internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb1	[222, 1]	[223, 8]	sorry	α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) = x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb2	[225, 1]	[226, 8]	sorry	α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y = x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb1αα	[230, 1]	[235, 20]	apply le_antisymm	α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) = x	case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) ≤ x case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊓ (x ⊔ y)
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb1αα	[230, 1]	[235, 20]	apply le_inf	case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊓ (x ⊔ y)	case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb1αα	[230, 1]	[235, 20]	apply le_sup_left	case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ y	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb1αα	[230, 1]	[235, 20]	apply inf_le_left	case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) ≤ x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb1αα	[230, 1]	[235, 20]	apply le_refl	case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb2αα	[237, 1]	[242, 20]	apply le_antisymm	α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y = x	case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y ≤ x case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ x ⊓ y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb2αα	[237, 1]	[242, 20]	apply le_sup_left	case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ x ⊓ y	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb2αα	[237, 1]	[242, 20]	apply sup_le	case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y ≤ x	case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ y ≤ x
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb2αα	[237, 1]	[242, 20]	apply inf_le_left	case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ y ≤ x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	absorb2αα	[237, 1]	[242, 20]	apply le_refl	case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	aux1	[350, 1]	[352, 26]	rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b]	R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : a ≤ b ⊢ 0 ≤ b - a	R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : a ≤ b ⊢ -a + a ≤ -a + b
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	aux1	[350, 1]	[352, 26]	apply add_le_add_left h	R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : a ≤ b ⊢ -a + a ≤ -a + b	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	aux2	[354, 1]	[356, 26]	rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)]	R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : 0 ≤ b - a ⊢ a ≤ b	R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : 0 ≤ b - a ⊢ a + 0 ≤ a + (b - a)
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean	aux2	[354, 1]	[356, 26]	apply add_le_add_left h	R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : 0 ≤ b - a ⊢ a + 0 ≤ a + (b - a)	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S03_Topological_Spaces.lean	aux	[347, 1]	[355, 89]	simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in	X✝ : Type u_1 Y✝ : Type u_2 X : Type u_3 Y : Type u_4 A : Type u_5 inst✝ : TopologicalSpace X c : A → X f : A → Y x : X F : Filter Y h : Tendsto f (comap c (𝓝 x)) F V' : Set Y V'_in : V' ∈ F ⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V'	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C07_Hierarchies/S01_Basics.lean	left_inv_eq_right_inv₁	[269, 1]	[270, 90]	rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b]	M : Type inst✝ : Monoid₁ M a b c : M hba : b ⋄ a = 𝟙 hac : a ⋄ c = 𝟙 ⊢ b = c	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C07_Hierarchies/S01_Basics.lean	dia_inv	[306, 1]	[310, 51]	rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)]	G : Type inst✝ : Group₁ G a : G ⊢ a ⋄ a⁻¹ = 𝟙	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C07_Hierarchies/S01_Basics.lean	left_inv_eq_right_inv'	[355, 1]	[357, 54]	rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b]	M : Type inst✝ : Monoid₃ M a b c : M hba : b * a = 1 hac : a * c = 1 ⊢ b = c	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C07_Hierarchies/S01_Basics.lean	Group₃.mul_inv	[418, 1]	[423, 48]	rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)]	G : Type inst✝ : Group₃ G a : G ⊢ a * a⁻¹ = 1	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	have : x ∈ g '' univ := by contrapose! hx rw [sbSet, mem_iUnion] use 0 rw [sbAux, mem_diff] exact ⟨mem_univ _, hx⟩	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g ⊢ g (invFun g x) = x	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : x ∈ g '' univ ⊢ g (invFun g x) = x
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	have : ∃ y, g y = x := by simp at this assumption	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : x ∈ g '' univ ⊢ g (invFun g x) = x	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this✝ : x ∈ g '' univ this : ∃ y, g y = x ⊢ g (invFun g x) = x
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	exact invFun_eq this	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this✝ : x ∈ g '' univ this : ∃ y, g y = x ⊢ g (invFun g x) = x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	contrapose! hx	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g ⊢ x ∈ g '' univ	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbSet f g
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	rw [sbSet, mem_iUnion]	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbSet f g	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ ∃ i, x ∈ sbAux f g i
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	use 0	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ ∃ i, x ∈ sbAux f g i	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbAux f g 0
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	rw [sbAux, mem_diff]	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbAux f g 0	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ univ ∧ x ∉ g '' univ
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	exact ⟨mem_univ _, hx⟩	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ univ ∧ x ∉ g '' univ	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	simp at this	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : x ∈ g '' univ ⊢ ∃ y, g y = x	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : ∃ y, g y = x ⊢ ∃ y, g y = x
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_right_inv	[188, 1]	[206, 23]	assumption	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : ∃ y, g y = x ⊢ ∃ y, g y = x	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	set A := sbSet f g with A_def	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f ⊢ Injective (sbFun f g)	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Injective (sbFun f g)
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	set h := sbFun f g with h_def	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Injective (sbFun f g)	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Injective h
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	intro x₁ x₂	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Injective h	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α ⊢ h x₁ = h x₂ → x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	intro (hxeq : h x₁ = h x₂)	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α ⊢ h x₁ = h x₂ → x₁ = x₂	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : h x₁ = h x₂ ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	simp only [h_def, sbFun, ← A_def] at hxeq	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : h x₁ = h x₂ ⊢ x₁ = x₂	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	by_cases xA : x₁ ∈ A ∨ x₂ ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ ⊢ x₁ = x₂	case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A ⊢ x₁ = x₂ case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : ¬(x₁ ∈ A ∨ x₂ ∈ A) ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	push_neg at xA	case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : ¬(x₁ ∈ A ∨ x₂ ∈ A) ⊢ x₁ = x₂	case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rw [if_neg xA.1, if_neg xA.2] at hxeq	case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂	case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : invFun g x₁ = invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2]	case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : invFun g x₁ = invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA	case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A ⊢ x₁ = x₂	case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₁ = x₂ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	have x₂A : x₂ ∈ A := by apply _root_.not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A have x₂eq : x₂ = g (f x₁) := by rw [hxeq, sb_right_inv f g x₂nA] rcases x₁A with ⟨n, hn⟩ rw [A_def, sbSet, mem_iUnion] use n + 1 simp [sbAux] exact ⟨x₁, hn, x₂eq.symm⟩	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₁ = x₂	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rw [if_pos x₁A, if_pos x₂A] at hxeq	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = f x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	exact hf hxeq	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = f x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	symm	case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₁ = x₂	case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₂ = x₁
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	apply this hxeq.symm xA.symm (xA.resolve_left x₁A)	case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₂ = x₁	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	apply _root_.not_imp_self.mp	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₂ ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₂ ∉ A → x₂ ∈ A
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	intro (x₂nA : x₂ ∉ A)	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₂ ∉ A → x₂ ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rw [if_pos x₁A, if_neg x₂nA] at hxeq	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rw [A_def, sbSet, mem_iUnion] at x₁A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A ⊢ x₂ ∈ A
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	have x₂eq : x₂ = g (f x₁) := by rw [hxeq, sb_right_inv f g x₂nA]	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A ⊢ x₂ ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) ⊢ x₂ ∈ A
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rcases x₁A with ⟨n, hn⟩	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) ⊢ x₂ ∈ A	case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ A
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rw [A_def, sbSet, mem_iUnion]	case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ A	case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ i, x₂ ∈ sbAux f g i
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	use n + 1	case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ i, x₂ ∈ sbAux f g i	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ sbAux f g (n + 1)
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	simp [sbAux]	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ sbAux f g (n + 1)	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ a ∈ sbAux f g n, g (f a) = x₂
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	exact ⟨x₁, hn, x₂eq.symm⟩	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ a ∈ sbAux f g n, g (f a) = x₂	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_injective	[236, 1]	[274, 60]	rw [hxeq, sb_right_inv f g x₂nA]	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A ⊢ x₂ = g (f x₁)	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	set A := sbSet f g with A_def	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g ⊢ Surjective (sbFun f g)	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Surjective (sbFun f g)
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	set h := sbFun f g with h_def	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Surjective (sbFun f g)	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Surjective h
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	intro y	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Surjective h	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β ⊢ ∃ a, h a = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	by_cases gyA : g y ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β ⊢ ∃ a, h a = y	case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∈ A ⊢ ∃ a, h a = y case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ ∃ a, h a = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	use g y	case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ ∃ a, h a = y	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ h (g y) = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	simp only [h_def, sbFun, if_neg gyA]	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ h (g y) = y	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ invFun g (g y) = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	apply leftInverse_invFun hg	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ invFun g (g y) = y	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	rw [A_def, sbSet, mem_iUnion] at gyA	case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∈ A ⊢ ∃ a, h a = y	case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : ∃ i, g y ∈ sbAux f g i ⊢ ∃ a, h a = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	rcases gyA with ⟨n, hn⟩	case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : ∃ i, g y ∈ sbAux f g i ⊢ ∃ a, h a = y	case pos.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g n ⊢ ∃ a, h a = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	rcases n with _ \| n	case pos.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g n ⊢ ∃ a, h a = y	case pos.intro.zero α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β hn : g y ∈ sbAux f g 0 ⊢ ∃ a, h a = y case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g (n + 1) ⊢ ∃ a, h a = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	simp [sbAux] at hn	case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g (n + 1) ⊢ ∃ a, h a = y	case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : ∃ a ∈ sbAux f g n, g (f a) = g y ⊢ ∃ a, h a = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	rcases hn with ⟨x, xmem, hx⟩	case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : ∃ a ∈ sbAux f g n, g (f a) = g y ⊢ ∃ a, h a = y	case pos.intro.succ.intro.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ a, h a = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	use x	case pos.intro.succ.intro.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ a, h a = y	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ h x = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	have : x ∈ A := by rw [A_def, sbSet, mem_iUnion] exact ⟨n, xmem⟩	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ h x = y	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ h x = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	simp only [h_def, sbFun, if_pos this]	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ h x = y	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ f x = y
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	exact hg hx	case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ f x = y	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	simp [sbAux] at hn	case pos.intro.zero α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β hn : g y ∈ sbAux f g 0 ⊢ ∃ a, h a = y	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	rw [A_def, sbSet, mem_iUnion]	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ x ∈ A	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ i, x ∈ sbAux f g i
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean	sb_surjective	[315, 1]	[337, 30]	exact ⟨n, xmem⟩	α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ i, x ∈ sbAux f g i	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	even_of_even_sqr	[118, 1]	[120, 25]	rw [pow_two, Nat.prime_two.dvd_mul] at h	m : ℕ h : 2 ∣ m ^ 2 ⊢ 2 ∣ m	m : ℕ h : 2 ∣ m ∨ 2 ∣ m ⊢ 2 ∣ m
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	even_of_even_sqr	[118, 1]	[120, 25]	cases h <;> assumption	m : ℕ h : 2 ∣ m ∨ 2 ∣ m ⊢ 2 ∣ m	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	factorization_mul'	[284, 1]	[287, 6]	rw [Nat.factorization_mul mnez nnez]	m n : ℕ mnez : m ≠ 0 nnez : n ≠ 0 p : ℕ ⊢ (m * n).factorization p = m.factorization p + n.factorization p	m n : ℕ mnez : m ≠ 0 nnez : n ≠ 0 p : ℕ ⊢ (m.factorization + n.factorization) p = m.factorization p + n.factorization p
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	factorization_mul'	[284, 1]	[287, 6]	rfl	m n : ℕ mnez : m ≠ 0 nnez : n ≠ 0 p : ℕ ⊢ (m.factorization + n.factorization) p = m.factorization p + n.factorization p	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	factorization_pow'	[289, 1]	[292, 6]	rw [Nat.factorization_pow]	n k p : ℕ ⊢ (n ^ k).factorization p = k * n.factorization p	n k p : ℕ ⊢ (k • n.factorization) p = k * n.factorization p
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	factorization_pow'	[289, 1]	[292, 6]	rfl	n k p : ℕ ⊢ (k • n.factorization) p = k * n.factorization p	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	Nat.Prime.factorization'	[294, 1]	[297, 7]	rw [prime_p.factorization]	p : ℕ prime_p : p.Prime ⊢ p.factorization p = 1	p : ℕ prime_p : p.Prime ⊢ (Finsupp.single p 1) p = 1
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean	Nat.Prime.factorization'	[294, 1]	[297, 7]	simp	p : ℕ prime_p : p.Prime ⊢ (Finsupp.single p 1) p = 1	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	rw [Metric.cauchySeq_iff']	X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ⊢ CauchySeq u	X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	intro ε ε_pos	X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε	X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry	X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε	case intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	use N	case intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε	case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∀ n ≥ N, dist (u n) (u N) < ε
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	intro n hn	case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∀ n ≥ N, dist (u n) (u N) < ε	case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε n : ℕ hn : n ≥ N ⊢ dist (u n) (u N) < ε
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn	case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε n : ℕ hn : n ≥ N ⊢ dist (u n) (u N) < ε	case h.intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε k : ℕ hn : N + k ≥ N ⊢ dist (u (N + k)) (u N) < ε
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := sorry _ ≤ 1 / 2 ^ N * 2 := sorry _ < ε := sorry	case h.intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε k : ℕ hn : N + k ≥ N ⊢ dist (u (N + k)) (u N) < ε	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C09_Topology/S02_Metric_Spaces.lean	cauchySeq_of_le_geometric_two'	[423, 1]	[437, 19]	sorry	X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 ⊢ ∃ N, 1 / 2 ^ N * 2 < ε	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	chineseMap_inj	[260, 1]	[264, 58]	rw [chineseMap, injective_lift_iff, ker_Pi_Quotient_mk]	ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : ι → Ideal R ⊢ Injective ⇑(chineseMap I)	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	isCoprime_Inf	[291, 1]	[311, 87]	classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with \| empty => simp \| @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K * (I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1 + K) * I + K * J i := by ring _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf	ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, IsCoprime I (J j) ⊢ IsCoprime I (⨅ j ∈ s, J j)	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	isCoprime_Inf	[291, 1]	[311, 87]	simp_rw [isCoprime_iff_add] at *	ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, IsCoprime I (J j) ⊢ IsCoprime I (⨅ j ∈ s, J j)	ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, I + J j = 1 ⊢ I + ⨅ j ∈ s, J j = 1
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	isCoprime_Inf	[291, 1]	[311, 87]	induction s using Finset.induction with \| empty => simp \| @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K * (I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1 + K) * I + K * J i := by ring _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf	ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, I + J j = 1 ⊢ I + ⨅ j ∈ s, J j = 1	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	isCoprime_Inf	[291, 1]	[311, 87]	simp	case empty ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R hf : ∀ j ∈ ∅, I + J j = 1 ⊢ I + ⨅ j ∈ ∅, J j = 1	no goals
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	isCoprime_Inf	[291, 1]	[311, 87]	rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]	case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1 hf : ∀ j ∈ insert i s, I + J j = 1 ⊢ I + ⨅ j ∈ insert i s, J j = 1	case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1 hf : ∀ j ∈ insert i s, I + J j = 1 ⊢ 1 ≤ I + (⨅ x ∈ s, J x) ⊓ J i
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	isCoprime_Inf	[291, 1]	[311, 87]	set K := ⨅ j ∈ s, J j	case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1 hf : ∀ j ∈ insert i s, I + J j = 1 ⊢ 1 ≤ I + (⨅ x ∈ s, J x) ⊓ J i	case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ 1 ≤ I + K ⊓ J i
https://github.com/avigad/mathematics_in_lean_source.git	3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3	MIL/C08_Groups_and_Rings/S02_Rings.lean	isCoprime_Inf	[291, 1]	[311, 87]	calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K * (I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1 + K) * I + K * J i := by ring _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf	case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ 1 ≤ I + K ⊓ J i	no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb1

[222, 1]

[223, 8]

sorry

α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) = x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb2

[225, 1]

[226, 8]

sorry

α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y = x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb1αα

[230, 1]

[235, 20]

apply le_antisymm

α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) = x

case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) ≤ x case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊓ (x ⊔ y)

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb1αα

[230, 1]

[235, 20]

apply le_inf

case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊓ (x ⊔ y)

case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb1αα

[230, 1]

[235, 20]

apply le_sup_left

case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ y

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb1αα

[230, 1]

[235, 20]

apply inf_le_left

case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ (x ⊔ y) ≤ x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb1αα

[230, 1]

[235, 20]

apply le_refl

case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb2αα

[237, 1]

[242, 20]

apply le_antisymm

α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y = x

case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y ≤ x case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ x ⊓ y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb2αα

[237, 1]

[242, 20]

apply le_sup_left

case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x ⊔ x ⊓ y

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb2αα

[237, 1]

[242, 20]

apply sup_le

case a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊔ x ⊓ y ≤ x

case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ y ≤ x

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb2αα

[237, 1]

[242, 20]

apply inf_le_left

case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ⊓ y ≤ x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

absorb2αα

[237, 1]

[242, 20]

apply le_refl

case a.a α : Type u_1 inst✝ : Lattice α x y z : α ⊢ x ≤ x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

aux1

[350, 1]

[352, 26]

rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b]

R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : a ≤ b ⊢ 0 ≤ b - a

R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : a ≤ b ⊢ -a + a ≤ -a + b

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

aux1

[350, 1]

[352, 26]

apply add_le_add_left h

R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : a ≤ b ⊢ -a + a ≤ -a + b

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

aux2

[354, 1]

[356, 26]

rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)]

R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : 0 ≤ b - a ⊢ a ≤ b

R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : 0 ≤ b - a ⊢ a + 0 ≤ a + (b - a)

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean

aux2

[354, 1]

[356, 26]

apply add_le_add_left h

R : Type u_1 inst✝ : StrictOrderedRing R a b c : R h : 0 ≤ b - a ⊢ a + 0 ≤ a + (b - a)

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S03_Topological_Spaces.lean

aux

[347, 1]

[355, 89]

simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in

X✝ : Type u_1 Y✝ : Type u_2 X : Type u_3 Y : Type u_4 A : Type u_5 inst✝ : TopologicalSpace X c : A → X f : A → Y x : X F : Filter Y h : Tendsto f (comap c (𝓝 x)) F V' : Set Y V'_in : V' ∈ F ⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V'

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C07_Hierarchies/S01_Basics.lean

left_inv_eq_right_inv₁

[269, 1]

[270, 90]

rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b]

M : Type inst✝ : Monoid₁ M a b c : M hba : b ⋄ a = 𝟙 hac : a ⋄ c = 𝟙 ⊢ b = c

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C07_Hierarchies/S01_Basics.lean

dia_inv

[306, 1]

[310, 51]

rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)]

G : Type inst✝ : Group₁ G a : G ⊢ a ⋄ a⁻¹ = 𝟙

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C07_Hierarchies/S01_Basics.lean

left_inv_eq_right_inv'

[355, 1]

[357, 54]

rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b]

M : Type inst✝ : Monoid₃ M a b c : M hba : b * a = 1 hac : a * c = 1 ⊢ b = c

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C07_Hierarchies/S01_Basics.lean

Group₃.mul_inv

[418, 1]

[423, 48]

rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)]

G : Type inst✝ : Group₃ G a : G ⊢ a * a⁻¹ = 1

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

have : x ∈ g '' univ := by contrapose! hx rw [sbSet, mem_iUnion] use 0 rw [sbAux, mem_diff] exact ⟨mem_univ _, hx⟩

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g ⊢ g (invFun g x) = x

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : x ∈ g '' univ ⊢ g (invFun g x) = x

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

have : ∃ y, g y = x := by simp at this assumption

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : x ∈ g '' univ ⊢ g (invFun g x) = x

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this✝ : x ∈ g '' univ this : ∃ y, g y = x ⊢ g (invFun g x) = x

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

exact invFun_eq this

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this✝ : x ∈ g '' univ this : ∃ y, g y = x ⊢ g (invFun g x) = x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

contrapose! hx

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g ⊢ x ∈ g '' univ

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbSet f g

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

rw [sbSet, mem_iUnion]

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbSet f g

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ ∃ i, x ∈ sbAux f g i

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

use 0

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ ∃ i, x ∈ sbAux f g i

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbAux f g 0

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

rw [sbAux, mem_diff]

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ sbAux f g 0

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ univ ∧ x ∉ g '' univ

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

exact ⟨mem_univ _, hx⟩

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ g '' univ ⊢ x ∈ univ ∧ x ∉ g '' univ

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

simp at this

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : x ∈ g '' univ ⊢ ∃ y, g y = x

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : ∃ y, g y = x ⊢ ∃ y, g y = x

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_right_inv

[188, 1]

[206, 23]

assumption

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α x : α hx : x ∉ sbSet f g this : ∃ y, g y = x ⊢ ∃ y, g y = x

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

set A := sbSet f g with A_def

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f ⊢ Injective (sbFun f g)

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Injective (sbFun f g)

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

set h := sbFun f g with h_def

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Injective (sbFun f g)

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Injective h

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

intro x₁ x₂

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Injective h

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α ⊢ h x₁ = h x₂ → x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

intro (hxeq : h x₁ = h x₂)

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α ⊢ h x₁ = h x₂ → x₁ = x₂

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : h x₁ = h x₂ ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

simp only [h_def, sbFun, ← A_def] at hxeq

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : h x₁ = h x₂ ⊢ x₁ = x₂

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

by_cases xA : x₁ ∈ A ∨ x₂ ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ ⊢ x₁ = x₂

case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A ⊢ x₁ = x₂ case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : ¬(x₁ ∈ A ∨ x₂ ∈ A) ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

push_neg at xA

case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : ¬(x₁ ∈ A ∨ x₂ ∈ A) ⊢ x₁ = x₂

case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rw [if_neg xA.1, if_neg xA.2] at hxeq

case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂

case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : invFun g x₁ = invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2]

case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : invFun g x₁ = invFun g x₂ xA : x₁ ∉ A ∧ x₂ ∉ A ⊢ x₁ = x₂

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA

case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A ⊢ x₁ = x₂

case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₁ = x₂ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

have x₂A : x₂ ∈ A := by apply _root_.not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A have x₂eq : x₂ = g (f x₁) := by rw [hxeq, sb_right_inv f g x₂nA] rcases x₁A with ⟨n, hn⟩ rw [A_def, sbSet, mem_iUnion] use n + 1 simp [sbAux] exact ⟨x₁, hn, x₂eq.symm⟩

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₁ = x₂

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rw [if_pos x₁A, if_pos x₂A] at hxeq

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = f x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

exact hf hxeq

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = f x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂A : x₂ ∈ A ⊢ x₁ = x₂

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

symm

case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₁ = x₂

case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₂ = x₁

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

apply this hxeq.symm xA.symm (xA.resolve_left x₁A)

case pos.inr α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A this : ∀ ⦃x₁ x₂ : α⦄, ((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂ x₁A : x₁ ∉ A ⊢ x₂ = x₁

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

apply _root_.not_imp_self.mp

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₂ ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₂ ∉ A → x₂ ∈ A

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

intro (x₂nA : x₂ ∉ A)

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A ⊢ x₂ ∉ A → x₂ ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rw [if_pos x₁A, if_neg x₂nA] at hxeq

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rw [A_def, sbSet, mem_iUnion] at x₁A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : x₁ ∈ A x₂nA : x₂ ∉ A ⊢ x₂ ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A ⊢ x₂ ∈ A

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

have x₂eq : x₂ = g (f x₁) := by rw [hxeq, sb_right_inv f g x₂nA]

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A ⊢ x₂ ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) ⊢ x₂ ∈ A

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rcases x₁A with ⟨n, hn⟩

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) ⊢ x₂ ∈ A

case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ A

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rw [A_def, sbSet, mem_iUnion]

case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ A

case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ i, x₂ ∈ sbAux f g i

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

use n + 1

case intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ i, x₂ ∈ sbAux f g i

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ sbAux f g (n + 1)

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

simp [sbAux]

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ x₂ ∈ sbAux f g (n + 1)

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ a ∈ sbAux f g n, g (f a) = x₂

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

exact ⟨x₁, hn, x₂eq.symm⟩

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₂nA : x₂ ∉ A x₂eq : x₂ = g (f x₁) n : ℕ hn : x₁ ∈ sbAux f g n ⊢ ∃ a ∈ sbAux f g n, g (f a) = x₂

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_injective

[236, 1]

[274, 60]

rw [hxeq, sb_right_inv f g x₂nA]

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g x₁ x₂ : α hxeq : f x₁ = invFun g x₂ xA : x₁ ∈ A ∨ x₂ ∈ A x₁A : ∃ i, x₁ ∈ sbAux f g i x₂nA : x₂ ∉ A ⊢ x₂ = g (f x₁)

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

set A := sbSet f g with A_def

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g ⊢ Surjective (sbFun f g)

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Surjective (sbFun f g)

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

set h := sbFun f g with h_def

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g ⊢ Surjective (sbFun f g)

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Surjective h

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

intro y

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g ⊢ Surjective h

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β ⊢ ∃ a, h a = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

by_cases gyA : g y ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β ⊢ ∃ a, h a = y

case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∈ A ⊢ ∃ a, h a = y case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ ∃ a, h a = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

use g y

case neg α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ ∃ a, h a = y

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ h (g y) = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

simp only [h_def, sbFun, if_neg gyA]

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ h (g y) = y

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ invFun g (g y) = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

apply leftInverse_invFun hg

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∉ A ⊢ invFun g (g y) = y

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

rw [A_def, sbSet, mem_iUnion] at gyA

case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : g y ∈ A ⊢ ∃ a, h a = y

case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : ∃ i, g y ∈ sbAux f g i ⊢ ∃ a, h a = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

rcases gyA with ⟨n, hn⟩

case pos α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β gyA : ∃ i, g y ∈ sbAux f g i ⊢ ∃ a, h a = y

case pos.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g n ⊢ ∃ a, h a = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

rcases n with _ | n

case pos.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g n ⊢ ∃ a, h a = y

case pos.intro.zero α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β hn : g y ∈ sbAux f g 0 ⊢ ∃ a, h a = y case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g (n + 1) ⊢ ∃ a, h a = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

simp [sbAux] at hn

case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : g y ∈ sbAux f g (n + 1) ⊢ ∃ a, h a = y

case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : ∃ a ∈ sbAux f g n, g (f a) = g y ⊢ ∃ a, h a = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

rcases hn with ⟨x, xmem, hx⟩

case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ hn : ∃ a ∈ sbAux f g n, g (f a) = g y ⊢ ∃ a, h a = y

case pos.intro.succ.intro.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ a, h a = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

use x

case pos.intro.succ.intro.intro α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ a, h a = y

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ h x = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

have : x ∈ A := by rw [A_def, sbSet, mem_iUnion] exact ⟨n, xmem⟩

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ h x = y

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ h x = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

simp only [h_def, sbFun, if_pos this]

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ h x = y

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ f x = y

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

exact hg hx

case h α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y this : x ∈ A ⊢ f x = y

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

simp [sbAux] at hn

case pos.intro.zero α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β hn : g y ∈ sbAux f g 0 ⊢ ∃ a, h a = y

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

rw [A_def, sbSet, mem_iUnion]

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ x ∈ A

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ i, x ∈ sbAux f g i

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean

sb_surjective

[315, 1]

[337, 30]

exact ⟨n, xmem⟩

α : Type u_1 β : Type u_2 inst✝ : Nonempty β f : α → β g : β → α hf : Injective f hg : Injective g A : Set α := sbSet f g A_def : A = sbSet f g h : α → β := sbFun f g h_def : h = sbFun f g y : β n : ℕ x : α xmem : x ∈ sbAux f g n hx : g (f x) = g y ⊢ ∃ i, x ∈ sbAux f g i

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

even_of_even_sqr

[118, 1]

[120, 25]

rw [pow_two, Nat.prime_two.dvd_mul] at h

m : ℕ h : 2 ∣ m ^ 2 ⊢ 2 ∣ m

m : ℕ h : 2 ∣ m ∨ 2 ∣ m ⊢ 2 ∣ m

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

even_of_even_sqr

[118, 1]

[120, 25]

cases h <;> assumption

m : ℕ h : 2 ∣ m ∨ 2 ∣ m ⊢ 2 ∣ m

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

factorization_mul'

[284, 1]

[287, 6]

rw [Nat.factorization_mul mnez nnez]

m n : ℕ mnez : m ≠ 0 nnez : n ≠ 0 p : ℕ ⊢ (m * n).factorization p = m.factorization p + n.factorization p

m n : ℕ mnez : m ≠ 0 nnez : n ≠ 0 p : ℕ ⊢ (m.factorization + n.factorization) p = m.factorization p + n.factorization p

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

factorization_mul'

[284, 1]

[287, 6]

rfl

m n : ℕ mnez : m ≠ 0 nnez : n ≠ 0 p : ℕ ⊢ (m.factorization + n.factorization) p = m.factorization p + n.factorization p

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

factorization_pow'

[289, 1]

[292, 6]

rw [Nat.factorization_pow]

n k p : ℕ ⊢ (n ^ k).factorization p = k * n.factorization p

n k p : ℕ ⊢ (k • n.factorization) p = k * n.factorization p

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

factorization_pow'

[289, 1]

[292, 6]

rfl

n k p : ℕ ⊢ (k • n.factorization) p = k * n.factorization p

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

Nat.Prime.factorization'

[294, 1]

[297, 7]

rw [prime_p.factorization]

p : ℕ prime_p : p.Prime ⊢ p.factorization p = 1

p : ℕ prime_p : p.Prime ⊢ (Finsupp.single p 1) p = 1

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean

Nat.Prime.factorization'

[294, 1]

[297, 7]

simp

p : ℕ prime_p : p.Prime ⊢ (Finsupp.single p 1) p = 1

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

rw [Metric.cauchySeq_iff']

X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ⊢ CauchySeq u

X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

intro ε ε_pos

X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε

X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry

X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε

case intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

use N

case intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε

case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∀ n ≥ N, dist (u n) (u N) < ε

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

intro n hn

case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε ⊢ ∀ n ≥ N, dist (u n) (u N) < ε

case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε n : ℕ hn : n ≥ N ⊢ dist (u n) (u N) < ε

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn

case h X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε n : ℕ hn : n ≥ N ⊢ dist (u n) (u N) < ε

case h.intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε k : ℕ hn : N + k ≥ N ⊢ dist (u (N + k)) (u N) < ε

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := sorry _ ≤ 1 / 2 ^ N * 2 := sorry _ < ε := sorry

case h.intro X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 N : ℕ hN : 1 / 2 ^ N * 2 < ε k : ℕ hn : N + k ≥ N ⊢ dist (u (N + k)) (u N) < ε

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C09_Topology/S02_Metric_Spaces.lean

cauchySeq_of_le_geometric_two'

[423, 1]

[437, 19]

sorry

X : Type u_1 inst✝ : MetricSpace X a b c : X r : ℝ u : ℕ → X hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n ε : ℝ ε_pos : ε > 0 ⊢ ∃ N, 1 / 2 ^ N * 2 < ε

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

chineseMap_inj

[260, 1]

[264, 58]

rw [chineseMap, injective_lift_iff, ker_Pi_Quotient_mk]

ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : ι → Ideal R ⊢ Injective ⇑(chineseMap I)

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

isCoprime_Inf

[291, 1]

[311, 87]

classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K * (I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1 + K) * I + K * J i := by ring _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf

ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, IsCoprime I (J j) ⊢ IsCoprime I (⨅ j ∈ s, J j)

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

isCoprime_Inf

[291, 1]

[311, 87]

simp_rw [isCoprime_iff_add] at *

ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, IsCoprime I (J j) ⊢ IsCoprime I (⨅ j ∈ s, J j)

ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, I + J j = 1 ⊢ I + ⨅ j ∈ s, J j = 1

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

isCoprime_Inf

[291, 1]

[311, 87]

induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K * (I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1 + K) * I + K * J i := by ring _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf

ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R s : Finset ι hf : ∀ j ∈ s, I + J j = 1 ⊢ I + ⨅ j ∈ s, J j = 1

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

isCoprime_Inf

[291, 1]

[311, 87]

simp

case empty ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R hf : ∀ j ∈ ∅, I + J j = 1 ⊢ I + ⨅ j ∈ ∅, J j = 1

no goals

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

isCoprime_Inf

[291, 1]

[311, 87]

rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]

case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1 hf : ∀ j ∈ insert i s, I + J j = 1 ⊢ I + ⨅ j ∈ insert i s, J j = 1

case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1 hf : ∀ j ∈ insert i s, I + J j = 1 ⊢ 1 ≤ I + (⨅ x ∈ s, J x) ⊓ J i

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

isCoprime_Inf

[291, 1]

[311, 87]

set K := ⨅ j ∈ s, J j

case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1 hf : ∀ j ∈ insert i s, I + J j = 1 ⊢ 1 ≤ I + (⨅ x ∈ s, J x) ⊓ J i

case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ 1 ≤ I + K ⊓ J i

https://github.com/avigad/mathematics_in_lean_source.git

3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3

MIL/C08_Groups_and_Rings/S02_Rings.lean

isCoprime_Inf

[291, 1]

[311, 87]

calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K * (I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1 + K) * I + K * J i := by ring _ ≤ I + K ⊓ J i := by gcongr ; apply mul_le_left ; apply mul_le_inf

case insert ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ 1 ≤ I + K ⊓ J i

no goals