} : b < -a + c → a + b < c :=
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iff.mpr !add_lt_iff_lt_neg_add_left
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theorem lt_neg_add_left_of_add_lt : a + b < c → b < -a + c :=
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theorem lt_neg_add_left_of_add_lt {a b c : A} : a + b < c → b < -a + c :=
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iff.mp !add_lt_iff_lt_neg_add_left
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theorem add_lt_iff_lt_neg_add_right : a + b < c ↔ a < -b + c :=
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by rewrite add.comm; apply add_lt_iff_lt_neg_add_left
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theorem add_lt_of_lt_neg_add_right : a < -b + c → a + b < c :=
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theorem add_lt_of_lt_neg_add_right {a b c : A} : a < -b + c → a + b < c :=
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iff.mpr !add_lt_iff_lt_neg_add_right
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theorem lt_neg_add_right_of_add_lt : a + b < c → a < -b + c :=
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theorem lt_neg_add_right_of_add_lt {a b c : A} : a + b < c → a < -b + c :=
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iff.mp !add_lt_iff_lt_neg_add_right
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theorem add_lt_iff_lt_sub_left : a + b < c ↔ b < c - a :=
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@ -440,71 +440,71 @@ section
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apply add_lt_iff_lt_neg_add_left
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end
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theorem add_lt_of_lt_sub_left : b < c - a → a + b < c :=
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theorem add_lt_of_lt_sub_left {a b c : A} : b < c - a → a + b < c :=
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iff.mpr !add_lt_iff_lt_sub_left
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theorem lt_sub_left_of_add_lt : a + b < c → b < c - a :=
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theorem lt_sub_left_of_add_lt {a b c : A} : a + b < c → b < c - a :=
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iff.mp !add_lt_iff_lt_sub_left
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theorem add_lt_iff_lt_sub_right : a + b < c ↔ a < c - b :=
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assert H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_add_right_iff),
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by rewrite add_neg_cancel_right at H; exact H
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theorem add_lt_of_lt_sub_right : a < c - b → a + b < c :=
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theorem add_lt_of_lt_sub_right {a b c : A} : a < c - b → a + b < c :=
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iff.mpr !add_lt_iff_lt_sub_right
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theorem lt_sub_right_of_add_lt : a + b < c → a < c - b :=
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theorem lt_sub_right_of_add_lt {a b c : A} : a + b < c → a < c - b :=
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iff.mp !add_lt_iff_lt_sub_right
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theorem lt_add_iff_neg_add_lt_left : a < b + c ↔ -b + a < c :=
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assert H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_add_left_iff),
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by rewrite neg_add_cancel_left at H; exact H
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theorem lt_add_of_neg_add_lt_left : -b + a < c → a < b + c :=
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theorem lt_add_of_neg_add_lt_left {a b c : A} : -b + a < c → a < b + c :=
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iff.mpr !lt_add_iff_neg_add_lt_left
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theorem neg_add_lt_left_of_lt_add : a < b + c → -b + a < c :=
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theorem neg_add_lt_left_of_lt_add {a b c : A} : a < b + c → -b + a < c :=
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iff.mp !lt_add_iff_neg_add_lt_left
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theorem lt_add_iff_neg_add_lt_right : a < b + c ↔ -c + a < b :=
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by rewrite add.comm; apply lt_add_iff_neg_add_lt_left
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theorem lt_add_of_neg_add_lt_right : -c + a < b → a < b + c :=
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theorem lt_add_of_neg_add_lt_right {a b c : A} : -c + a < b → a < b + c :=
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iff.mpr !lt_add_iff_neg_add_lt_right
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theorem neg_add_lt_right_of_lt_add : a < b + c → -c + a < b :=
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theorem neg_add_lt_right_of_lt_add {a b c : A} : a < b + c → -c + a < b :=
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iff.mp !lt_add_iff_neg_add_lt_right
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theorem lt_add_iff_sub_lt_left : a < b + c ↔ a - b < c :=
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by rewrite [sub_eq_add_neg, {a + _}add.comm]; apply lt_add_iff_neg_add_lt_left
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theorem lt_add_of_sub_lt_left : a - b < c → a < b + c :=
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theorem lt_add_of_sub_lt_left {a b c : A} : a - b < c → a < b + c :=
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iff.mpr !lt_add_iff_sub_lt_left
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theorem sub_lt_left_of_lt_add : a < b + c → a - b < c :=
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theorem sub_lt_left_of_lt_add {a b c : A} : a < b + c → a - b < c :=
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iff.mp !lt_add_iff_sub_lt_left
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theorem lt_add_iff_sub_lt_right : a < b + c ↔ a - c < b :=
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by rewrite add.comm; apply lt_add_iff_sub_lt_left
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theorem lt_add_of_sub_lt_right : a - c < b → a < b + c :=
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theorem lt_add_of_sub_lt_right {a b c : A} : a - c < b → a < b + c :=
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iff.mpr !lt_add_iff_sub_lt_right
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theorem sub_lt_right_of_lt_add : a < b + c → a - c < b :=
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theorem sub_lt_right_of_lt_add {a b c : A} : a < b + c → a - c < b :=
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iff.mp !lt_add_iff_sub_lt_right
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theorem sub_lt_of_sub_lt : a - b < c → a - c < b :=
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theorem sub_lt_of_sub_lt {a b c : A} : a - b < c → a - c < b :=
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begin
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intro H,
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apply sub_lt_left_of_lt_add,
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apply lt_add_of_sub_lt_right _ _ _ H
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apply lt_add_of_sub_lt_right H
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end
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theorem sub_le_of_sub_le : a - b ≤ c → a - c ≤ b :=
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theorem sub_le_of_sub_le {a b c : A} : a - b ≤ c → a - c ≤ b :=
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begin
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intro H,
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apply sub_left_le_of_le_add,
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apply le_add_of_sub_right_le _ _ _ H
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apply le_add_of_sub_right_le H
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end
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-- TODO: the Isabelle library has varations on a + b ≤ b ↔ a ≤ 0
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apply le.refl
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end
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theorem sub_le_of_nonneg (H : b ≥ 0) : a - b ≤ a :=
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theorem sub_le_of_nonneg {b : A} (H : b ≥ 0) : a - b ≤ a :=
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add_le_of_le_of_nonpos (le.refl a) (neg_nonpos_of_nonneg H)
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theorem sub_lt_of_pos {b : A} (H : b > 0) : a - b < a :=
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@ -663,7 +663,7 @@ section
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if Hz : 0 ≤ a - b then
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(calc
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a ≥ b : (iff.mp !sub_nonneg_iff_le) Hz
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... ≥ b - c : sub_le_of_nonneg _ _ (le.trans !abs_nonneg H))
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... ≥ b - c : sub_le_of_nonneg _ (le.trans !abs_nonneg H))
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else
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(have Habs : b - a ≤ c, by rewrite [abs_of_neg (lt_of_not_ge Hz) at H, neg_sub at H]; apply H,
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have Habs' : b ≤ c + a, from (iff.mpr !le_add_iff_sub_right_le) Habs,
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@ -679,8 +679,8 @@ section
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... > b - c : sub_lt_of_pos _ (lt_of_le_of_lt !abs_nonneg H))
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else
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(have Habs : b - a < c, by rewrite [abs_of_neg (lt_of_not_ge Hz) at H, neg_sub at H]; apply H,
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have Habs' : b < c + a, from lt_add_of_sub_lt_right _ _ _ Habs,
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sub_lt_left_of_lt_add _ _ _ Habs')
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have Habs' : b < c + a, from lt_add_of_sub_lt_right Habs,
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sub_lt_left_of_lt_add Habs')
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theorem sub_lt_of_abs_sub_lt_right (H : abs (a - b) < c) : a - c < b :=
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sub_lt_of_abs_sub_lt_left (!abs_sub ▸ H)
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|
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|
@ -351,6 +351,16 @@ section
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-- TODO: do we want the iff versions?
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|
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theorem eq_zero_of_mul_eq_self_right {a b : A} (H₁ : b ≠ 1) (H₂ : a * b = a) : a = 0 :=
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have b - 1 ≠ 0, from
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suppose b - 1 = 0, H₁ (!zero_add ▸ eq_add_of_sub_eq this),
|
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have a * b - a = 0, by rewrite H₂; apply sub_self,
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||||
have a * (b - 1) = 0, by+ rewrite [mul_sub_left_distrib, mul_one]; apply this,
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show a = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero this) `b - 1 ≠ 0`
|
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|
||||
theorem eq_zero_of_mul_eq_self_left {a b : A} (H₁ : b ≠ 1) (H₂ : b * a = a) : a = 0 :=
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eq_zero_of_mul_eq_self_right H₁ (!mul.comm ▸ H₂)
|
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theorem mul_self_eq_mul_self_iff (a b : A) : a * a = b * b ↔ a = b ∨ a = -b :=
|
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iff.intro
|
||||
(suppose a * a = b * b,
|
||||
|
|
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Reference in a new issue