Split (1)

train · 219k rows

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	input stringlengths 73 2.09M
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	{ assumption }	case mp.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ ¬B s case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t	case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mp.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ ¬B s case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	{ rw [BigStepEquiv] at hT rw [← hT] assumption }	case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	assumption	case mp.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ ¬B s	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mp.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ ¬B s TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	rw [BigStepEquiv] at hT	case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t	case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	rw [← hT]	case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t	case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₂, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	assumption	case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₁, s) ⟹ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mp.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₁ : (T₁, s) ⟹ t ⊢ (T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	intro hif	case mpr B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State ⊢ (Stmt.ifThenElse B S₂ T₂, s) ⟹ t → (Stmt.ifThenElse B S₁ T₁, s) ⟹ t	case mpr B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hif : (Stmt.ifThenElse B S₂ T₂, s) ⟹ t ⊢ (Stmt.ifThenElse B S₁ T₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State ⊢ (Stmt.ifThenElse B S₂ T₂, s) ⟹ t → (Stmt.ifThenElse B S₁ T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	cases hif with \| if_true _ _ _ _ _ hB hS₂ => apply BigStep.if_true { assumption } { rw [BigStepEquiv] at hS rw [hS] assumption } \| if_false _ _ _ _ _ hB hT₂ => apply BigStep.if_false { assumption } { rw [BigStepEquiv] at hT rw [hT] assumption }	case mpr B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hif : (Stmt.ifThenElse B S₂ T₂, s) ⟹ t ⊢ (Stmt.ifThenElse B S₁ T₁, s) ⟹ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hif : (Stmt.ifThenElse B S₂ T₂, s) ⟹ t ⊢ (Stmt.ifThenElse B S₁ T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	apply BigStep.if_true	case mpr.if_true B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (Stmt.ifThenElse B S₁ T₁, s) ⟹ t	case mpr.if_true.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ B s case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_true B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (Stmt.ifThenElse B S₁ T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	{ assumption }	case mpr.if_true.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ B s case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t	case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_true.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ B s case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	{ rw [BigStepEquiv] at hS rw [hS] assumption }	case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	assumption	case mpr.if_true.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ B s	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_true.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ B s TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	rw [BigStepEquiv] at hS	case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t	case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : ∀ (s t : State), (S₁, s) ⟹ t ↔ (S₂, s) ⟹ t hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	rw [hS]	case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : ∀ (s t : State), (S₁, s) ⟹ t ↔ (S₂, s) ⟹ t hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t	case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : ∀ (s t : State), (S₁, s) ⟹ t ↔ (S₂, s) ⟹ t hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₂, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : ∀ (s t : State), (S₁, s) ⟹ t ↔ (S₂, s) ⟹ t hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	assumption	case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : ∀ (s t : State), (S₁, s) ⟹ t ↔ (S₂, s) ⟹ t hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₂, s) ⟹ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_true.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : ∀ (s t : State), (S₁, s) ⟹ t ↔ (S₂, s) ⟹ t hT : T₁ ~ T₂ s t : State hB : B s hS₂ : (S₂, s) ⟹ t ⊢ (S₂, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	apply BigStep.if_false	case mpr.if_false B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (Stmt.ifThenElse B S₁ T₁, s) ⟹ t	case mpr.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ ¬B s case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_false B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (Stmt.ifThenElse B S₁ T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	{ assumption }	case mpr.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ ¬B s case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t	case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ ¬B s case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	{ rw [BigStepEquiv] at hT rw [hT] assumption }	case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	assumption	case mpr.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ ¬B s	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_false.hcond B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ ¬B s TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	rw [BigStepEquiv] at hT	case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t	case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : T₁ ~ T₂ s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	rw [hT]	case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t	case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₂, s) ⟹ t	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₁, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.BigStepEquiv.if_congr	[335, 1]	[367, 25]	assumption	case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₂, s) ⟹ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr.if_false.hbody B : State → Prop S₁ S₂ T₁ T₂ : Stmt hS : S₁ ~ S₂ hT : ∀ (s t : State), (T₁, s) ⟹ t ↔ (T₂, s) ⟹ t s t : State hB : ¬B s hT₂ : (T₂, s) ⟹ t ⊢ (T₂, s) ⟹ t TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	intro htp hstep	e e' : FnExp τ : FnType ⊢ e ∶ τ → e ⇒ e' → e' ∶ τ	e e' : FnExp τ : FnType htp : e ∶ τ hstep : e ⇒ e' ⊢ e' ∶ τ	Please generate a tactic in lean4 to solve the state. STATE: e e' : FnExp τ : FnType ⊢ e ∶ τ → e ⇒ e' → e' ∶ τ TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	induction hstep generalizing τ with \| @appArg f f' x h ih => cases htp with \| appZero hf hx => apply HasType.appZero . exact ih hf . exact hx \| @appSucc _ _ n hf hx => apply HasType.appSucc . exact ih hf . exact hx \| @appZero x => cases htp with \| appZero => assumption \| appSucc hf hx => cases hf \| @appSucc _ n => cases htp with \| appZero hf hx => cases hf \| appSucc hf hx => cases hf exact HasType.fn	e e' : FnExp τ : FnType htp : e ∶ τ hstep : e ⇒ e' ⊢ e' ∶ τ	no goals	Please generate a tactic in lean4 to solve the state. STATE: e e' : FnExp τ : FnType htp : e ∶ τ hstep : e ⇒ e' ⊢ e' ∶ τ TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	cases htp with \| appZero hf hx => apply HasType.appZero . exact ih hf . exact hx \| @appSucc _ _ n hf hx => apply HasType.appSucc . exact ih hf . exact hx	case appArg e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ τ : FnType htp : app f x ∶ τ ⊢ app f' x ∶ τ	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appArg e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ τ : FnType htp : app f x ∶ τ ⊢ app f' x ∶ τ TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	apply HasType.appZero	case appArg.appZero e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ app f' x ∶ argument	case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ f' ∶ function 0 case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appZero e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ app f' x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	. exact ih hf	case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ f' ∶ function 0 case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument	case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ f' ∶ function 0 case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	. exact hx	case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	exact ih hf	case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ f' ∶ function 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ f' ∶ function 0 TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	exact hx	case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appZero.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ hf : f ∶ function 0 hx : x ∶ argument ⊢ x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	apply HasType.appSucc	case appArg.appSucc e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ app f' x ∶ function n	case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ f' ∶ function (Nat.succ n) case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appSucc e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ app f' x ∶ function n TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	. exact ih hf	case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ f' ∶ function (Nat.succ n) case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument	case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ f' ∶ function (Nat.succ n) case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	. exact hx	case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	exact ih hf	case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ f' ∶ function (Nat.succ n)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ f' ∶ function (Nat.succ n) TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	exact hx	case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appArg.appSucc.a e e' f f' x : FnExp h : f ⇒ f' ih : ∀ {τ : FnType}, f ∶ τ → f' ∶ τ n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ⊢ x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	cases htp with \| appZero => assumption \| appSucc hf hx => cases hf	case appZero e e' x : FnExp τ : FnType htp : app (fn 0) x ∶ τ ⊢ x ∶ τ	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero e e' x : FnExp τ : FnType htp : app (fn 0) x ∶ τ ⊢ x ∶ τ TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	assumption	case appZero.appZero e e' x : FnExp a✝¹ : fn 0 ∶ function 0 a✝ : x ∶ argument ⊢ x ∶ argument	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.appZero e e' x : FnExp a✝¹ : fn 0 ∶ function 0 a✝ : x ∶ argument ⊢ x ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	cases hf	case appZero.appSucc e e' x : FnExp n✝ : ℕ hf : fn 0 ∶ function (Nat.succ n✝) hx : x ∶ argument ⊢ x ∶ function n✝	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.appSucc e e' x : FnExp n✝ : ℕ hf : fn 0 ∶ function (Nat.succ n✝) hx : x ∶ argument ⊢ x ∶ function n✝ TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	cases htp with \| appZero hf hx => cases hf \| appSucc hf hx => cases hf exact HasType.fn	case appSucc e e' x✝ : FnExp n : ℕ τ : FnType htp : app (fn (Nat.succ n)) x✝ ∶ τ ⊢ fn n ∶ τ	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc e e' x✝ : FnExp n : ℕ τ : FnType htp : app (fn (Nat.succ n)) x✝ ∶ τ ⊢ fn n ∶ τ TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	cases hf	case appSucc.appZero e e' x✝ : FnExp n : ℕ hf : fn (Nat.succ n) ∶ function 0 hx : x✝ ∶ argument ⊢ fn n ∶ argument	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.appZero e e' x✝ : FnExp n : ℕ hf : fn (Nat.succ n) ∶ function 0 hx : x✝ ∶ argument ⊢ fn n ∶ argument TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	cases hf	case appSucc.appSucc e e' x✝ : FnExp n n✝ : ℕ hf : fn (Nat.succ n) ∶ function (Nat.succ n✝) hx : x✝ ∶ argument ⊢ fn n ∶ function n✝	case appSucc.appSucc.fn e e' x✝ : FnExp n : ℕ hx : x✝ ∶ argument ⊢ fn n ∶ function n	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.appSucc e e' x✝ : FnExp n n✝ : ℕ hf : fn (Nat.succ n) ∶ function (Nat.succ n✝) hx : x✝ ∶ argument ⊢ fn n ∶ function n✝ TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.preservation	[523, 1]	[547, 23]	exact HasType.fn	case appSucc.appSucc.fn e e' x✝ : FnExp n : ℕ hx : x✝ ∶ argument ⊢ fn n ∶ function n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.appSucc.fn e e' x✝ : FnExp n : ℕ hx : x✝ ∶ argument ⊢ fn n ∶ function n TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	intro htp	e : FnExp τ : FnType ⊢ e ∶ τ → Value e ∨ ∃ e', e ⇒ e'	e : FnExp τ : FnType htp : e ∶ τ ⊢ Value e ∨ ∃ e', e ⇒ e'	Please generate a tactic in lean4 to solve the state. STATE: e : FnExp τ : FnType ⊢ e ∶ τ → Value e ∨ ∃ e', e ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	induction htp with \| fn => exact Or.inl Value.fn \| arg => exact Or.inl Value.arg \| @appZero f x hf hx ihf ihx => apply Or.inr cases ihf with \| inl hval => cases hval with \| @fn n => cases hf apply Exists.intro x apply Step.appZero \| arg => cases hf \| inr hstep => cases hstep with \| intro f' hf' => apply Exists.intro (app f' x) exact Step.appArg hf' \| @appSucc f x n hf hx ihf ihx => apply Or.inr cases ihf with \| inl hval => cases hf with \| fn => apply Exists.intro (fn n) exact Step.appSucc \| appSucc => cases hval \| inr hstep => cases hstep with \| intro f' hf' => apply Exists.intro (app f' x) exact Step.appArg hf'	e : FnExp τ : FnType htp : e ∶ τ ⊢ Value e ∨ ∃ e', e ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: e : FnExp τ : FnType htp : e ∶ τ ⊢ Value e ∨ ∃ e', e ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	exact Or.inl Value.fn	case fn e : FnExp τ : FnType n✝ : ℕ ⊢ Value (fn n✝) ∨ ∃ e', fn n✝ ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case fn e : FnExp τ : FnType n✝ : ℕ ⊢ Value (fn n✝) ∨ ∃ e', fn n✝ ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	exact Or.inl Value.arg	case arg e : FnExp τ : FnType ⊢ Value arg ∨ ∃ e', arg ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case arg e : FnExp τ : FnType ⊢ Value arg ∨ ∃ e', arg ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	apply Or.inr	case appZero e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ Value (app f x) ∨ ∃ e', app f x ⇒ e'	case appZero.h e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app f x ⇒ e'	Please generate a tactic in lean4 to solve the state. STATE: case appZero e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ Value (app f x) ∨ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases ihf with \| inl hval => cases hval with \| @fn n => cases hf apply Exists.intro x apply Step.appZero \| arg => cases hf \| inr hstep => cases hstep with \| intro f' hf' => apply Exists.intro (app f' x) exact Step.appArg hf'	case appZero.h e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app f x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases hval with \| @fn n => cases hf apply Exists.intro x apply Step.appZero \| arg => cases hf	case appZero.h.inl e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value f ⊢ ∃ e', app f x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inl e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value f ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases hf	case appZero.h.inl.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' n : ℕ hf : fn n ∶ function 0 ⊢ ∃ e', app (fn n) x ⇒ e'	case appZero.h.inl.fn.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app (fn 0) x ⇒ e'	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inl.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' n : ℕ hf : fn n ∶ function 0 ⊢ ∃ e', app (fn n) x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	apply Exists.intro x	case appZero.h.inl.fn.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app (fn 0) x ⇒ e'	case appZero.h.inl.fn.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ app (fn 0) x ⇒ x	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inl.fn.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app (fn 0) x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	apply Step.appZero	case appZero.h.inl.fn.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ app (fn 0) x ⇒ x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inl.fn.fn e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ app (fn 0) x ⇒ x TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases hf	case appZero.h.inl.arg e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hf : arg ∶ function 0 ⊢ ∃ e', app arg x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inl.arg e : FnExp τ : FnType x : FnExp hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hf : arg ∶ function 0 ⊢ ∃ e', app arg x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases hstep with \| intro f' hf' => apply Exists.intro (app f' x) exact Step.appArg hf'	case appZero.h.inr e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hstep : ∃ e', f ⇒ e' ⊢ ∃ e', app f x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inr e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hstep : ∃ e', f ⇒ e' ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	apply Exists.intro (app f' x)	case appZero.h.inr.intro e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ ∃ e', app f x ⇒ e'	case appZero.h.inr.intro e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ app f x ⇒ app f' x	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inr.intro e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	exact Step.appArg hf'	case appZero.h.inr.intro e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ app f x ⇒ app f' x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appZero.h.inr.intro e : FnExp τ : FnType f x : FnExp hf : f ∶ function 0 hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ app f x ⇒ app f' x TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	apply Or.inr	case appSucc e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ Value (app f x) ∨ ∃ e', app f x ⇒ e'	case appSucc.h e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app f x ⇒ e'	Please generate a tactic in lean4 to solve the state. STATE: case appSucc e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ Value (app f x) ∨ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases ihf with \| inl hval => cases hf with \| fn => apply Exists.intro (fn n) exact Step.appSucc \| appSucc => cases hval \| inr hstep => cases hstep with \| intro f' hf' => apply Exists.intro (app f' x) exact Step.appArg hf'	case appSucc.h e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app f x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihf : Value f ∨ ∃ e', f ⇒ e' ihx : Value x ∨ ∃ e', x ⇒ e' ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases hf with \| fn => apply Exists.intro (fn n) exact Step.appSucc \| appSucc => cases hval	case appSucc.h.inl e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value f ⊢ ∃ e', app f x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h.inl e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value f ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	apply Exists.intro (fn n)	case appSucc.h.inl.fn e : FnExp τ : FnType x : FnExp n : ℕ hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value (fn (Nat.succ n)) ⊢ ∃ e', app (fn (Nat.succ n)) x ⇒ e'	case appSucc.h.inl.fn e : FnExp τ : FnType x : FnExp n : ℕ hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value (fn (Nat.succ n)) ⊢ app (fn (Nat.succ n)) x ⇒ fn n	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h.inl.fn e : FnExp τ : FnType x : FnExp n : ℕ hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value (fn (Nat.succ n)) ⊢ ∃ e', app (fn (Nat.succ n)) x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	exact Step.appSucc	case appSucc.h.inl.fn e : FnExp τ : FnType x : FnExp n : ℕ hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value (fn (Nat.succ n)) ⊢ app (fn (Nat.succ n)) x ⇒ fn n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h.inl.fn e : FnExp τ : FnType x : FnExp n : ℕ hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hval : Value (fn (Nat.succ n)) ⊢ app (fn (Nat.succ n)) x ⇒ fn n TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases hval	case appSucc.h.inl.appSucc e : FnExp τ : FnType x : FnExp n : ℕ hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f✝ x✝ : FnExp a✝¹ : x✝ ∶ argument hval : Value (app f✝ x✝) a✝ : f✝ ∶ function (Nat.succ (Nat.succ n)) ⊢ ∃ e', app (app f✝ x✝) x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h.inl.appSucc e : FnExp τ : FnType x : FnExp n : ℕ hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f✝ x✝ : FnExp a✝¹ : x✝ ∶ argument hval : Value (app f✝ x✝) a✝ : f✝ ∶ function (Nat.succ (Nat.succ n)) ⊢ ∃ e', app (app f✝ x✝) x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	cases hstep with \| intro f' hf' => apply Exists.intro (app f' x) exact Step.appArg hf'	case appSucc.h.inr e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hstep : ∃ e', f ⇒ e' ⊢ ∃ e', app f x ⇒ e'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h.inr e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' hstep : ∃ e', f ⇒ e' ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	apply Exists.intro (app f' x)	case appSucc.h.inr.intro e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ ∃ e', app f x ⇒ e'	case appSucc.h.inr.intro e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ app f x ⇒ app f' x	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h.inr.intro e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ ∃ e', app f x ⇒ e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.progress	[562, 1]	[595, 28]	exact Step.appArg hf'	case appSucc.h.inr.intro e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ app f x ⇒ app f' x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case appSucc.h.inr.intro e : FnExp τ : FnType f x : FnExp n : ℕ hf : f ∶ function (Nat.succ n) hx : x ∶ argument ihx : Value x ∨ ∃ e', x ⇒ e' f' : FnExp hf' : f ⇒ f' ⊢ app f x ⇒ app f' x TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.negation_of_coherence_property	[606, 1]	[621, 14]	intro hneg	⊢ ¬∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e'	hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ¬∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.negation_of_coherence_property	[606, 1]	[621, 14]	have htp : app (app (fn 1) arg) arg ∶ argument := HasType.appZero (HasType.appSucc HasType.fn HasType.arg) HasType.arg	hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' ⊢ False	hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.negation_of_coherence_property	[606, 1]	[621, 14]	have hexp_step := hneg htp	hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument ⊢ False	hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument hexp_step : Value (app (app (fn 1) arg) arg) ∨ ∃ e', BadStep (app (app (fn 1) arg) arg) e' ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.negation_of_coherence_property	[606, 1]	[621, 14]	cases hexp_step with \| inl hval => cases hval \| inr hstep => cases hstep with \| intro e' he' => cases he'	hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument hexp_step : Value (app (app (fn 1) arg) arg) ∨ ∃ e', BadStep (app (app (fn 1) arg) arg) e' ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument hexp_step : Value (app (app (fn 1) arg) arg) ∨ ∃ e', BadStep (app (app (fn 1) arg) arg) e' ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.negation_of_coherence_property	[606, 1]	[621, 14]	cases hval	case inl hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument hval : Value (app (app (fn 1) arg) arg) ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inl hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument hval : Value (app (app (fn 1) arg) arg) ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.negation_of_coherence_property	[606, 1]	[621, 14]	cases hstep with \| intro e' he' => cases he'	case inr hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument hstep : ∃ e', BadStep (app (app (fn 1) arg) arg) e' ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inr hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument hstep : ∃ e', BadStep (app (app (fn 1) arg) arg) e' ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab6Solution.lean	LoVe.FnExp.negation_of_coherence_property	[606, 1]	[621, 14]	cases he'	case inr.intro hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument e' : FnExp he' : BadStep (app (app (fn 1) arg) arg) e' ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inr.intro hneg : ∀ {e : FnExp} {τ : FnType}, e ∶ τ → Value e ∨ ∃ e', BadStep e e' htp : app (app (fn 1) arg) arg ∶ argument e' : FnExp he' : BadStep (app (app (fn 1) arg) arg) e' ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab7.lean	LoVe.List.elems_mirror_counterexample	[117, 1]	[121, 22]	apply Exists.intro badTree	⊢ ∃ t, elems t ≠ elems (mirror t)	⊢ elems badTree ≠ elems (mirror badTree)	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∃ t, elems t ≠ elems (mirror t) TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Labs/Lab7.lean	LoVe.List.elems_mirror_counterexample	[117, 1]	[121, 22]	simp [List.elems]	⊢ elems badTree ≠ elems (mirror badTree)	⊢ ¬elems badTree = elems (mirror badTree)	Please generate a tactic in lean4 to solve the state. STATE: ⊢ elems badTree ≠ elems (mirror badTree) TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	intro M	⊢ ∀ (N : ℕ), ∃ p, p ≥ N ∧ Nat.Prime p	M : ℕ ⊢ ∃ p, p ≥ M ∧ Nat.Prime p	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (N : ℕ), ∃ p, p ≥ N ∧ Nat.Prime p TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	let F := M ! + 1	M : ℕ ⊢ ∃ p, p ≥ M ∧ Nat.Prime p	M : ℕ F : ℕ := M ! + 1 ⊢ ∃ p, p ≥ M ∧ Nat.Prime p	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ ⊢ ∃ p, p ≥ M ∧ Nat.Prime p TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	let q := minFac F	M : ℕ F : ℕ := M ! + 1 ⊢ ∃ p, p ≥ M ∧ Nat.Prime p	M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ ∃ p, p ≥ M ∧ Nat.Prime p	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ F : ℕ := M ! + 1 ⊢ ∃ p, p ≥ M ∧ Nat.Prime p TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	use q	M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ ∃ p, p ≥ M ∧ Nat.Prime p	case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ q ≥ M ∧ Nat.Prime q	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ ∃ p, p ≥ M ∧ Nat.Prime p TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	have qPrime : Nat.Prime q	case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ q ≥ M ∧ Nat.Prime q	case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ Nat.Prime q case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M ∧ Nat.Prime q	Please generate a tactic in lean4 to solve the state. STATE: case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ q ≥ M ∧ Nat.Prime q TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	{ refine' minFac_prime _ have hn : M ! > 0 := factorial_pos M linarith }	case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ Nat.Prime q case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M ∧ Nat.Prime q	case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M ∧ Nat.Prime q	Please generate a tactic in lean4 to solve the state. STATE: case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ Nat.Prime q case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M ∧ Nat.Prime q TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	apply And.intro	case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M ∧ Nat.Prime q	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q	Please generate a tactic in lean4 to solve the state. STATE: case h M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M ∧ Nat.Prime q TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	{ by_contra hqM have h1 : q ∣ M ! + 1 := minFac_dvd F have hqM2 : q ≤ M := by linarith have hqM3 : q ∣ M ! := Iff.mpr (Prime.dvd_factorial qPrime) hqM2 have hq1 : q ∣ 1 := Iff.mp (Nat.dvd_add_right hqM3) h1 apply Nat.Prime.not_dvd_one qPrime hq1 }	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q	case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q	Please generate a tactic in lean4 to solve the state. STATE: case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	{ assumption }	case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	refine' minFac_prime _	case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ Nat.Prime q	case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ F ≠ 1	Please generate a tactic in lean4 to solve the state. STATE: case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ Nat.Prime q TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	have hn : M ! > 0 := factorial_pos M	case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ F ≠ 1	case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F hn : M ! > 0 ⊢ F ≠ 1	Please generate a tactic in lean4 to solve the state. STATE: case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F ⊢ F ≠ 1 TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	linarith	case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F hn : M ! > 0 ⊢ F ≠ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: case qPrime M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F hn : M ! > 0 ⊢ F ≠ 1 TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	by_contra hqM	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ q ≥ M TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	have h1 : q ∣ M ! + 1 := minFac_dvd F	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M ⊢ False	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	have hqM2 : q ≤ M := by linarith	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 ⊢ False	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	have hqM3 : q ∣ M ! := Iff.mpr (Prime.dvd_factorial qPrime) hqM2	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M ⊢ False	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M hqM3 : q ∣ M ! ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	have hq1 : q ∣ 1 := Iff.mp (Nat.dvd_add_right hqM3) h1	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M hqM3 : q ∣ M ! ⊢ False	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M hqM3 : q ∣ M ! hq1 : q ∣ 1 ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M hqM3 : q ∣ M ! ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	apply Nat.Prime.not_dvd_one qPrime hq1	case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M hqM3 : q ∣ M ! hq1 : q ∣ 1 ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.left M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 hqM2 : q ≤ M hqM3 : q ∣ M ! hq1 : q ∣ 1 ⊢ False TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	linarith	M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 ⊢ q ≤ M	no goals	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q hqM : ¬q ≥ M h1 : q ∣ M ! + 1 ⊢ q ≤ M TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	infinitude_of_primes	[117, 1]	[139, 7]	assumption	case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.right M : ℕ F : ℕ := M ! + 1 q : ℕ := minFac F qPrime : Nat.Prime q ⊢ Nat.Prime q TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	biggerPrimeIsPrime	[149, 1]	[154, 7]	intro M	⊢ ∀ (N : ℕ), Nat.Prime (biggerPrime N)	M : ℕ ⊢ Nat.Prime (biggerPrime M)	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (N : ℕ), Nat.Prime (biggerPrime N) TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	biggerPrimeIsPrime	[149, 1]	[154, 7]	refine' minFac_prime _	M : ℕ ⊢ Nat.Prime (biggerPrime M)	M : ℕ ⊢ M ! + 1 ≠ 1	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ ⊢ Nat.Prime (biggerPrime M) TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	biggerPrimeIsPrime	[149, 1]	[154, 7]	have hn : M ! > 0 := factorial_pos M	M : ℕ ⊢ M ! + 1 ≠ 1	M : ℕ hn : M ! > 0 ⊢ M ! + 1 ≠ 1	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ ⊢ M ! + 1 ≠ 1 TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	biggerPrimeIsPrime	[149, 1]	[154, 7]	linarith	M : ℕ hn : M ! > 0 ⊢ M ! + 1 ≠ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ hn : M ! > 0 ⊢ M ! + 1 ≠ 1 TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	biggerPrimeIsBigger	[156, 1]	[165, 7]	intro M	⊢ ∀ (N : ℕ), biggerPrime N ≥ N	M : ℕ ⊢ biggerPrime M ≥ M	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (N : ℕ), biggerPrime N ≥ N TACTIC:
https://github.com/BrownCS1951x/fpv2023.git	9aaf6b5c454aa9a70fc4e6807adf3123b001ea66	LoVe/Lectures/LoVe00_Preface_Demo.lean	biggerPrimeIsBigger	[156, 1]	[165, 7]	by_contra hqM	M : ℕ ⊢ biggerPrime M ≥ M	M : ℕ hqM : ¬biggerPrime M ≥ M ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: M : ℕ ⊢ biggerPrime M ≥ M TACTIC:

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