Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
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https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp only [Forall_] at h1
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (Forall_ [] P) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (List.foldr forall_ P []) ⊒ IsDeduct Ξ” P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (Forall_ [] P) ⊒ IsDeduct Ξ” P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp at h1
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (List.foldr forall_ P []) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” P ⊒ IsDeduct Ξ” P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (List.foldr forall_ P []) ⊒ IsDeduct Ξ” P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
exact h1
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” P ⊒ IsDeduct Ξ” P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” P ⊒ IsDeduct Ξ” P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp only [Forall_] at h1
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (Forall_ (xs_hd :: xs_tl) P) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (List.foldr forall_ P (xs_hd :: xs_tl)) ⊒ IsDeduct Ξ” P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (Forall_ (xs_hd :: xs_tl) P) ⊒ IsDeduct Ξ” P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp at h1
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (List.foldr forall_ P (xs_hd :: xs_tl)) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (List.foldr forall_ P (xs_hd :: xs_tl)) ⊒ IsDeduct Ξ” P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
apply xs_ih
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (Forall_ xs_tl P)
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp only [Forall_]
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (Forall_ xs_tl P)
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (List.foldr forall_ P xs_tl)
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (Forall_ xs_tl P) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
apply specId xs_hd
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (List.foldr forall_ P xs_tl)
case h1 P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl))
Please generate a tactic in lean4 to solve the state. STATE: P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (List.foldr forall_ P xs_tl) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
exact h1
case h1 P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h1 P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp only [Formula.Forall_]
P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (Forall_ xs P) ↔ x ∈ xs ∨ isBoundIn x P
P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (List.foldr forall_ P xs) ↔ x ∈ xs ∨ isBoundIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (Forall_ xs P) ↔ x ∈ xs ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
induction xs
P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (List.foldr forall_ P xs) ↔ x ∈ xs ∨ isBoundIn x P
case nil P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P case cons P : Formula x head✝ : VarName tail✝ : List VarName tail_ih✝ : isBoundIn x (List.foldr forall_ P tail✝) ↔ x ∈ tail✝ ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (head✝ :: tail✝)) ↔ x ∈ head✝ :: tail✝ ∨ isBoundIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (List.foldr forall_ P xs) ↔ x ∈ xs ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
case nil => simp
P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
case cons xs_hd xs_tl xs_ih => simp simp only [isBoundIn] simp only [xs_ih] tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x ∈ xs_hd :: xs_tl ∨ isBoundIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x ∈ xs_hd :: xs_tl ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp
P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x ∈ xs_hd :: xs_tl ∨ isBoundIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x ∈ xs_hd :: xs_tl ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp only [isBoundIn]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ isBoundIn x (List.foldr forall_ P xs_tl) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp only [xs_ih]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ isBoundIn x (List.foldr forall_ P xs_tl) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ x ∈ xs_tl ∨ isBoundIn x P ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ isBoundIn x (List.foldr forall_ P xs_tl) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ x ∈ xs_tl ∨ isBoundIn x P ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ x ∈ xs_tl ∨ isBoundIn x P ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp only [Formula.Forall_]
P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (Forall_ xs P) ↔ x βˆ‰ xs ∧ isFreeIn x P
P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (List.foldr forall_ P xs) ↔ x βˆ‰ xs ∧ isFreeIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (Forall_ xs P) ↔ x βˆ‰ xs ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
induction xs
P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (List.foldr forall_ P xs) ↔ x βˆ‰ xs ∧ isFreeIn x P
case nil P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P case cons P : Formula x head✝ : VarName tail✝ : List VarName tail_ih✝ : isFreeIn x (List.foldr forall_ P tail✝) ↔ x βˆ‰ tail✝ ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (head✝ :: tail✝)) ↔ x βˆ‰ head✝ :: tail✝ ∧ isFreeIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (List.foldr forall_ P xs) ↔ x βˆ‰ xs ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
case nil => simp
P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
case cons xs_hd xs_tl xs_ih => simp simp only [isFreeIn] simp only [xs_ih] tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x βˆ‰ xs_hd :: xs_tl ∧ isFreeIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x βˆ‰ xs_hd :: xs_tl ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp
P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x βˆ‰ xs_hd :: xs_tl ∧ isFreeIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x βˆ‰ xs_hd :: xs_tl ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp only [isFreeIn]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ isFreeIn x (List.foldr forall_ P xs_tl) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp only [xs_ih]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ isFreeIn x (List.foldr forall_ P xs_tl) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ x βˆ‰ xs_tl ∧ isFreeIn x P ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ isFreeIn x (List.foldr forall_ P xs_tl) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ x βˆ‰ xs_tl ∧ isFreeIn x P ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ x βˆ‰ xs_tl ∧ isFreeIn x P ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
induction h1
U V P_U P_V : Formula l : List VarName h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_U.iff_ P_V))
case same_ U V P_U P_V : Formula l : List VarName P_u✝ P_v✝ : Formula a✝ : P_u✝ = P_v✝ h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) case diff_ U V P_U P_V : Formula l : List VarName P_u✝ P_v✝ : Formula a✝¹ : P_u✝ = U a✝ : P_v✝ = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) case not_ U V P_U P_V : Formula l : List VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝.not_ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.not_.iff_ P_v✝.not_)) case imp_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.imp_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.imp_ Q_u✝).iff_ (P_v✝.imp_ Q_v✝))) case and_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.and_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝))) case or_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.or_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝))) case iff_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.iff_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝))) case forall_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (forall_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ x✝ P_u✝).iff_ (forall_ x✝ P_v✝))) case exists_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (exists_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝)))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_U.iff_ P_V)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
case same_ h1_P h1_P' h1_1 => subst h1_1 simp only [def_iff_] simp only [def_and_] SC
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = h1_P' h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
no goals
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = h1_P' h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
case diff_ h1_P h1_P' h1_1 h1_2 => subst h1_1 subst h1_2 apply Forall_spec_id
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = U h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
no goals
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = U h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
all_goals sorry
case and_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.and_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝))) case or_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.or_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝))) case iff_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.iff_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝))) case exists_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (exists_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝)))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case and_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.and_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝))) case or_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.or_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝))) case iff_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.iff_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝))) case exists_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (exists_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
subst h1_1
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = h1_P' h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = h1_P' h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_iff_]
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P))
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P).and_ (h1_P.imp_ h1_P)))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_and_]
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P).and_ (h1_P.imp_ h1_P)))
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P).imp_ (h1_P.imp_ h1_P).not_).not_)
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P).and_ (h1_P.imp_ h1_P))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
SC
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P).imp_ (h1_P.imp_ h1_P).not_).not_)
no goals
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P).imp_ (h1_P.imp_ h1_P).not_).not_) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
subst h1_1
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = U h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ V)).imp_ (h1_P.iff_ h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = U h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
subst h1_2
V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ V)).imp_ (h1_P.iff_ h1_P'))
P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v h1_P') ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ h1_P')).imp_ (h1_P.iff_ h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply Forall_spec_id
P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v h1_P') ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ h1_P')).imp_ (h1_P.iff_ h1_P'))
no goals
Please generate a tactic in lean4 to solve the state. STATE: P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v h1_P') ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ h1_P')).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [isBoundIn] at h2
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P.not_ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P.not_ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply IsDeduct.mp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))) case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_iff_]
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_)))
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.not_.imp_ h1_P'.not_).and_ (h1_P'.not_.imp_ h1_P.not_))))
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_and_]
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.not_.imp_ h1_P'.not_).and_ (h1_P'.not_.imp_ h1_P.not_))))
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.not_.imp_ h1_P'.not_).imp_ (h1_P'.not_.imp_ h1_P.not_).not_).not_))
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.not_.imp_ h1_P'.not_).and_ (h1_P'.not_.imp_ h1_P.not_)))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
SC
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.not_.imp_ h1_P'.not_).imp_ (h1_P'.not_.imp_ h1_P.not_).not_).not_))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.not_.imp_ h1_P'.not_).imp_ (h1_P'.not_.imp_ h1_P.not_).not_).not_)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
exact h1_ih h2
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [isBoundIn] at h2
U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (h1_P.imp_ h1_Q) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))
U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (h1_P.imp_ h1_Q) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q'))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply IsDeduct.mp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))) case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q'))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply IsDeduct.mp_ ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q'))))
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')).imp_ (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q'))))) case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_iff_]
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')).imp_ (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))))
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_Q.imp_ h1_Q').and_ (h1_Q'.imp_ h1_Q))).imp_ (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ (((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).and_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q))))))
Please generate a tactic in lean4 to solve the state. STATE: case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')).imp_ (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q'))))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_and_]
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_Q.imp_ h1_Q').and_ (h1_Q'.imp_ h1_Q))).imp_ (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ (((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).and_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q))))))
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_Q.imp_ h1_Q').imp_ (h1_Q'.imp_ h1_Q).not_).not_).imp_ (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ (((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).imp_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q)).not_).not_)))
Please generate a tactic in lean4 to solve the state. STATE: case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_Q.imp_ h1_Q').and_ (h1_Q'.imp_ h1_Q))).imp_ (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ (((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).and_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q)))))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
SC
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_Q.imp_ h1_Q').imp_ (h1_Q'.imp_ h1_Q).not_).not_).imp_ (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ (((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).imp_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q)).not_).not_)))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_Q.imp_ h1_Q').imp_ (h1_Q'.imp_ h1_Q).not_).not_).imp_ (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ (((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).imp_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q)).not_).not_))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply h1_ih_2
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l
Please generate a tactic in lean4 to solve the state. STATE: case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
intro v a2
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a2 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q ⊒ v ∈ l
Please generate a tactic in lean4 to solve the state. STATE: case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply h2 v
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a2 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q ⊒ v ∈ l
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a2 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q)
Please generate a tactic in lean4 to solve the state. STATE: case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a2 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q ⊒ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
tauto
case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a2 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a2 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply h1_ih_1
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
intro v a1
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ v ∈ l
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply h2 v
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ v ∈ l
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q)
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
cases a1
case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q)
case a.intro U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName left✝ : isFreeIn v U ∨ isFreeIn v V right✝ : isBoundIn v h1_P ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q)
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
constructor
U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q)
case left U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isFreeIn v U ∨ isFreeIn v V case right U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P ∨ isBoundIn v h1_Q
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
exact a1_left
case left U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isFreeIn v U ∨ isFreeIn v V
no goals
Please generate a tactic in lean4 to solve the state. STATE: case left U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isFreeIn v U ∨ isFreeIn v V TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
left
case right U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P ∨ isBoundIn v h1_Q
case right.h U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P
Please generate a tactic in lean4 to solve the state. STATE: case right U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P ∨ isBoundIn v h1_Q TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
exact a1_right
case right.h U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P
no goals
Please generate a tactic in lean4 to solve the state. STATE: case right.h U V P_U P_V : Formula l : List VarName h1_P h1_Q h1_P' h1_Q' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q' h1_ih_1 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h1_ih_2 : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_Q β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (isBoundIn v h1_P ∨ isBoundIn v h1_Q) β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [isBoundIn] at h2
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (forall_ h1_x h1_P) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (v = h1_x ∨ isBoundIn v h1_P) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (forall_ h1_x h1_P) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp at h2
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (v = h1_x ∨ isBoundIn v h1_P) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ (v = h1_x ∨ isBoundIn v h1_P) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply deduction_theorem
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
case h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct (βˆ… βˆͺ {Forall_ l (U.iff_ V)}) ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp
case h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct (βˆ… βˆͺ {Forall_ l (U.iff_ V)}) ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
case h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: case h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct (βˆ… βˆͺ {Forall_ l (U.iff_ V)}) ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply IsDeduct.mp_ (forall_ h1_x (h1_P.iff_ h1_P'))
case h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
case h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))) case h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (forall_ h1_x (h1_P.iff_ h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: case h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply proof_imp_deduct
case h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
case h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
Please generate a tactic in lean4 to solve the state. STATE: case h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply T_18_1
case h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply generalization
case h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (forall_ h1_x (h1_P.iff_ h1_P'))
case h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (h1_P.iff_ h1_P') case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ βˆ€ H ∈ {Forall_ l (U.iff_ V)}, Β¬isFreeIn h1_x H
Please generate a tactic in lean4 to solve the state. STATE: case h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (forall_ h1_x (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply IsDeduct.mp_ (Forall_ l (U.iff_ V))
case h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (h1_P.iff_ h1_P')
case h1.a.h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) case h1.a.h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (Forall_ l (U.iff_ V))
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (h1_P.iff_ h1_P') TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply proof_imp_deduct
case h1.a.h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply h1_ih
case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
intro v a1
case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l
case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ v ∈ l
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
cases a1
case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ v ∈ l
case h1.a.h1.a.h1.intro U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName left✝ : isFreeIn v U ∨ isFreeIn v V right✝ : isBoundIn v h1_P ⊒ v ∈ l
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1.a.h1 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P ⊒ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
case _ a1_left a1_right => apply h2 v a1_left right apply a1_right
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ v ∈ l
no goals
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply h2 v a1_left
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ v ∈ l
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ v = h1_x ∨ isBoundIn v h1_P
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ v ∈ l TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
right
U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ v = h1_x ∨ isBoundIn v h1_P
case h U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ v = h1_x ∨ isBoundIn v h1_P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply a1_right
case h U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l v : VarName a1_left : isFreeIn v U ∨ isFreeIn v V a1_right : isBoundIn v h1_P ⊒ isBoundIn v h1_P TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply IsDeduct.assume_
case h1.a.h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (Forall_ l (U.iff_ V))
case h1.a.h1.a.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Forall_ l (U.iff_ V) ∈ {Forall_ l (U.iff_ V)}
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct {Forall_ l (U.iff_ V)} (Forall_ l (U.iff_ V)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp
case h1.a.h1.a.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Forall_ l (U.iff_ V) ∈ {Forall_ l (U.iff_ V)}
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h1.a.a U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Forall_ l (U.iff_ V) ∈ {Forall_ l (U.iff_ V)} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
intro H a1
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ βˆ€ H ∈ {Forall_ l (U.iff_ V)}, Β¬isFreeIn h1_x H
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l H : Formula a1 : H ∈ {Forall_ l (U.iff_ V)} ⊒ Β¬isFreeIn h1_x H
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ βˆ€ H ∈ {Forall_ l (U.iff_ V)}, Β¬isFreeIn h1_x H TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp at a1
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l H : Formula a1 : H ∈ {Forall_ l (U.iff_ V)} ⊒ Β¬isFreeIn h1_x H
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l H : Formula a1 : H = Forall_ l (U.iff_ V) ⊒ Β¬isFreeIn h1_x H
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l H : Formula a1 : H ∈ {Forall_ l (U.iff_ V)} ⊒ Β¬isFreeIn h1_x H TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
subst a1
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l H : Formula a1 : H = Forall_ l (U.iff_ V) ⊒ Β¬isFreeIn h1_x H
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬isFreeIn h1_x (Forall_ l (U.iff_ V))
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l H : Formula a1 : H = Forall_ l (U.iff_ V) ⊒ Β¬isFreeIn h1_x H TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [Forall_isFreeIn]
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬isFreeIn h1_x (Forall_ l (U.iff_ V))
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x (U.iff_ V))
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬isFreeIn h1_x (Forall_ l (U.iff_ V)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_iff_]
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x (U.iff_ V))
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x ((U.imp_ V).and_ (V.imp_ U)))
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x (U.iff_ V)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_and_]
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x ((U.imp_ V).and_ (V.imp_ U)))
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x ((U.imp_ V).imp_ (V.imp_ U).not_).not_)
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x ((U.imp_ V).and_ (V.imp_ U))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [isFreeIn]
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x ((U.imp_ V).imp_ (V.imp_ U).not_).not_)
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ ((isFreeIn h1_x U ∨ isFreeIn h1_x V) ∨ isFreeIn h1_x V ∨ isFreeIn h1_x U))
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ isFreeIn h1_x ((U.imp_ V).imp_ (V.imp_ U).not_).not_) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
sorry
case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ ((isFreeIn h1_x U ∨ isFreeIn h1_x V) ∨ isFreeIn h1_x V ∨ isFreeIn h1_x U))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h1.a.h2 U V P_U P_V : Formula l : List VarName h1_x : VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), isFreeIn v U ∨ isFreeIn v V β†’ v = h1_x ∨ isBoundIn v h1_P β†’ v ∈ l ⊒ Β¬(h1_x βˆ‰ l ∧ ((isFreeIn h1_x U ∨ isFreeIn h1_x V) ∨ isFreeIn h1_x V ∨ isFreeIn h1_x U)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
sorry
case exists_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (exists_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝)))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case exists_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (exists_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝))) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
apply IsDeduct.mp_ (Forall_ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList (U.iff_ V))
U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsProof (P_U.iff_ P_V)
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… ((Forall_ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList (U.iff_ V)).imp_ (P_U.iff_ P_V)) case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (Forall_ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList (U.iff_ V))
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsProof (P_U.iff_ P_V) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
apply T_18_2 U V P_U P_V ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList h1
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… ((Forall_ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList (U.iff_ V)).imp_ (P_U.iff_ P_V))
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U β†’ v ∈ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… ((Forall_ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList (U.iff_ V)).imp_ (P_U.iff_ P_V)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
intro v a1
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U β†’ v ∈ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U ⊒ v ∈ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U β†’ v ∈ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
simp
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U ⊒ v ∈ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U ⊒ v ∈ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
simp only [isFreeIn_iff_mem_freeVarSet] at a1
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ isBoundIn v P_U ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
simp only [isBoundIn_iff_mem_boundVarSet] at a1
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ isBoundIn v P_U ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ isBoundIn v P_U ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
exact a1
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) v : VarName a1 : (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet ⊒ (v ∈ U.freeVarSet ∨ v ∈ V.freeVarSet) ∧ v ∈ P_U.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
simp only [Formula.Forall_]
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (Forall_ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList (U.iff_ V))
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList)
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (Forall_ ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList (U.iff_ V)) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
induction ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList
case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList)
case a.nil U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) []) case a.cons U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) head✝ : VarName tail✝ : List VarName tail_ih✝ : IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) tail✝) ⊒ IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) (head✝ :: tail✝))
Please generate a tactic in lean4 to solve the state. STATE: case a U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) ((U.freeVarSet βˆͺ V.freeVarSet) ∩ P_U.boundVarSet).toList) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.C_18_3
[889, 1]
[913, 13]
case _ => simp exact h2
U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) [])
no goals
Please generate a tactic in lean4 to solve the state. STATE: U V P_U P_V : Formula h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : IsProof (U.iff_ V) ⊒ IsDeduct βˆ… (List.foldr forall_ (U.iff_ V) []) TACTIC: