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Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case _ c2 => subst c1 simp only [Function.updateITE_idem]	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 c2 : x = v ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case _ c2 => subst c1 simp only [Function.updateITE_idem]	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : x = v c2 : NotFree Γ v phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	subst c1	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 c2 : x = v ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) x : VarName phi : Formula V : Valuation D d d' : D h2 : ∀ (X : MetaVarName), (x, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) x phi_ih : NotFree Γ x phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V x d) M E phi c2 : x = x ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [Function.updateITE_idem]	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) x : VarName phi : Formula V : Valuation D d d' : D h2 : ∀ (X : MetaVarName), (x, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) x phi_ih : NotFree Γ x phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V x d) M E phi c2 : x = x ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	subst c1	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : x = v c2 : NotFree Γ v phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) x : VarName phi : Formula V : Valuation D d d' : D h2 : ∀ (X : MetaVarName), (x, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) x phi_ih : NotFree Γ x phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V x d) M E phi c2 : NotFree Γ x phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [Function.updateITE_idem]	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) x : VarName phi : Formula V : Valuation D d d' : D h2 : ∀ (X : MetaVarName), (x, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) x phi_ih : NotFree Γ x phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V x d) M E phi c2 : NotFree Γ x phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	cases h1	case neg D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi h1 : x = v ∨ NotFree Γ v phi V : Valuation D d d' : D c1 : ¬x = v ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	case neg.inl D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v h✝ : x = v ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi case neg.inr D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v h✝ : NotFree Γ v phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case _ c2 => contradiction	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v c2 : x = v ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case _ c2 => simp only [← Function.updateITE_comm V x v d d' c1] exact phi_ih c2 (Function.updateITE V x d') d	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v c2 : NotFree Γ v phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	contradiction	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v c2 : x = v ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [← Function.updateITE_comm V x v d d' c1]	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v c2 : NotFree Γ v phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v c2 : NotFree Γ v phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V x d') v d) M E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	exact phi_ih c2 (Function.updateITE V x d') d	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x : VarName phi : Formula phi_ih : NotFree Γ v phi → ∀ (V : Valuation D) (d : D), Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi V : Valuation D d d' : D c1 : ¬x = v c2 : NotFree Γ v phi ⊢ Holds D I (Function.updateITE V x d') M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V x d') v d) M E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	induction E	D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) ⊢ IsNotFree D I M E (def_ X xs) v	case nil D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M [] (meta_var_ X) v ⊢ IsNotFree D I M [] (def_ X xs) v case cons D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) head✝ : Definition tail✝ : List Definition tail_ih✝ : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M tail✝ (meta_var_ X) v) → IsNotFree D I M tail✝ (def_ X xs) v h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M (head✝ :: tail✝) (meta_var_ X) v ⊢ IsNotFree D I M (head✝ :: tail✝) (def_ X xs) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case nil => unfold IsNotFree intro V d simp only [Holds]	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M [] (meta_var_ X) v ⊢ IsNotFree D I M [] (def_ X xs) v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	unfold IsNotFree	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M [] (meta_var_ X) v ⊢ IsNotFree D I M [] (def_ X xs) v	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M [] (meta_var_ X) v ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M [] (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M [] (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	intro V d	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M [] (meta_var_ X) v ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M [] (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M [] (def_ X xs)	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M [] (meta_var_ X) v V : Valuation D d : D ⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M [] (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [Holds]	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M [] (meta_var_ X) v V : Valuation D d : D ⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M [] (def_ X xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	unfold NotFree at h1	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName h1 : NotFree Γ v (def_ X xs) E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E_tl (meta_var_ X) v) → IsNotFree D I M E_tl (def_ X xs) v h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M (E_hd :: E_tl) (meta_var_ X) v ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E_tl (meta_var_ X) v) → IsNotFree D I M E_tl (def_ X xs) v h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M (E_hd :: E_tl) (meta_var_ X) v h1 : v ∉ xs ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	unfold IsNotFree at h2	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E_tl (meta_var_ X) v) → IsNotFree D I M E_tl (def_ X xs) v h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M (E_hd :: E_tl) (meta_var_ X) v h1 : v ∉ xs ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E_tl (meta_var_ X) v) → IsNotFree D I M E_tl (def_ X xs) v h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), Holds D I V M (E_hd :: E_tl) (meta_var_ X) ↔ Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (meta_var_ X) h1 : v ∉ xs ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [Holds] at h2	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E_tl (meta_var_ X) v) → IsNotFree D I M E_tl (def_ X xs) v h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), Holds D I V M (E_hd :: E_tl) (meta_var_ X) ↔ Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (meta_var_ X) h1 : v ∉ xs ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E_tl (meta_var_ X) v) → IsNotFree D I M E_tl (def_ X xs) v h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	unfold IsNotFree at E_ih	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E_tl (meta_var_ X) v) → IsNotFree D I M E_tl (def_ X xs) v h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (meta_var_ X) ↔ Holds D I (Function.updateITE V v d) M E_tl (meta_var_ X)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [Holds] at E_ih	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (meta_var_ X) ↔ Holds D I (Function.updateITE V v d) M E_tl (meta_var_ X)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	unfold IsNotFree	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) ⊢ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [Holds]	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (def_ X xs)	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) ⊢ ∀ (V : Valuation D) (d : D), (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F else Holds D I V M E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (Function.updateITE V v d) E_hd.args (List.map (Function.updateITE V v d) xs)) M E_tl E_hd.F else Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	intro V a	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) ⊢ ∀ (V : Valuation D) (d : D), (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F else Holds D I V M E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (Function.updateITE V v d) E_hd.args (List.map (Function.updateITE V v d) xs)) M E_tl E_hd.F else Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D ⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F else Holds D I V M E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M E_tl E_hd.F else Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	split_ifs	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D ⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F else Holds D I V M E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M E_tl E_hd.F else Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)	case pos D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M E_tl E_hd.F case neg D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case _ c1 => exact E_ih h2 V a	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	apply holds_coincide_var D I (Function.updateListITE V E_hd.args (List.map V xs)) (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M E_tl E_hd.F E_hd.args E_hd.nf	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M E_tl E_hd.F	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ ∀ v_1 ∈ E_hd.args, Function.updateListITE V E_hd.args (List.map V xs) v_1 = Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v_1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	intro v' a1	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ ∀ v_1 ∈ E_hd.args, Function.updateListITE V E_hd.args (List.map V xs) v_1 = Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v_1	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ Function.updateListITE V E_hd.args (List.map V xs) v' = Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	apply Function.updateListITE_map_updateIte V (Function.updateITE V v a)	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ Function.updateListITE V E_hd.args (List.map V xs) v' = Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v'	case h1 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ ∀ y ∈ xs, ¬y = v case h2 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ E_hd.args.length = xs.length case h3 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ v' ∈ E_hd.args
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	intro y a2 contra	case h1 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ ∀ y ∈ xs, ¬y = v	case h1 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args y : VarName a2 : y ∈ xs contra : y = v ⊢ False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	subst contra	case h1 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args y : VarName a2 : y ∈ xs contra : y = v ⊢ False	case h1 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args y : VarName a2 : y ∈ xs h1 : y ∉ xs h2 : ∀ (X : MetaVarName), (y, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V y d) E_ih : (∀ (X : MetaVarName), (y, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V y d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V y d) M E_tl (def_ X xs) ⊢ False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	contradiction	case h1 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args y : VarName a2 : y ∈ xs h1 : y ∉ xs h2 : ∀ (X : MetaVarName), (y, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V y d) E_ih : (∀ (X : MetaVarName), (y, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V y d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V y d) M E_tl (def_ X xs) ⊢ False	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	cases c1	case h2 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ E_hd.args.length = xs.length	case h2.intro D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D v' : VarName a1 : v' ∈ E_hd.args left✝ : X = E_hd.name right✝ : xs.length = E_hd.args.length ⊢ E_hd.args.length = xs.length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case intro c1_left c1_right => simp only [c1_right]	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D v' : VarName a1 : v' ∈ E_hd.args c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊢ E_hd.args.length = xs.length	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [c1_right]	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D v' : VarName a1 : v' ∈ E_hd.args c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊢ E_hd.args.length = xs.length	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	exact a1	case h3 D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : X = E_hd.name ∧ xs.length = E_hd.args.length v' : VarName a1 : v' ∈ E_hd.args ⊢ v' ∈ E_hd.args	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	exact E_ih h2 V a	D : Type I : Interpretation D M : MetaValuation D Γ : List (VarName × MetaVarName) v : VarName X : DefName xs : List VarName E_hd : Definition E_tl : List Definition h1 : v ∉ xs h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d) E_ih : (∀ (X : MetaVarName), (v, X) ∈ Γ → ∀ (V : Valuation D) (d : D), M X V ↔ M X (Function.updateITE V v d)) → ∀ (V : Valuation D) (d : D), Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v d) M E_tl (def_ X xs) V : Valuation D a : D c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	intro v X a1	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) ⊢ ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I (fun X V' => Holds D I (V' ∘ σ') M E (τ X)) E (meta_var_ X) v	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ ⊢ IsNotFree D I (fun X V' => Holds D I (V' ∘ σ') M E (τ X)) E (meta_var_ X) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	unfold IsNotFree	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ ⊢ IsNotFree D I (fun X V' => Holds D I (V' ∘ σ') M E (τ X)) E (meta_var_ X) v	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ ⊢ ∀ (V : Valuation D) (d : D), Holds D I V (fun X V' => Holds D I (V' ∘ σ') M E (τ X)) E (meta_var_ X) ↔ Holds D I (Function.updateITE V v d) (fun X V' => Holds D I (V' ∘ σ') M E (τ X)) E (meta_var_ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	simp only [Holds]	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ ⊢ ∀ (V : Valuation D) (d : D), Holds D I V (fun X V' => Holds D I (V' ∘ σ') M E (τ X)) E (meta_var_ X) ↔ Holds D I (Function.updateITE V v d) (fun X V' => Holds D I (V' ∘ σ') M E (τ X)) E (meta_var_ X)	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ ⊢ ∀ (V : Valuation D) (d : D), Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	intro V d	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ ⊢ ∀ (V : Valuation D) (d : D), Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D ⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	cases h1	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D ⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)	case intro D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D left✝ : ↑σ ∘ σ' = id right✝ : σ' ∘ ↑σ = id ⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	simp only [Function.updateITE_comp_right σ' σ.1 V v d h1_left h1_right]	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id ⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id ⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE (V ∘ σ') (↑σ v) d) M E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	apply not_free_imp_is_not_free D I M E (τ X) Γ' (σ.1 v)	D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id ⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE (V ∘ σ') (↑σ v) d) M E (τ X)	case h1 D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id ⊢ NotFree Γ' (↑σ v) (τ X) case h2 D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id ⊢ ∀ (X : MetaVarName), (↑σ v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) (↑σ v)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	exact h3 v X a1	case h1 D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id ⊢ NotFree Γ' (↑σ v) (τ X)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	intro X' a2	case h2 D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id ⊢ ∀ (X : MetaVarName), (↑σ v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) (↑σ v)	case h2 D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id X' : MetaVarName a2 : (↑σ v, X') ∈ Γ' ⊢ IsNotFree D I M E (meta_var_ X') (↑σ v)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_1	[1737, 1]	[1763, 29]	exact h2 (σ.1 v) X' a2	case h2 D : Type I : Interpretation D M : MetaValuation D E : Env Γ Γ' : List (VarName × MetaVarName) σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X) v : VarName X : MetaVarName a1 : (v, X) ∈ Γ V : Valuation D d : D h1_left : ↑σ ∘ σ' = id h1_right : σ' ∘ ↑σ = id X' : MetaVarName a2 : (↑σ v, X') ∈ Γ' ⊢ IsNotFree D I M E (meta_var_ X') (↑σ v)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	induction F	E : Env σ : Instantiation τ : MetaInstantiation F : Formula h1 : IsMetaVarOrAllDefInEnv E F h2 : ∀ X ∈ F.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ F)	case meta_var_ E : Env σ : Instantiation τ : MetaInstantiation a✝ : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ a✝) h2 : ∀ X ∈ (meta_var_ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ a✝)) case pred_ E : Env σ : Instantiation τ : MetaInstantiation a✝¹ : PredName a✝ : List VarName h1 : IsMetaVarOrAllDefInEnv E (pred_ a✝¹ a✝) h2 : ∀ X ∈ (pred_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (pred_ a✝¹ a✝)) case eq_ E : Env σ : Instantiation τ : MetaInstantiation a✝¹ a✝ : VarName h1 : IsMetaVarOrAllDefInEnv E (eq_ a✝¹ a✝) h2 : ∀ X ∈ (eq_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (eq_ a✝¹ a✝)) case true_ E : Env σ : Instantiation τ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E true_ h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ true_) case not_ E : Env σ : Instantiation τ : MetaInstantiation a✝ : Formula a_ih✝ : IsMetaVarOrAllDefInEnv E a✝ → (∀ X ∈ a✝.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝) h1 : IsMetaVarOrAllDefInEnv E a✝.not_ h2 : ∀ X ∈ a✝.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ a✝.not_) case imp_ E : Env σ : Instantiation τ : MetaInstantiation a✝¹ a✝ : Formula a_ih✝¹ : IsMetaVarOrAllDefInEnv E a✝¹ → (∀ X ∈ a✝¹.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝¹) a_ih✝ : IsMetaVarOrAllDefInEnv E a✝ → (∀ X ∈ a✝.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝) h1 : IsMetaVarOrAllDefInEnv E (a✝¹.imp_ a✝) h2 : ∀ X ∈ (a✝¹.imp_ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (a✝¹.imp_ a✝)) case forall_ E : Env σ : Instantiation τ : MetaInstantiation a✝¹ : VarName a✝ : Formula a_ih✝ : IsMetaVarOrAllDefInEnv E a✝ → (∀ X ∈ a✝.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝) h1 : IsMetaVarOrAllDefInEnv E (forall_ a✝¹ a✝) h2 : ∀ X ∈ (forall_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ a✝¹ a✝)) case def_ E : Env σ : Instantiation τ : MetaInstantiation a✝¹ : DefName a✝ : List VarName h1 : IsMetaVarOrAllDefInEnv E (def_ a✝¹ a✝) h2 : ∀ X ∈ (def_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	case meta_var_ X => unfold Formula.metaVarSet at h2 simp at h2 unfold sub exact h2	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : ∀ X_1 ∈ (meta_var_ X).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	case pred_ X xs => unfold sub unfold IsMetaVarOrAllDefInEnv simp only	E : Env σ : Instantiation τ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (pred_ X xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	case eq_ x y => unfold sub unfold IsMetaVarOrAllDefInEnv simp only	E : Env σ : Instantiation τ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (eq_ x y))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	case true_ => unfold sub unfold IsMetaVarOrAllDefInEnv simp only	E : Env σ : Instantiation τ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E true_ h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ true_)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	case not_ phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h1 unfold Formula.metaVarSet at h2 unfold sub unfold IsMetaVarOrAllDefInEnv exact phi_ih h1 h2	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi.not_ h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	case forall_ x phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h1 unfold Formula.metaVarSet at h2 unfold sub unfold IsMetaVarOrAllDefInEnv exact phi_ih h1 h2	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi) h2 : ∀ X ∈ (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ x phi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	case def_ X xs => unfold IsMetaVarOrAllDefInEnv at h1 unfold sub unfold IsMetaVarOrAllDefInEnv simp exact h1	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (def_ X xs) h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (def_ X xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold Formula.metaVarSet at h2	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : ∀ X_1 ∈ (meta_var_ X).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : ∀ X_1 ∈ {X}, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	simp at h2	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : ∀ X_1 ∈ {X}, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	exact h2	E : Env σ : Instantiation τ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2 : IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (τ X)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (pred_ X xs))	E : Env σ : Instantiation τ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv	E : Env σ : Instantiation τ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (↑σ) xs))	E : Env σ : Instantiation τ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ True
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	simp only	E : Env σ : Instantiation τ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ True	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (eq_ x y))	E : Env σ : Instantiation τ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (eq_ (↑σ x) (↑σ y))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv	E : Env σ : Instantiation τ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (eq_ (↑σ x) (↑σ y))	E : Env σ : Instantiation τ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ True
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	simp only	E : Env σ : Instantiation τ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ True	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E true_ h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ true_)	E : Env σ : Instantiation τ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E true_ h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E true_
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv	E : Env σ : Instantiation τ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E true_ h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E true_	E : Env σ : Instantiation τ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E true_ h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ True
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	simp only	E : Env σ : Instantiation τ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E true_ h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ True	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv at h1	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi.not_ h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) h1 : IsMetaVarOrAllDefInEnv E phi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold Formula.metaVarSet at h2	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) h1 : IsMetaVarOrAllDefInEnv E phi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi).not_
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi).not_	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	exact phi_ih h1 h2	E : Env σ : Instantiation τ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv at h1	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E (phi.imp_ psi) h2 : ∀ X ∈ (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ X ∈ (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold Formula.metaVarSet at h2	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ X ∈ (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ X ∈ phi.metaVarSet ∪ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	simp at h2	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ X ∈ phi.metaVarSet ∪ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E ((sub σ τ phi).imp_ (sub σ τ psi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E ((sub σ τ phi).imp_ (sub σ τ psi))	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi) ∧ IsMetaVarOrAllDefInEnv E (sub σ τ psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	cases h1	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi) ∧ IsMetaVarOrAllDefInEnv E (sub σ τ psi)	case intro E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) left✝ : IsMetaVarOrAllDefInEnv E phi right✝ : IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi) ∧ IsMetaVarOrAllDefInEnv E (sub σ τ psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	constructor	E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi) ∧ IsMetaVarOrAllDefInEnv E (sub σ τ psi)	case left E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi) case right E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	apply phi_ih h1_left	case left E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi)	case left E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	intro X a1	case left E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)	case left E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ phi.metaVarSet ⊢ IsMetaVarOrAllDefInEnv E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	apply h2	case left E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ phi.metaVarSet ⊢ IsMetaVarOrAllDefInEnv E (τ X)	case left.a E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ phi.metaVarSet ⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	left	case left.a E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ phi.metaVarSet ⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet	case left.a.h E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ phi.metaVarSet ⊢ X ∈ phi.metaVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	exact a1	case left.a.h E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ phi.metaVarSet ⊢ X ∈ phi.metaVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	apply psi_ih h1_right	case right E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ psi)	case right E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ ∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	intro X a1	case right E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊢ ∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)	case right E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ psi.metaVarSet ⊢ IsMetaVarOrAllDefInEnv E (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	apply h2	case right E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ psi.metaVarSet ⊢ IsMetaVarOrAllDefInEnv E (τ X)	case right.a E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ psi.metaVarSet ⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	right	case right.a E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ psi.metaVarSet ⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet	case right.a.h E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ psi.metaVarSet ⊢ X ∈ psi.metaVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	exact a1	case right.a.h E : Env σ : Instantiation τ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi → (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi) h2 : ∀ (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → IsMetaVarOrAllDefInEnv E (τ X) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi X : MetaVarName a1 : X ∈ psi.metaVarSet ⊢ X ∈ psi.metaVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv at h1	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi) h2 : ∀ X ∈ (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ x phi))	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h2 : ∀ X ∈ (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) h1 : IsMetaVarOrAllDefInEnv E phi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ x phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold Formula.metaVarSet at h2	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h2 : ∀ X ∈ (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) h1 : IsMetaVarOrAllDefInEnv E phi ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ x phi))	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ x phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ x phi))	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (forall_ (↑σ x) (sub σ τ phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (forall_ (↑σ x) (sub σ τ phi))	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	exact phi_ih h1 h2	E : Env σ : Instantiation τ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi → (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi) h1 : IsMetaVarOrAllDefInEnv E phi h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv at h1	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (def_ X xs) h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (def_ X xs))	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) h1 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (def_ X xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold sub	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) h1 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (def_ X xs))	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) h1 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊢ IsMetaVarOrAllDefInEnv E (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	unfold IsMetaVarOrAllDefInEnv	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) h1 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊢ IsMetaVarOrAllDefInEnv E (def_ X (List.map (↑σ) xs))	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) h1 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊢ ∃ d ∈ E, X = d.name ∧ (List.map (↑σ) xs).length = d.args.length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.lem_2_a	[1766, 1]	[1837, 13]	simp	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) h1 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊢ ∃ d ∈ E, X = d.name ∧ (List.map (↑σ) xs).length = d.args.length	E : Env σ : Instantiation τ : MetaInstantiation X : DefName xs : List VarName h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1) h1 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊢ ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length

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