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stringlengths 6
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https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | contradiction | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ j ∉ f j | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | intro f surjf | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | let S := { i | i ∉ f i } | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | rcases surjf S with ⟨j, h⟩ | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | have h₁ : j ∉ f j := by
intro h'
have : j ∉ f j := by rwa [h] at h'
contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | have h₂ : j ∈ S := h₁ | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | have h₃ : j ∉ S := by rwa [h] at h₁ | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | intro h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ j ∉ f j | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | have : j ∉ f j := by rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | contradiction | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ j ∉ f j | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantorαα | [688, 1] | [698, 16] | rwa [h] at h₁ | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [158, 1] | [159, 8] | sorry | x✝ y x : ℝ
⊢ x ≤ |x| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [161, 1] | [162, 8] | sorry | x✝ y x : ℝ
⊢ -x ≤ |x| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_add | [164, 1] | [165, 8] | sorry | x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_selfαα | [169, 1] | [173, 13] | rcases le_or_gt 0 x with h | h | x✝ y x : ℝ
⊢ x ≤ |x| | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ x ≤ |x|
case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_selfαα | [169, 1] | [173, 13] | . rw [abs_of_neg h]
linarith | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_selfαα | [169, 1] | [173, 13] | rw [abs_of_nonneg h] | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ x ≤ |x| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_selfαα | [169, 1] | [173, 13] | rw [abs_of_neg h] | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x| | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ -x |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_selfαα | [169, 1] | [173, 13] | linarith | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ -x | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_selfαα | [175, 1] | [179, 22] | rcases le_or_gt 0 x with h | h | x✝ y x : ℝ
⊢ -x ≤ |x| | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ |x|
case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_selfαα | [175, 1] | [179, 22] | . rw [abs_of_neg h] | case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_selfαα | [175, 1] | [179, 22] | rw [abs_of_nonneg h] | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ |x| | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ x |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_selfαα | [175, 1] | [179, 22] | linarith | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ x | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_selfαα | [175, 1] | [179, 22] | rw [abs_of_neg h] | case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_addαα | [181, 1] | [186, 52] | rcases le_or_gt 0 (x + y) with h | h | x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y| | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y|
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_addαα | [181, 1] | [186, 52] | . rw [abs_of_neg h]
linarith [neg_le_abs_self x, neg_le_abs_self y] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_addαα | [181, 1] | [186, 52] | rw [abs_of_nonneg h] | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y| | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_addαα | [181, 1] | [186, 52] | linarith [le_abs_self x, le_abs_self y] | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_addαα | [181, 1] | [186, 52] | rw [abs_of_neg h] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_addαα | [181, 1] | [186, 52] | linarith [neg_le_abs_self x, neg_le_abs_self y] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [194, 1] | [195, 8] | sorry | x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [197, 1] | [198, 8] | sorry | x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | rcases le_or_gt 0 y with h | h | x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y
case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | rw [abs_of_neg h] | case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y | case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | constructor | case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y | case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y
case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | rw [abs_of_nonneg h] | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | constructor | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y | case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y
case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | intro h' | case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y | case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | left | case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y | case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | exact h' | case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | intro h' | case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y | case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | rcases h' with h' | h' | case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y | case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y
case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | . linarith | case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | exact h' | case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | linarith | case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | intro h' | case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y | case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | right | case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y | case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | exact h' | case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | intro h' | case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y | case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | rcases h' with h' | h' | case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y | case inr.mpr.inl
x y : ℝ
h : 0 > y
h' : x < y
⊢ x < -y
case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | . exact h' | case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | linarith | case inr.mpr.inl
x y : ℝ
h : 0 > y
h' : x < y
⊢ x < -y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_absαα | [202, 1] | [221, 15] | exact h' | case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | rcases le_or_gt 0 x with h | h | x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y | case inl
x y : ℝ
h : 0 ≤ x
⊢ |x| < y ↔ -y < x ∧ x < y
case inr
x y : ℝ
h : 0 > x
⊢ |x| < y ↔ -y < x ∧ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | rw [abs_of_nonneg h] | case inl
x y : ℝ
h : 0 ≤ x
⊢ |x| < y ↔ -y < x ∧ x < y | case inl
x y : ℝ
h : 0 ≤ x
⊢ x < y ↔ -y < x ∧ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | constructor | case inl
x y : ℝ
h : 0 ≤ x
⊢ x < y ↔ -y < x ∧ x < y | case inl.mp
x y : ℝ
h : 0 ≤ x
⊢ x < y → -y < x ∧ x < y
case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | . intro h'
rcases h' with ⟨h1, h2⟩
exact h2 | case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | intro h' | case inl.mp
x y : ℝ
h : 0 ≤ x
⊢ x < y → -y < x ∧ x < y | case inl.mp
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x ∧ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | constructor | case inl.mp
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x ∧ x < y | case inl.mp.left
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x
case inl.mp.right
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | exact h' | case inl.mp.right
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | linarith | case inl.mp.left
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | intro h' | case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y | case inl.mpr
x y : ℝ
h : 0 ≤ x
h' : -y < x ∧ x < y
⊢ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | rcases h' with ⟨h1, h2⟩ | case inl.mpr
x y : ℝ
h : 0 ≤ x
h' : -y < x ∧ x < y
⊢ x < y | case inl.mpr.intro
x y : ℝ
h : 0 ≤ x
h1 : -y < x
h2 : x < y
⊢ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | exact h2 | case inl.mpr.intro
x y : ℝ
h : 0 ≤ x
h1 : -y < x
h2 : x < y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | rw [abs_of_neg h] | case inr
x y : ℝ
h : 0 > x
⊢ |x| < y ↔ -y < x ∧ x < y | case inr
x y : ℝ
h : 0 > x
⊢ -x < y ↔ -y < x ∧ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | constructor | case inr
x y : ℝ
h : 0 > x
⊢ -x < y ↔ -y < x ∧ x < y | case inr.mp
x y : ℝ
h : 0 > x
⊢ -x < y → -y < x ∧ x < y
case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | . intro h'
linarith | case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | intro h' | case inr.mp
x y : ℝ
h : 0 > x
⊢ -x < y → -y < x ∧ x < y | case inr.mp
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x ∧ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | constructor | case inr.mp
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x ∧ x < y | case inr.mp.left
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x
case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | . linarith | case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | linarith | case inr.mp.left
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | linarith | case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | intro h' | case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y | case inr.mpr
x y : ℝ
h : 0 > x
h' : -y < x ∧ x < y
⊢ -x < y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_ltαα | [223, 1] | [241, 15] | linarith | case inr.mpr
x y : ℝ
h : 0 > x
h' : -y < x ∧ x < y
⊢ -x < y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_zero | [138, 1] | [138, 67] | rw [add_comm, zero_add] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a + 0 = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_neg | [140, 1] | [140, 77] | rw [add_comm, add_left_neg] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a + -a = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_add_cancel_left | [170, 1] | [171, 43] | rw [← add_assoc, add_left_neg, zero_add] | R : Type u_1
inst✝ : Ring R
a b : R
⊢ -a + (a + b) = b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_neg_cancel_right | [179, 1] | [180, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_neg_cancel_rightαα | [184, 1] | [185, 42] | rw [add_assoc, add_right_neg, add_zero] | R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_left_cancel | [191, 1] | [192, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_cancel | [194, 1] | [195, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_left_cancelαα | [199, 1] | [200, 57] | rw [← neg_add_cancel_left a b, h, neg_add_cancel_left] | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_cancelαα | [202, 1] | [203, 59] | rw [← add_neg_cancel_right a b, h, add_neg_cancel_right] | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [237, 1] | [240, 25] | have h : a * 0 + a * 0 = a * 0 + 0 := by
rw [← mul_add, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 = 0 | R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [237, 1] | [240, 25] | rw [add_left_cancel h] | R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [237, 1] | [240, 25] | rw [← mul_add, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 + a * 0 = a * 0 + 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mul | [279, 1] | [280, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mulαα | [284, 1] | [286, 25] | have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0 | R : Type u_1
inst✝ : Ring R
a : R
h : 0 * a + 0 * a = 0 * a + 0
⊢ 0 * a = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mulαα | [284, 1] | [286, 25] | rw [add_left_cancel h] | R : Type u_1
inst✝ : Ring R
a : R
h : 0 * a + 0 * a = 0 * a + 0
⊢ 0 * a = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mulαα | [284, 1] | [286, 25] | rw [← add_mul, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a + 0 * a = 0 * a + 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_eq_of_add_eq_zero | [295, 1] | [296, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zero | [298, 1] | [299, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [301, 1] | [303, 16] | apply neg_eq_of_add_eq_zero | R : Type u_1
inst✝ : Ring R
⊢ -0 = 0 | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [301, 1] | [303, 16] | rw [add_zero] | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_neg | [305, 1] | [306, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ - -a = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_eq_of_add_eq_zeroαα | [310, 1] | [311, 46] | rw [← neg_add_cancel_left a b, h, add_zero] | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zeroαα | [313, 1] | [316, 19] | symm | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -b = a |
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