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https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_injective
[37, 1]
[64, 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...
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 +...
Please generate a tactic in lean4 to solve the state. STATE: 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 ...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_injective
[37, 1]
[64, 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 +...
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, ...
Please generate a tactic in lean4 to solve the state. STATE: 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...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_injective
[37, 1]
[64, 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, ...
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_injective
[37, 1]
[64, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 x₂...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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)
Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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 : Injecti...
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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 Nat.zero ⊢ ∃ a, h a = y case pos.intro.succ α : Type u_1 β : Type u_2 inst✝ : Nonem...
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTI...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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 (Nat.succ n) ⊢ ∃ 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
Please generate a tactic in lean4 to solve the state. STATE: 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 (Nat.succ n) ⊢ ∃ ...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 ...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 thi...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 thi...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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 Nat.zero ⊢ ∃ a, h a = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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 Nat.zero ⊢ ∃ a, h a = y...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TA...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean
sb_surjective
[66, 1]
[85, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 ∈...
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
inverse_spec
[201, 1]
[203, 32]
rw [inverse, dif_pos h]
α : Type u_1 β : Type u_2 inst✝ : Inhabited α f : α → β y : β h : ∃ x, f x = y ⊢ f (inverse f y) = y
α : Type u_1 β : Type u_2 inst✝ : Inhabited α f : α → β y : β h : ∃ x, f x = y ⊢ f (choose h) = y
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 β : Type u_2 inst✝ : Inhabited α f : α → β y : β h : ∃ x, f x = y ⊢ f (inverse f y) = y TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
inverse_spec
[201, 1]
[203, 32]
exact Classical.choose_spec h
α : Type u_1 β : Type u_2 inst✝ : Inhabited α f : α → β y : β h : ∃ x, f x = y ⊢ f (choose h) = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 β : Type u_2 inst✝ : Inhabited α f : α → β y : β h : ∃ x, f x = y ⊢ f (choose h) = y TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
intro f surjf
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 ⊢ ∀ (f : α → Set α), ¬Surjective f
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 ⊢ ∀ (f : α → Set α), ¬Surjective f TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
let S := { i | i ∉ f i }
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f ⊢ False
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
rcases surjf S with ⟨j, h⟩
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} ⊢ False
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
have h₁ : j ∉ f j := by intro h' have : j ∉ f j := by rwa [h] at h' contradiction
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S ⊢ False
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
have h₂ : j ∈ S := h₁
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j ⊢ False
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
have h₃ : j ∉ S := by rwa [h] at h₁
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S ⊢ False
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S h₃ : j ∉ S ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
contradiction
case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S h₃ : j ∉ S ⊢ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S h₃ : j ∉ S ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
intro h'
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S ⊢ j ∉ f j
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S ⊢ j ∉ f j TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
have : j ∉ f j := by rwa [h] at h'
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j ⊢ False
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j this : j ∉ f j ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
contradiction
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j this : j ∉ f j ⊢ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j this : j ∉ f j ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
rwa [h] at h'
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j ⊢ j ∉ f j
no goals
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h' : j ∈ f j ⊢ j ∉ f j TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
Cantor
[239, 1]
[249, 16]
rwa [h] at h₁
α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S ⊢ j ∉ S
no goals
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 β : Type u_2 inst✝ : Inhabited α✝ α : Type u_3 f : α → Set α surjf : Surjective f S : Set α := {i | i ∉ f i} j : α h : f j = S h₁ : j ∉ f j h₂ : j ∈ S ⊢ j ∉ S TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
C02S04.aux
[41, 1]
[46, 21]
apply le_min
a b c d : ℝ ⊢ min a b + c ≤ min (a + c) (b + c)
case h₁ a b c d : ℝ ⊢ min a b + c ≤ a + c case h₂ a b c d : ℝ ⊢ min a b + c ≤ b + c
Please generate a tactic in lean4 to solve the state. STATE: a b c d : ℝ ⊢ min a b + c ≤ min (a + c) (b + c) TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
C02S04.aux
[41, 1]
[46, 21]
apply add_le_add_right
case h₂ a b c d : ℝ ⊢ min a b + c ≤ b + c
case h₂.bc a b c d : ℝ ⊢ min a b ≤ b
Please generate a tactic in lean4 to solve the state. STATE: case h₂ a b c d : ℝ ⊢ min a b + c ≤ b + c TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
C02S04.aux
[41, 1]
[46, 21]
apply min_le_right
case h₂.bc a b c d : ℝ ⊢ min a b ≤ b
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₂.bc a b c d : ℝ ⊢ min a b ≤ b TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
C02S04.aux
[41, 1]
[46, 21]
apply add_le_add_right
case h₁ a b c d : ℝ ⊢ min a b + c ≤ a + c
case h₁.bc a b c d : ℝ ⊢ min a b ≤ a
Please generate a tactic in lean4 to solve the state. STATE: case h₁ a b c d : ℝ ⊢ min a b + c ≤ a + c TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
C02S04.aux
[41, 1]
[46, 21]
apply min_le_left
case h₁.bc a b c d : ℝ ⊢ min a b ≤ a
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁.bc a b c d : ℝ ⊢ min a b ≤ a TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S5_Topology/solutions/Solutions_S02_Metric_Spaces.lean
cauchySeq_of_le_geometric_two'
[185, 1]
[199, 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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.add_zero
[246, 1]
[246, 67]
rw [add_comm, zero_add]
R : Type u_1 inst✝ : Ring R a : R ⊢ a + 0 = a
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊢ a + 0 = a TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.add_right_neg
[248, 1]
[248, 77]
rw [add_comm, add_left_neg]
R : Type u_1 inst✝ : Ring R a : R ⊢ a + -a = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊢ a + -a = 0 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.mul_zero
[261, 1]
[264, 25]
have h : a * 0 + a * 0 = a * 0 + 0 := by rw [← mul_add, add_zero, add_zero]
R : Type u_1 inst✝ : Ring R a : R ⊢ a * 0 = 0
R : Type u_1 inst✝ : Ring R a : R h : a * 0 + a * 0 = a * 0 + 0 ⊢ a * 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊢ a * 0 = 0 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.mul_zero
[261, 1]
[264, 25]
rw [add_left_cancel h]
R : Type u_1 inst✝ : Ring R a : R h : a * 0 + a * 0 = a * 0 + 0 ⊢ a * 0 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R h : a * 0 + a * 0 = a * 0 + 0 ⊢ a * 0 = 0 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.mul_zero
[261, 1]
[264, 25]
rw [← mul_add, add_zero, add_zero]
R : Type u_1 inst✝ : Ring R a : R ⊢ a * 0 + a * 0 = a * 0 + 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊢ a * 0 + a * 0 = a * 0 + 0 TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.neg_eq_of_add_eq_zero
[266, 1]
[267, 46]
rw [← neg_add_cancel_left a b, h, add_zero]
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ -a = b
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ -a = b TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.eq_neg_of_add_eq_zero
[281, 1]
[285, 19]
symm
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ a = -b
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ -b = a
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ a = -b TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.eq_neg_of_add_eq_zero
[281, 1]
[285, 19]
apply neg_eq_of_add_eq_zero
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ -b = a
case h R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ b + a = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ -b = a TACTIC:
https://github.com/fpvandoorn/LeanInRome.git
55e064179515cdc8f96fd8e82b2c106fb80e5c0e
LeanInRome/S2_Basics/tutorial.lean
MyRing.eq_neg_of_add_eq_zero
[281, 1]
[285, 19]
rw [add_comm, h]
case h R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ b + a = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊢ b + a = 0 TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.fwd_rev
[15, 9]
[16, 31]
rw [Equiv.fwd_eq_iff_rev_eq]
α : Sort u_1 β : Sort u_2 e : Equiv α β x : β ⊢ e.fwd (e.rev x) = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e : Equiv α β x : β ⊢ e.fwd (e.rev x) = x TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.rev_fwd
[18, 9]
[19, 32]
rw [←Equiv.fwd_eq_iff_rev_eq]
α : Sort u_1 β : Sort u_2 e : Equiv α β x : α ⊢ e.rev (e.fwd x) = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e : Equiv α β x : α ⊢ e.rev (e.fwd x) = x TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext
[25, 18]
[27, 65]
have h' : e₁.rev = e₂.rev := by funext; rw [← fwd_eq_iff_rev_eq, h, fwd_rev]
α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd ⊢ e₁ = e₂
α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd h' : e₁.rev = e₂.rev ⊢ e₁ = e₂
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd ⊢ e₁ = e₂ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext
[25, 18]
[27, 65]
match e₁, e₂ with | ⟨_,_,_⟩, ⟨_,_,_⟩ => cases h; cases h'; rfl
α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd h' : e₁.rev = e₂.rev ⊢ e₁ = e₂
no goals
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd h' : e₁.rev = e₂.rev ⊢ e₁ = e₂ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext
[25, 18]
[27, 65]
funext
α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd ⊢ e₁.rev = e₂.rev
case h α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd x✝ : β ⊢ e₁.rev x✝ = e₂.rev x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd ⊢ e₁.rev = e₂.rev TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext
[25, 18]
[27, 65]
rw [← fwd_eq_iff_rev_eq, h, fwd_rev]
case h α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd x✝ : β ⊢ e₁.rev x✝ = e₂.rev x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : e₁.fwd = e₂.fwd x✝ : β ⊢ e₁.rev x✝ = e₂.rev x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext
[25, 18]
[27, 65]
cases h
α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝¹ : α → β rev✝¹ : β → α fwd_eq_iff_rev_eq✝¹ : ∀ {x : α} {y : β}, fwd✝¹ x = y ↔ rev✝¹ y = x fwd✝ : α → β rev✝ : β → α fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x h : { fwd := fwd✝¹, rev := rev✝¹, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝¹ }.fwd = { ...
case refl α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝ : α → β rev✝¹ : β → α fwd_eq_iff_rev_eq✝¹ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝¹ y = x rev✝ : β → α fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x h' : { fwd := fwd✝, rev := rev✝¹, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝¹ }.rev = { fwd :...
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝¹ : α → β rev✝¹ : β → α fwd_eq_iff_rev_eq✝¹ : ∀ {x : α} {y : β}, fwd✝¹ x = y ↔ rev✝¹ y = x fwd✝ : α → β rev✝ : β → α fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x h : { fwd := fwd✝¹, rev := r...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext
[25, 18]
[27, 65]
cases h'
case refl α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝ : α → β rev✝¹ : β → α fwd_eq_iff_rev_eq✝¹ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝¹ y = x rev✝ : β → α fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x h' : { fwd := fwd✝, rev := rev✝¹, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝¹ }.rev = { fwd :...
case refl.refl α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝ : α → β rev✝ : β → α fwd_eq_iff_rev_eq✝¹ fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x ⊢ { fwd := fwd✝, rev := rev✝, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝¹ } = { fwd := fwd✝, rev := rev✝, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝ }
Please generate a tactic in lean4 to solve the state. STATE: case refl α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝ : α → β rev✝¹ : β → α fwd_eq_iff_rev_eq✝¹ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝¹ y = x rev✝ : β → α fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x h' : { fwd := fwd✝, rev := rev✝¹,...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext
[25, 18]
[27, 65]
rfl
case refl.refl α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝ : α → β rev✝ : β → α fwd_eq_iff_rev_eq✝¹ fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x ⊢ { fwd := fwd✝, rev := rev✝, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝¹ } = { fwd := fwd✝, rev := rev✝, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝ }
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refl.refl α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β fwd✝ : α → β rev✝ : β → α fwd_eq_iff_rev_eq✝¹ fwd_eq_iff_rev_eq✝ : ∀ {x : α} {y : β}, fwd✝ x = y ↔ rev✝ y = x ⊢ { fwd := fwd✝, rev := rev✝, fwd_eq_iff_rev_eq := fwd_eq_iff_rev_eq✝¹ } = { fwd := fw...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext'
[29, 11]
[30, 18]
ext
α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : ∀ (x : α), e₁.fwd x = e₂.fwd x ⊢ e₁ = e₂
case h.h α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : ∀ (x : α), e₁.fwd x = e₂.fwd x x✝ : α ⊢ e₁.fwd x✝ = e₂.fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : ∀ (x : α), e₁.fwd x = e₂.fwd x ⊢ e₁ = e₂ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.ext'
[29, 11]
[30, 18]
exact h ..
case h.h α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : ∀ (x : α), e₁.fwd x = e₂.fwd x x✝ : α ⊢ e₁.fwd x✝ = e₂.fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h α : Sort u_1 β : Sort u_2 e₁ e₂ : Equiv α β h : ∀ (x : α), e₁.fwd x = e₂.fwd x x✝ : α ⊢ e₁.fwd x✝ = e₂.fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_assoc
[66, 11]
[67, 81]
ext
γ : Sort u_1 δ : Sort u_2 β : Sort u_3 α : Sort u_4 g : Equiv γ δ f : Equiv β γ e : Equiv α β ⊢ (g.comp f).comp e = g.comp (f.comp e)
case h.h γ : Sort u_1 δ : Sort u_2 β : Sort u_3 α : Sort u_4 g : Equiv γ δ f : Equiv β γ e : Equiv α β x✝ : α ⊢ ((g.comp f).comp e).fwd x✝ = (g.comp (f.comp e)).fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: γ : Sort u_1 δ : Sort u_2 β : Sort u_3 α : Sort u_4 g : Equiv γ δ f : Equiv β γ e : Equiv α β ⊢ (g.comp f).comp e = g.comp (f.comp e) TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_assoc
[66, 11]
[67, 81]
rfl
case h.h γ : Sort u_1 δ : Sort u_2 β : Sort u_3 α : Sort u_4 g : Equiv γ δ f : Equiv β γ e : Equiv α β x✝ : α ⊢ ((g.comp f).comp e).fwd x✝ = (g.comp (f.comp e)).fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h γ : Sort u_1 δ : Sort u_2 β : Sort u_3 α : Sort u_4 g : Equiv γ δ f : Equiv β γ e : Equiv α β x✝ : α ⊢ ((g.comp f).comp e).fwd x✝ = (g.comp (f.comp e)).fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_id_right
[69, 19]
[69, 99]
ext
α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.comp Equiv.id = e
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.comp Equiv.id).fwd x✝ = e.fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.comp Equiv.id = e TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_id_right
[69, 19]
[69, 99]
rfl
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.comp Equiv.id).fwd x✝ = e.fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.comp Equiv.id).fwd x✝ = e.fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_id_left
[71, 19]
[71, 98]
ext
α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.comp Equiv.id = e
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.comp Equiv.id).fwd x✝ = e.fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.comp Equiv.id = e TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_id_left
[71, 19]
[71, 98]
rfl
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.comp Equiv.id).fwd x✝ = e.fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.comp Equiv.id).fwd x✝ = e.fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_inv_right
[73, 19]
[74, 58]
ext
α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.comp e.inv = Equiv.id
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : β ⊢ (e.comp e.inv).fwd x✝ = Equiv.id.fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.comp e.inv = Equiv.id TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_inv_right
[73, 19]
[74, 58]
simp
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : β ⊢ (e.comp e.inv).fwd x✝ = Equiv.id.fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : β ⊢ (e.comp e.inv).fwd x✝ = Equiv.id.fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_inv_left
[76, 19]
[77, 56]
ext
α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.inv.comp e = Equiv.id
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.inv.comp e).fwd x✝ = Equiv.id.fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.inv.comp e = Equiv.id TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.comp_inv_left
[76, 19]
[77, 56]
simp
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.inv.comp e).fwd x✝ = Equiv.id.fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ (e.inv.comp e).fwd x✝ = Equiv.id.fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.inv_id
[79, 19]
[79, 89]
ext
α : Sort u_1 ⊢ Equiv.id.inv = Equiv.id
case h.h α : Sort u_1 x✝ : α ⊢ Equiv.id.inv.fwd x✝ = Equiv.id.fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 ⊢ Equiv.id.inv = Equiv.id TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.inv_id
[79, 19]
[79, 89]
rfl
case h.h α : Sort u_1 x✝ : α ⊢ Equiv.id.inv.fwd x✝ = Equiv.id.fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h α : Sort u_1 x✝ : α ⊢ Equiv.id.inv.fwd x✝ = Equiv.id.fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.inv_inv
[81, 19]
[81, 95]
ext
α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.inv.inv = e
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ e.inv.inv.fwd x✝ = e.fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: α : Sort u_1 β : Sort u_2 e : Equiv α β ⊢ e.inv.inv = e TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.inv_inv
[81, 19]
[81, 95]
rfl
case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ e.inv.inv.fwd x✝ = e.fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h α : Sort u_1 β : Sort u_2 e : Equiv α β x✝ : α ⊢ e.inv.inv.fwd x✝ = e.fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.inv_comp
[83, 11]
[84, 87]
ext
β : Sort u_1 γ : Sort u_2 α : Sort u_3 f : Equiv β γ e : Equiv α β ⊢ (f.comp e).inv = e.inv.comp f.inv
case h.h β : Sort u_1 γ : Sort u_2 α : Sort u_3 f : Equiv β γ e : Equiv α β x✝ : γ ⊢ (f.comp e).inv.fwd x✝ = (e.inv.comp f.inv).fwd x✝
Please generate a tactic in lean4 to solve the state. STATE: β : Sort u_1 γ : Sort u_2 α : Sort u_3 f : Equiv β γ e : Equiv α β ⊢ (f.comp e).inv = e.inv.comp f.inv TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Equiv/Basic.lean
Equiv.inv_comp
[83, 11]
[84, 87]
rfl
case h.h β : Sort u_1 γ : Sort u_2 α : Sort u_3 f : Equiv β γ e : Equiv α β x✝ : γ ⊢ (f.comp e).inv.fwd x✝ = (e.inv.comp f.inv).fwd x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h β : Sort u_1 γ : Sort u_2 α : Sort u_3 f : Equiv β γ e : Equiv α β x✝ : γ ⊢ (f.comp e).inv.fwd x✝ = (e.inv.comp f.inv).fwd x✝ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
let _ : Inhabited α := ⟨(xs.push x)[0]⟩
α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α ⊢ recPush empty push (xs.push x) = push xs x (recPush empty push xs)
α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush empty push (xs.push x) = push xs x (recPush empty push xs)
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α ⊢ recPush empty push (xs.push x) = push xs x (recPush empty push xs) TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
rw [recPush]
α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush empty push (xs.push x) = push xs x (recPush empty push xs)
α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.push x).size ⋯ = push xs x (recPush empty push xs)
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush empty push (xs.push x) = push xs x (recPush empty push ...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
trans (recPush.aux empty push (xs.push x) (xs.size + 1) (size_push xs x))
α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.push x).size ⋯ = push xs x (recPush empty push xs)
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.push x).size ⋯ = recPush.aux empty push (xs.push x) (xs.size + 1) ⋯ case a α ...
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.push x).size ⋯ = push xs...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
congr
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.push x).size ⋯ = recPush.aux empty push (xs.push x) (xs.size + 1) ⋯
case a.e_n α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).size = xs.size + 1
Please generate a tactic in lean4 to solve the state. STATE: case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.push x).size ⋯ = ...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
exact size_push ..
case a.e_n α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).size = xs.size + 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.e_n α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).size = xs.size + 1 TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
rw [recPush.aux]
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.size + 1) ⋯ = push xs x (recPush empty push xs)
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (let_fun this := ⋯; let_fun x_1 := { default := (xs.push x)[0] }; ⋯ ▸ push (xs.push x).pop (xs.push x).back (...
Please generate a tactic in lean4 to solve the state. STATE: case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ recPush.aux empty push (xs.push x) (xs.size + 1) ⋯ = pus...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
simp
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (let_fun this := ⋯; let_fun x_1 := { default := (xs.push x)[0] }; ⋯ ▸ push (xs.push x).pop (xs.push x).back (...
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ ⋯ ▸ push (xs.push x).pop (xs.push x).back (recPush.aux empty push (xs.push x).pop xs.size ⋯) = push xs x (recPush...
Please generate a tactic in lean4 to solve the state. STATE: case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (let_fun this := ⋯; let_fun x_1 := { default := (xs....
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
elim_cast
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ ⋯ ▸ push (xs.push x).pop (xs.push x).back (recPush.aux empty push (xs.push x).pop xs.size ⋯) = push xs x (recPush...
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ HEq (push (xs.push x).pop (xs.push x).back (recPush.aux empty push (xs.push x).pop xs.size ⋯)) (push xs x (recPus...
Please generate a tactic in lean4 to solve the state. STATE: case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ ⋯ ▸ push (xs.push x).pop (xs.push x).back (recPush.aux e...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
congr
case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ HEq (push (xs.push x).pop (xs.push x).back (recPush.aux empty push (xs.push x).pop xs.size ⋯)) (push xs x (recPus...
case a.e_1 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).pop = xs case a.e_2 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α...
Please generate a tactic in lean4 to solve the state. STATE: case a α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ HEq (push (xs.push x).pop (xs.push x).back (recPush.aux ...
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
exact pop_push ..
case a.e_1 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).pop = xs
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.e_1 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).pop = xs TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
exact back_push ..
case a.e_2 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).back = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.e_2 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).back = x TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
exact pop_push ..
case a.e_3.e_5.h α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).pop = xs
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.e_3.e_5.h α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ (xs.push x).pop = xs TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Array.lean
Array.recPush_push
[34, 1]
[47, 31]
exact proof_irrel_heq ..
case a.e_3.e_7 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ HEq ⋯ ⋯
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.e_3.e_7 α : Type u_1 motive : Array α → Sort u_2 empty : motive #[] push : (xs : Array α) → (x : α) → motive xs → motive (xs.push x) xs : Array α x : α x✝ : Inhabited α := { default := (xs.push x)[0] } ⊢ HEq ⋯ ⋯ TACTIC:
https://github.com/fgdorais/extra4.git
eb1c6c30f5790bed1bb4b01534d271a2490d9730
Extra/Index/Join.lean
List.Index.unjoin_join
[16, 1]
[22, 72]
induction xss with | nil => cases i; contradiction | cons xs xss ih => match i with | ⟨head, _⟩ => simp only [join, unjoin, unappend_append] | ⟨tail _, _⟩ => simp only [join, unjoin, unappend_append]; rw [ih]
α✝ : Type u_1 xss : List (List α✝) i : (i : xss.Index) × i.val.Index ⊢ (join i).unjoin = i
no goals
Please generate a tactic in lean4 to solve the state. STATE: α✝ : Type u_1 xss : List (List α✝) i : (i : xss.Index) × i.val.Index ⊢ (join i).unjoin = i TACTIC: