pkuAI4M/lean_github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	let F := def_ X xs	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))	Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	simp only [Formula.primeSet] at h1	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑{def_ X xs} ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))	Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	simp at h1	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑{def_ X xs} ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))	Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑{def_ X xs} ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	simp only [evalPrimeFfToNot]	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_)	Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	simp only [Formula.evalPrime]	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_)	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_)	Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	apply IsDeduct.assume_	Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_)	case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U	Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	simp	case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U	case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_	Please generate a tactic in lean4 to solve the state. STATE: case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	apply Exists.intro F	case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_	case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_	Please generate a tactic in lean4 to solve the state. STATE: case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	tauto	case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.L_15_7	[773, 1]	[916, 10]	sorry	Δ_U : Set Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝) h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝))	no goals	Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝) h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	apply IsDeduct.mp_ (U.not_.imp_ P)	P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ P	case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P) case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.not_.imp_ P)	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	apply IsDeduct.mp_ (U.imp_ P)	case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P)	case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P)) case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.imp_ P)	Please generate a tactic in lean4 to solve the state. STATE: case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	apply proof_imp_deduct	case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))	case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))	Please generate a tactic in lean4 to solve the state. STATE: case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	apply T_14_9	case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	apply deduction_theorem	case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.imp_ P)	case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U}) P	Please generate a tactic in lean4 to solve the state. STATE: case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.imp_ P) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	exact h1	case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U}) P	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	apply deduction_theorem	case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.not_.imp_ P)	case a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U.not_}) P	Please generate a tactic in lean4 to solve the state. STATE: case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.not_.imp_ P) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.T_14_9_Deduct	[919, 1]	[933, 13]	exact h2	case a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U.not_}) P	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U.not_}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	induction F	F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) F = Function.updateITE (evalPrimeFfToNot V) F' F F	case pred_const_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_const_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (pred_const_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ a✝¹ a✝) (pred_const_ a✝¹ a✝) case pred_var_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_var_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (pred_var_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ a✝¹ a✝) (pred_var_ a✝¹ a✝) case eq_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : VarName h1 : (eq_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (eq_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (eq_ a✝¹ a✝) (eq_ a✝¹ a✝) case true_ F' : Formula V : VarBoolAssignment h1 : true_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_ true_ case false_ F' : Formula V : VarBoolAssignment h1 : false_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) false_ = Function.updateITE (evalPrimeFfToNot V) F' false_ false_ case not_ F' : Formula V : VarBoolAssignment a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : a✝.not_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) a✝.not_ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝.not_ case imp_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.imp_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.imp_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.imp_ a✝) (a✝¹.imp_ a✝) case and_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.and_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.and_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.and_ a✝) (a✝¹.and_ a✝) case or_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.or_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.or_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.or_ a✝) (a✝¹.or_ a✝) case iff_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝) case forall_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (forall_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (forall_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (forall_ a✝¹ a✝) (forall_ a✝¹ a✝) case exists_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (exists_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (exists_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (exists_ a✝¹ a✝) (exists_ a✝¹ a✝) case def_ F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝)	Please generate a tactic in lean4 to solve the state. STATE: F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) F = Function.updateITE (evalPrimeFfToNot V) F' F F TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	case pred_const_ \| pred_var_ \| eq_ \| forall_ \| exists_ \| def_ => unfold Function.updateITE simp only [evalPrimeFfToNot] simp only [Formula.evalPrime] split_ifs <;> tauto	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝)	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	case true_ \| false_ \| not_ \| imp_ \| and_ \| or_ \| iff_ => simp only [Formula.IsPrime] at h1	F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝)	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	unfold Function.updateITE	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝)	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝)	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	simp only [evalPrimeFfToNot]	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝)	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	simp only [Formula.evalPrime]	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	split_ifs <;> tauto	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_true	[936, 1]	[950, 38]	simp only [Formula.IsPrime] at h1	F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝)	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	induction F	F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) F = Function.updateITE (evalPrimeFfToNot V) F' F.not_ F	case pred_const_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_const_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (pred_const_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ a✝¹ a✝).not_ (pred_const_ a✝¹ a✝) case pred_var_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_var_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (pred_var_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ a✝¹ a✝).not_ (pred_var_ a✝¹ a✝) case eq_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : VarName h1 : (eq_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (eq_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (eq_ a✝¹ a✝).not_ (eq_ a✝¹ a✝) case true_ F' : Formula V : VarBoolAssignment h1 : true_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_.not_ true_ case false_ F' : Formula V : VarBoolAssignment h1 : false_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) false_ = Function.updateITE (evalPrimeFfToNot V) F' false_.not_ false_ case not_ F' : Formula V : VarBoolAssignment a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : a✝.not_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) a✝.not_ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_.not_ a✝.not_ case imp_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.imp_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.imp_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.imp_ a✝).not_ (a✝¹.imp_ a✝) case and_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.and_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.and_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.and_ a✝).not_ (a✝¹.and_ a✝) case or_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.or_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.or_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.or_ a✝).not_ (a✝¹.or_ a✝) case iff_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝) case forall_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (forall_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (forall_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (forall_ a✝¹ a✝).not_ (forall_ a✝¹ a✝) case exists_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (exists_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (exists_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (exists_ a✝¹ a✝).not_ (exists_ a✝¹ a✝) case def_ F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)	Please generate a tactic in lean4 to solve the state. STATE: F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) F = Function.updateITE (evalPrimeFfToNot V) F' F.not_ F TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	case pred_const_ \| pred_var_ \| eq_ \| forall_ \| exists_ \| def_ => unfold Function.updateITE simp only [evalPrimeFfToNot] simp only [Formula.evalPrime] split_ifs <;> tauto	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	case true_ \| false_ \| not_ \| imp_ \| and_ \| or_ \| iff_ => simp only [Formula.IsPrime] at h1	F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝)	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	unfold Function.updateITE	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝)	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	simp only [evalPrimeFfToNot]	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝)	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	simp only [Formula.evalPrime]	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	split_ifs <;> tauto	F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.evalPrimeFfToNot_of_function_updateIte_false	[953, 1]	[967, 38]	simp only [Formula.IsPrime] at h1	F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝)	no goals	Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	apply Set.image_congr	U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) '' Δ = evalPrimeFfToNot V '' Δ	case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ ∀ a ∈ Δ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a	Please generate a tactic in lean4 to solve the state. STATE: U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) '' Δ = evalPrimeFfToNot V '' Δ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	intro U' a1	case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ ∀ a ∈ Δ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a	case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ ∀ a ∈ Δ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	specialize h1_Δ U' a1	case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'	case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	cases b	case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U' case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	simp only [evalPrimeFfToNot_of_function_updateIte_false U' U V h1_Δ]	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U'	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	simp only [Function.updateITE]	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U'	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	simp	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U'.not_ = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	intro a2	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U'.not_ = evalPrimeFfToNot V U'	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U'.not_ = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U'.not_ = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	subst a2	case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U'.not_ = evalPrimeFfToNot V U'	case h.false Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U'.not_ = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U'.not_ = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	contradiction	case h.false Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U'.not_ = evalPrimeFfToNot V U'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.false Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U'.not_ = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	simp only [evalPrimeFfToNot_of_function_updateIte_true U' U V h1_Δ]	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U'	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	simp only [Function.updateITE]	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U'	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	simp	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	intro a2	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U' = evalPrimeFfToNot V U'	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	subst a2	case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U' = evalPrimeFfToNot V U'	case h.true Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U' = evalPrimeFfToNot V U'	Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte	[970, 1]	[996, 18]	contradiction	case h.true Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U' = evalPrimeFfToNot V U'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.true Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U' = evalPrimeFfToNot V U' TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	intro V	P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ) P	P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ) P	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	apply T_14_9_Deduct P U (Δ.image (evalPrimeFfToNot V))	P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ) P	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	specialize h3 (Function.updateITE V U true)	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [image_of_evalPrimeFfToNot_of_function_updateIte U Δ V true h1_Δ h1_U h2] at h3	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [evalPrimeFfToNot_of_function_updateIte_true U U V h1_U] at h3	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [Function.updateITE] at h3	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [eq_self_iff_true, if_true] at h3	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	exact h3	case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	specialize h3 (Function.updateITE V U Bool.false)	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [image_of_evalPrimeFfToNot_of_function_updateIte U Δ V false h1_Δ h1_U h2] at h3	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [evalPrimeFfToNot_of_function_updateIte_false U U V h1_U] at h3	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [Function.updateITE] at h3	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	simp only [eq_self_iff_true, if_true] at h3	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAuxAux	[999, 1]	[1021, 13]	exact h3	case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	induction Δ_U using Finset.induction_on	P : Formula Δ_U : Finset Formula h1 : Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P ⊢ IsDeduct ∅ P	case empty P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P case insert P a✝² : Formula s✝ : Finset Formula a✝¹ : a✝² ∉ s✝ a✝ : s✝ ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) s✝)) P) → IsDeduct ∅ P h1 : insert a✝² s✝ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert a✝² s✝))) P ⊢ IsDeduct ∅ P	Please generate a tactic in lean4 to solve the state. STATE: P : Formula Δ_U : Finset Formula h1 : Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P ⊢ IsDeduct ∅ P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	case empty => simp at h2 exact h2	P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P	no goals	Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	simp at h2	P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P	P : Formula h1 : ∅ ⊆ P.primeSet h2 : IsDeduct ∅ P ⊢ IsDeduct ∅ P	Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	exact h2	P : Formula h1 : ∅ ⊆ P.primeSet h2 : IsDeduct ∅ P ⊢ IsDeduct ∅ P	no goals	Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : ∅ ⊆ P.primeSet h2 : IsDeduct ∅ P ⊢ IsDeduct ∅ P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	apply Δ_U_2	P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ IsDeduct ∅ P	case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ Δ_U ⊆ P.primeSet case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ IsDeduct ∅ P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	simp only [Finset.insert_subset_iff] at h1	case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ Δ_U ⊆ P.primeSet	case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ Δ_U ⊆ P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	cases h1	case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet	case h1.intro P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P left✝ : U ∈ P.primeSet right✝ : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet	Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	case intro h1_left h1_right => exact h1_right	P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	exact h1_right	P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	simp only [Finset.insert_subset_iff] at h1	case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	simp at h2	case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	cases h1	case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	case h2.intro P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P left✝ : U ∈ P.primeSet right✝ : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	simp	P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P	P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U) P	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	apply propCompleteAuxAux P U Δ_U	P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U) P	case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U.IsPrime case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∉ ↑Δ_U case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {evalPrimeFfToNot V U}) P	Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	intro U' a1	case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime	case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U'.IsPrime	Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	apply mem_primeSet_isPrime P U'	case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U'.IsPrime	case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ P.primeSet	Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U'.IsPrime TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	apply h1_right	case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ P.primeSet	case h1_Δ.a P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ Δ_U	Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	exact a1	case h1_Δ.a P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ Δ_U	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ.a P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ Δ_U TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	apply mem_primeSet_isPrime P U	case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U.IsPrime	case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∈ P.primeSet	Please generate a tactic in lean4 to solve the state. STATE: case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U.IsPrime TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	exact h1_left	case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∈ P.primeSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∈ P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	exact Δ_U_1	case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∉ ↑Δ_U	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∉ ↑Δ_U TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	simp	case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {evalPrimeFfToNot V U}) P	case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P	Please generate a tactic in lean4 to solve the state. STATE: case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {evalPrimeFfToNot V U}) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.propCompleteAux	[1024, 1]	[1057, 19]	exact h2	case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	simp only [IsProof]	P : Formula h1 : P.IsTautoPrime ⊢ IsProof P	P : Formula h1 : P.IsTautoPrime ⊢ IsDeduct ∅ P	Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : P.IsTautoPrime ⊢ IsProof P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	apply propCompleteAux P P.primeSet	P : Formula h1 : P.IsTautoPrime ⊢ IsDeduct ∅ P	case h1 P : Formula h1 : P.IsTautoPrime ⊢ P.primeSet ⊆ P.primeSet case h2 P : Formula h1 : P.IsTautoPrime ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P	Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : P.IsTautoPrime ⊢ IsDeduct ∅ P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	rfl	case h1 P : Formula h1 : P.IsTautoPrime ⊢ P.primeSet ⊆ P.primeSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h1 P : Formula h1 : P.IsTautoPrime ⊢ P.primeSet ⊆ P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	intro V	case h2 P : Formula h1 : P.IsTautoPrime ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P	case h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P : Formula h1 : P.IsTautoPrime ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	apply L_15_7 P P P.primeSet V (P.primeSet.image (evalPrimeFfToNot V))	case h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P	case h2.h1 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑P.primeSet ⊆ ↑P.primeSet case h2.h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.primeSet case h2.h3 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P	Please generate a tactic in lean4 to solve the state. STATE: case h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	rfl	case h2.h1 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑P.primeSet ⊆ ↑P.primeSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h2.h1 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑P.primeSet ⊆ ↑P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	simp only [Finset.coe_image]	case h2.h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.primeSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h2.h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.primeSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	simp only [Formula.IsTautoPrime] at h1	case h2.h3 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P	case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P	Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	simp only [evalPrimeFfToNot]	case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P	case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = if evalPrime V P then P else P.not_	Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	specialize h1 V	case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = if evalPrime V P then P else P.not_	case h2.h3 P : Formula V : VarBoolAssignment h1 : evalPrime V P ⊢ P = if evalPrime V P then P else P.not_	Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = if evalPrime V P then P else P.not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Prop.lean	FOL.NV.prop_complete	[1063, 1]	[1078, 28]	simp only [if_pos h1]	case h2.h3 P : Formula V : VarBoolAssignment h1 : evalPrime V P ⊢ P = if evalPrime V P then P else P.not_	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula V : VarBoolAssignment h1 : evalPrime V P ⊢ P = if evalPrime V P then P else P.not_ TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Finset.lean	Finset.union_subset_left_right	[8, 1]	[23, 44]	apply Finset.union_subset_iff.mpr	α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ∪ B ⊆ C ∪ D	α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D	Please generate a tactic in lean4 to solve the state. STATE: α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ∪ B ⊆ C ∪ D TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Finset.lean	Finset.union_subset_left_right	[8, 1]	[23, 44]	constructor	α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D	case left α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D case right α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ B ⊆ C ∪ D	Please generate a tactic in lean4 to solve the state. STATE: α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

let F := def_ X xs

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))

Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

simp only [Formula.primeSet] at h1

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑{def_ X xs} ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))

Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑(def_ X xs).primeSet ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

simp at h1

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑{def_ X xs} ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))

Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName h1 : ↑{def_ X xs} ⊆ Δ_U F : Formula := def_ X xs ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

simp only [evalPrimeFfToNot]

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_)

Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

simp only [Formula.evalPrime]

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_)

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_)

Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

apply IsDeduct.assume_

Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_)

case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U

Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

simp

case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U

case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_

Please generate a tactic in lean4 to solve the state. STATE: case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

apply Exists.intro F

case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_

case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_

Please generate a tactic in lean4 to solve the state. STATE: case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

tauto

case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_

no goals

Please generate a tactic in lean4 to solve the state. STATE: case a Δ_U : Set Formula V : VarBoolAssignment X : DefName xs : List VarName F : Formula := def_ X xs h1 : def_ X xs ∈ Δ_U ⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.L_15_7

[773, 1]

[916, 10]

sorry

Δ_U : Set Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝) h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝))

no goals

Please generate a tactic in lean4 to solve the state. STATE: Δ_U : Set Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝) h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U ⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

apply IsDeduct.mp_ (U.not_.imp_ P)

P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ P

case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P) case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.not_.imp_ P)

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

apply IsDeduct.mp_ (U.imp_ P)

case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P)

case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P)) case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.imp_ P)

Please generate a tactic in lean4 to solve the state. STATE: case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

apply proof_imp_deduct

case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))

case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))

Please generate a tactic in lean4 to solve the state. STATE: case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

apply T_14_9

case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))

no goals

Please generate a tactic in lean4 to solve the state. STATE: case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

apply deduction_theorem

case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.imp_ P)

case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U}) P

Please generate a tactic in lean4 to solve the state. STATE: case a.a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.imp_ P) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

exact h1

case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U}) P

no goals

Please generate a tactic in lean4 to solve the state. STATE: case a.a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

apply deduction_theorem

case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.not_.imp_ P)

case a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U.not_}) P

Please generate a tactic in lean4 to solve the state. STATE: case a P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct Δ (U.not_.imp_ P) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.T_14_9_Deduct

[919, 1]

[933, 13]

exact h2

case a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U.not_}) P

no goals

Please generate a tactic in lean4 to solve the state. STATE: case a.h1 P U : Formula Δ : Set Formula h1 : IsDeduct (Δ ∪ {U}) P h2 : IsDeduct (Δ ∪ {U.not_}) P ⊢ IsDeduct (Δ ∪ {U.not_}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

induction F

F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) F = Function.updateITE (evalPrimeFfToNot V) F' F F

case pred_const_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_const_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (pred_const_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ a✝¹ a✝) (pred_const_ a✝¹ a✝) case pred_var_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_var_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (pred_var_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ a✝¹ a✝) (pred_var_ a✝¹ a✝) case eq_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : VarName h1 : (eq_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (eq_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (eq_ a✝¹ a✝) (eq_ a✝¹ a✝) case true_ F' : Formula V : VarBoolAssignment h1 : true_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_ true_ case false_ F' : Formula V : VarBoolAssignment h1 : false_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) false_ = Function.updateITE (evalPrimeFfToNot V) F' false_ false_ case not_ F' : Formula V : VarBoolAssignment a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : a✝.not_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) a✝.not_ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝.not_ case imp_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.imp_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.imp_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.imp_ a✝) (a✝¹.imp_ a✝) case and_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.and_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.and_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.and_ a✝) (a✝¹.and_ a✝) case or_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.or_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.or_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.or_ a✝) (a✝¹.or_ a✝) case iff_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝) case forall_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (forall_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (forall_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (forall_ a✝¹ a✝) (forall_ a✝¹ a✝) case exists_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (exists_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (exists_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (exists_ a✝¹ a✝) (exists_ a✝¹ a✝) case def_ F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝)

Please generate a tactic in lean4 to solve the state. STATE: F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) F = Function.updateITE (evalPrimeFfToNot V) F' F F TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝)

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

case true_ | false_ | not_ | imp_ | and_ | or_ | iff_ => simp only [Formula.IsPrime] at h1

F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝)

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

unfold Function.updateITE

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝)

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝)

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

simp only [evalPrimeFfToNot]

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝)

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

simp only [Formula.evalPrime]

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

split_ifs <;> tauto

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_true

[936, 1]

[950, 38]

simp only [Formula.IsPrime] at h1

F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝)

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

induction F

F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) F = Function.updateITE (evalPrimeFfToNot V) F' F.not_ F

case pred_const_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_const_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (pred_const_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ a✝¹ a✝).not_ (pred_const_ a✝¹ a✝) case pred_var_ F' : Formula V : VarBoolAssignment a✝¹ : PredName a✝ : List VarName h1 : (pred_var_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (pred_var_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ a✝¹ a✝).not_ (pred_var_ a✝¹ a✝) case eq_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : VarName h1 : (eq_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (eq_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (eq_ a✝¹ a✝).not_ (eq_ a✝¹ a✝) case true_ F' : Formula V : VarBoolAssignment h1 : true_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_.not_ true_ case false_ F' : Formula V : VarBoolAssignment h1 : false_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) false_ = Function.updateITE (evalPrimeFfToNot V) F' false_.not_ false_ case not_ F' : Formula V : VarBoolAssignment a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : a✝.not_.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) a✝.not_ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_.not_ a✝.not_ case imp_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.imp_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.imp_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.imp_ a✝).not_ (a✝¹.imp_ a✝) case and_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.and_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.and_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.and_ a✝).not_ (a✝¹.and_ a✝) case or_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.or_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.or_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.or_ a✝).not_ (a✝¹.or_ a✝) case iff_ F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝) case forall_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (forall_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (forall_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (forall_ a✝¹ a✝).not_ (forall_ a✝¹ a✝) case exists_ F' : Formula V : VarBoolAssignment a✝¹ : VarName a✝ : Formula a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (exists_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (exists_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (exists_ a✝¹ a✝).not_ (exists_ a✝¹ a✝) case def_ F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)

Please generate a tactic in lean4 to solve the state. STATE: F F' : Formula V : VarBoolAssignment h1 : F.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) F = Function.updateITE (evalPrimeFfToNot V) F' F.not_ F TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

case true_ | false_ | not_ | imp_ | and_ | or_ | iff_ => simp only [Formula.IsPrime] at h1

F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝)

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

unfold Function.updateITE

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝)

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

simp only [evalPrimeFfToNot]

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝)

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

simp only [Formula.evalPrime]

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

split_ifs <;> tauto

F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ : DefName a✝ : List VarName h1 : (def_ a✝¹ a✝).IsPrime ⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) = if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.evalPrimeFfToNot_of_function_updateIte_false

[953, 1]

[967, 38]

simp only [Formula.IsPrime] at h1

F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝)

no goals

Please generate a tactic in lean4 to solve the state. STATE: F' : Formula V : VarBoolAssignment a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹ a_ih✝ : a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝ h1 : (a✝¹.iff_ a✝).IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) = Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

apply Set.image_congr

U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) '' Δ = evalPrimeFfToNot V '' Δ

case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ ∀ a ∈ Δ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a

Please generate a tactic in lean4 to solve the state. STATE: U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) '' Δ = evalPrimeFfToNot V '' Δ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

intro U' a1

case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ ∀ a ∈ Δ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a

case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ ⊢ ∀ a ∈ Δ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

specialize h1_Δ U' a1

case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'

case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

cases b

case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U' case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h U : Formula Δ : Set Formula V : VarBoolAssignment b : Bool h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

simp only [evalPrimeFfToNot_of_function_updateIte_false U' U V h1_Δ]

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U'

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

simp only [Function.updateITE]

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U'

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

simp

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U'.not_ = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

intro a2

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U'.not_ = evalPrimeFfToNot V U'

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U'.not_ = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U'.not_ = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

subst a2

case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U'.not_ = evalPrimeFfToNot V U'

case h.false Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U'.not_ = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.false U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U'.not_ = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

contradiction

case h.false Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U'.not_ = evalPrimeFfToNot V U'

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h.false Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U'.not_ = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

simp only [evalPrimeFfToNot_of_function_updateIte_true U' U V h1_Δ]

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U'

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

simp only [Function.updateITE]

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U'

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

simp

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

intro a2

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U' = evalPrimeFfToNot V U'

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime ⊢ U' = U → U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

subst a2

case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U' = evalPrimeFfToNot V U'

case h.true Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U' = evalPrimeFfToNot V U'

Please generate a tactic in lean4 to solve the state. STATE: case h.true U : Formula Δ : Set Formula V : VarBoolAssignment h1_U : U.IsPrime h2 : U ∉ Δ U' : Formula a1 : U' ∈ Δ h1_Δ : U'.IsPrime a2 : U' = U ⊢ U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte

[970, 1]

[996, 18]

contradiction

case h.true Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U' = evalPrimeFfToNot V U'

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h.true Δ : Set Formula V : VarBoolAssignment U' : Formula a1 : U' ∈ Δ h1_Δ h1_U : U'.IsPrime h2 : U' ∉ Δ ⊢ U' = evalPrimeFfToNot V U' TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

intro V

P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ) P

P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ) P

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

apply T_14_9_Deduct P U (Δ.image (evalPrimeFfToNot V))

P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ) P

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

specialize h3 (Function.updateITE V U true)

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [image_of_evalPrimeFfToNot_of_function_updateIte U Δ V true h1_Δ h1_U h2] at h3

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [evalPrimeFfToNot_of_function_updateIte_true U U V h1_U] at h3

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [Function.updateITE] at h3

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [eq_self_iff_true, if_true] at h3

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

exact h3

case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

specialize h3 (Function.updateITE V U Bool.false)

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P V : VarBoolAssignment ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [image_of_evalPrimeFfToNot_of_function_updateIte U Δ V false h1_Δ h1_U h2] at h3

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [evalPrimeFfToNot_of_function_updateIte_false U U V h1_U] at h3

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [Function.updateITE] at h3

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

simp only [eq_self_iff_true, if_true] at h3

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAuxAux

[999, 1]

[1021, 13]

exact h3

case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ : Set Formula h1_Δ : ∀ U' ∈ Δ, U'.IsPrime h1_U : U.IsPrime h2 : U ∉ Δ V : VarBoolAssignment h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P ⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

induction Δ_U using Finset.induction_on

P : Formula Δ_U : Finset Formula h1 : Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P ⊢ IsDeduct ∅ P

case empty P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P case insert P a✝² : Formula s✝ : Finset Formula a✝¹ : a✝² ∉ s✝ a✝ : s✝ ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) s✝)) P) → IsDeduct ∅ P h1 : insert a✝² s✝ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert a✝² s✝))) P ⊢ IsDeduct ∅ P

Please generate a tactic in lean4 to solve the state. STATE: P : Formula Δ_U : Finset Formula h1 : Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P ⊢ IsDeduct ∅ P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

case empty => simp at h2 exact h2

P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P

no goals

Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

simp at h2

P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P

P : Formula h1 : ∅ ⊆ P.primeSet h2 : IsDeduct ∅ P ⊢ IsDeduct ∅ P

Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : ∅ ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P ⊢ IsDeduct ∅ P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

exact h2

P : Formula h1 : ∅ ⊆ P.primeSet h2 : IsDeduct ∅ P ⊢ IsDeduct ∅ P

no goals

Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : ∅ ⊆ P.primeSet h2 : IsDeduct ∅ P ⊢ IsDeduct ∅ P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

apply Δ_U_2

P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ IsDeduct ∅ P

case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ Δ_U ⊆ P.primeSet case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ IsDeduct ∅ P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

simp only [Finset.insert_subset_iff] at h1

case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ Δ_U ⊆ P.primeSet

case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ Δ_U ⊆ P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

cases h1

case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet

case h1.intro P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P left✝ : U ∈ P.primeSet right✝ : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet

Please generate a tactic in lean4 to solve the state. STATE: case h1 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

case intro h1_left h1_right => exact h1_right

P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

exact h1_right

P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ Δ_U ⊆ P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

simp only [Finset.insert_subset_iff] at h1

case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : insert U Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

simp at h2

case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

cases h1

case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

case h2.intro P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P left✝ : U ∈ P.primeSet right✝ : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

simp

P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P

P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U) P

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

apply propCompleteAuxAux P U Δ_U

P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U) P

case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U.IsPrime case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∉ ↑Δ_U case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {evalPrimeFfToNot V U}) P

Please generate a tactic in lean4 to solve the state. STATE: P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

intro U' a1

case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime

case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U'.IsPrime

Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

apply mem_primeSet_isPrime P U'

case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U'.IsPrime

case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ P.primeSet

Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U'.IsPrime TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

apply h1_right

case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ P.primeSet

case h1_Δ.a P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ Δ_U

Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

exact a1

case h1_Δ.a P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ Δ_U

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h1_Δ.a P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet U' : Formula a1 : U' ∈ ↑Δ_U ⊢ U' ∈ Δ_U TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

apply mem_primeSet_isPrime P U

case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U.IsPrime

case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∈ P.primeSet

Please generate a tactic in lean4 to solve the state. STATE: case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U.IsPrime TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

exact h1_left

case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∈ P.primeSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h1_U P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∈ P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

exact Δ_U_1

case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∉ ↑Δ_U

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h2 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ U ∉ ↑Δ_U TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

simp

case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {evalPrimeFfToNot V U}) P

case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P

Please generate a tactic in lean4 to solve the state. STATE: case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {evalPrimeFfToNot V U}) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.propCompleteAux

[1024, 1]

[1057, 19]

exact h2

case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h3 P U : Formula Δ_U : Finset Formula Δ_U_1 : U ∉ Δ_U Δ_U_2 : Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P h1_left : U ∈ P.primeSet h1_right : Δ_U ⊆ P.primeSet ⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

simp only [IsProof]

P : Formula h1 : P.IsTautoPrime ⊢ IsProof P

P : Formula h1 : P.IsTautoPrime ⊢ IsDeduct ∅ P

Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : P.IsTautoPrime ⊢ IsProof P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

apply propCompleteAux P P.primeSet

P : Formula h1 : P.IsTautoPrime ⊢ IsDeduct ∅ P

case h1 P : Formula h1 : P.IsTautoPrime ⊢ P.primeSet ⊆ P.primeSet case h2 P : Formula h1 : P.IsTautoPrime ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P

Please generate a tactic in lean4 to solve the state. STATE: P : Formula h1 : P.IsTautoPrime ⊢ IsDeduct ∅ P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

rfl

case h1 P : Formula h1 : P.IsTautoPrime ⊢ P.primeSet ⊆ P.primeSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h1 P : Formula h1 : P.IsTautoPrime ⊢ P.primeSet ⊆ P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

intro V

case h2 P : Formula h1 : P.IsTautoPrime ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P

case h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P : Formula h1 : P.IsTautoPrime ⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

apply L_15_7 P P P.primeSet V (P.primeSet.image (evalPrimeFfToNot V))

case h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P

case h2.h1 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑P.primeSet ⊆ ↑P.primeSet case h2.h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.primeSet case h2.h3 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P

Please generate a tactic in lean4 to solve the state. STATE: case h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

rfl

case h2.h1 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑P.primeSet ⊆ ↑P.primeSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h2.h1 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑P.primeSet ⊆ ↑P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

simp only [Finset.coe_image]

case h2.h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.primeSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h2.h2 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.primeSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

simp only [Formula.IsTautoPrime] at h1

case h2.h3 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P

case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P

Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula h1 : P.IsTautoPrime V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

simp only [evalPrimeFfToNot]

case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P

case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = if evalPrime V P then P else P.not_

Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = evalPrimeFfToNot V P TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

specialize h1 V

case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = if evalPrime V P then P else P.not_

case h2.h3 P : Formula V : VarBoolAssignment h1 : evalPrime V P ⊢ P = if evalPrime V P then P else P.not_

Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula h1 : ∀ (V : VarBoolAssignment), evalPrime V P V : VarBoolAssignment ⊢ P = if evalPrime V P then P else P.not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Margaris/Prop.lean

FOL.NV.prop_complete

[1063, 1]

[1078, 28]

simp only [if_pos h1]

case h2.h3 P : Formula V : VarBoolAssignment h1 : evalPrime V P ⊢ P = if evalPrime V P then P else P.not_

no goals

Please generate a tactic in lean4 to solve the state. STATE: case h2.h3 P : Formula V : VarBoolAssignment h1 : evalPrime V P ⊢ P = if evalPrime V P then P else P.not_ TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Finset.lean

Finset.union_subset_left_right

[8, 1]

[23, 44]

apply Finset.union_subset_iff.mpr

α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ∪ B ⊆ C ∪ D

α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D

Please generate a tactic in lean4 to solve the state. STATE: α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ∪ B ⊆ C ∪ D TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Finset.lean

Finset.union_subset_left_right

[8, 1]

[23, 44]

constructor

α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D

case left α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D case right α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ B ⊆ C ∪ D

Please generate a tactic in lean4 to solve the state. STATE: α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D TACTIC: