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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.holds_sub	[1465, 1]	[1577, 15]	simp only [Holds]	D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E (forall_ x phi) ↔ Holds D I V M E (forall_ (↑σ x) (sub σ τ phi))	D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id ⊢ (∀ (d : D), Holds D I (Function.updateITE (V ∘ ↑σ) x d) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V (↑σ x) d) M E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	apply forall_congr'	D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id ⊢ (∀ (d : D), Holds D I (Function.updateITE (V ∘ ↑σ) x d) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V (↑σ x) d) M E (sub σ τ phi)	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id ⊢ ∀ (a : D), Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	intro a	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id ⊢ ∀ (a : D), Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	have s1 : Function.updateITE V (σ.val x) a ∘ σ.val = Function.updateITE (V ∘ σ.val) x a	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)	case s1 D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D ⊢ Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	apply Function.updateITE_comp_right_injective	case s1 D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D ⊢ Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)	case s1.h1 D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D ⊢ Function.Injective ↑σ case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	apply Instantiation.is_injective	case s1.h1 D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D ⊢ Function.Injective ↑σ case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp only [← s1]	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE (V ∘ ↑σ) x a) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE V (↑σ x) a ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	exact phi_ih (Function.updateITE V (σ.val x) a) h1	case h D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation x : VarName phi : Formula phi_ih : ∀ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I V M E (sub σ τ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ σ' = id h2_right : σ' ∘ ↑σ = id a : D s1 : Function.updateITE V (↑σ x) a ∘ ↑σ = Function.updateITE (V ∘ ↑σ) x a ⊢ Holds D I (Function.updateITE V (↑σ x) a ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E phi ↔ Holds D I (Function.updateITE V (↑σ x) a) M E (sub σ τ phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	induction E	D : Type I : Interpretation D M : MetaValuation D E : Env σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (def_ X xs) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E (τ X')) E (def_ X xs) ↔ Holds D I V M E (sub σ τ (def_ X xs))	case nil D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv [] (def_ X xs) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M [] (τ X')) [] (def_ X xs) ↔ Holds D I V M [] (sub σ τ (def_ X xs)) case cons D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D head✝ : Definition tail✝ : List Definition tail_ih✝ : IsMetaVarOrAllDefInEnv tail✝ (def_ X xs) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M tail✝ (τ X')) tail✝ (def_ X xs) ↔ Holds D I V M tail✝ (sub σ τ (def_ X xs))) h1 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (def_ X xs) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (head✝ :: tail✝) (τ X')) (head✝ :: tail✝) (def_ X xs) ↔ Holds D I V M (head✝ :: tail✝) (sub σ τ (def_ X xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	case nil => unfold IsMetaVarOrAllDefInEnv at h1 simp at h1	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv [] (def_ X xs) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M [] (τ X')) [] (def_ X xs) ↔ Holds D I V M [] (sub σ τ (def_ X xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	unfold IsMetaVarOrAllDefInEnv at h1	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv [] (def_ X xs) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M [] (τ X')) [] (def_ X xs) ↔ Holds D I V M [] (sub σ τ (def_ X xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D h1 : ∃ d ∈ [], X = d.name ∧ xs.length = d.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M [] (τ X')) [] (def_ X xs) ↔ Holds D I V M [] (sub σ τ (def_ X xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp at h1	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D h1 : ∃ d ∈ [], X = d.name ∧ xs.length = d.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M [] (τ X')) [] (def_ X xs) ↔ Holds D I V M [] (sub σ τ (def_ X xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	unfold IsMetaVarOrAllDefInEnv at E_ih	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : IsMetaVarOrAllDefInEnv E_tl (def_ X xs) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M (E_hd :: E_tl) (sub σ τ (def_ X xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M (E_hd :: E_tl) (sub σ τ (def_ X xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	have s1 : E_hd.F.metaVarSet = ∅ := no_meta_var_imp_meta_var_set_is_empty E_hd.F E_hd.args E_hd.nf	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M (E_hd :: E_tl) (sub σ τ (def_ X xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M (E_hd :: E_tl) (sub σ τ (def_ X xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	unfold sub	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M (E_hd :: E_tl) (sub σ τ (def_ X xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M (E_hd :: E_tl) (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp only [Holds]	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M (E_hd :: E_tl) (def_ X (List.map (↑σ) xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F else Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map (↑σ) xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map (↑σ) xs))) M E_tl E_hd.F else Holds D I V M E_tl (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F else Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map (↑σ) xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map (↑σ) xs))) M E_tl E_hd.F else Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F else Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F else Holds D I V M E_tl (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	split_ifs	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ ⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F else Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F else Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	case pos D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F case neg D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	case _ c1 => unfold IsMetaVarOrAllDefInEnv at h1 simp at h1 cases h1 case inl c2 => contradiction case inr c2 => unfold sub at E_ih simp only [← E_ih c2] apply holds_coincide_meta_var unfold Formula.metaVarSet simp	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	cases c1	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F	case intro D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ left✝ : X = E_hd.name right✝ : xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	have s2 : Holds D I (Function.updateListITE (V ∘ σ.val) E_hd.args (List.map (V ∘ σ.val) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ σ.val) xs)) M E_tl E_hd.F	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F	case s2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	apply holds_coincide_var D I (Function.updateListITE (V ∘ σ.val) E_hd.args (List.map (V ∘ σ.val) xs)) (Function.updateListITE V E_hd.args (List.map (V ∘ σ.val) xs)) M E_tl E_hd.F E_hd.args E_hd.nf	case s2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F	case s2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊢ ∀ v ∈ E_hd.args, Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs) v D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	intro v a1	case s2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊢ ∀ v ∈ E_hd.args, Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs) v D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F	case s2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.args ⊢ Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs) v D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	apply Function.updateListITE_mem_eq_len	case s2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.args ⊢ Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs) v D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F	case s2.h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.args ⊢ v ∈ E_hd.args case s2.h2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.args ⊢ E_hd.args.length = (List.map (V ∘ ↑σ) xs).length D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp only [← s2]	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	apply holds_coincide_meta_var_no_meta_var	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl E_hd.F ↔ Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F	case h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ E_hd.F.metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	exact s1	case h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length s2 : Holds D I (Function.updateListITE (V ∘ ↑σ) E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ ↑σ) xs)) M E_tl E_hd.F ⊢ E_hd.F.metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	exact a1	case s2.h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : 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.holds_sub	[1465, 1]	[1577, 15]	simp	case s2.h2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.args ⊢ E_hd.args.length = (List.map (V ∘ ↑σ) xs).length	case s2.h2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.args ⊢ E_hd.args.length = xs.length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp only [c1_right]	case s2.h2 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.args ⊢ E_hd.args.length = xs.length	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	unfold IsMetaVarOrAllDefInEnv at h1	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h1 : ∃ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp at h1	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h1 : ∃ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h1 : X = E_hd.name ∧ xs.length = E_hd.args.length ∨ ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	cases h1	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h1 : X = E_hd.name ∧ xs.length = E_hd.args.length ∨ ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	case inl D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs)) case inr D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h✝ : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	case inl c2 => contradiction	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	case inr c2 => unfold sub at E_ih simp only [← E_ih c2] apply holds_coincide_meta_var unfold Formula.metaVarSet simp	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	contradiction	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	unfold sub at E_ih	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (sub σ τ (def_ X xs))) s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp only [← E_ih c2]	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	apply holds_coincide_meta_var	D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X')) E_tl (def_ X xs) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs)	case h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X_1) ↔ Holds D I (V' ∘ σ') M E_tl (τ X_1)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	unfold Formula.metaVarSet	case h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X_1) ↔ Holds D I (V' ∘ σ') M E_tl (τ X_1)	case h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X) ↔ Holds D I (V' ∘ σ') M E_tl (τ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.holds_sub	[1465, 1]	[1577, 15]	simp	case h1 D : Type I : Interpretation D M : MetaValuation D σ : Instantiation σ' : VarName → VarName τ : MetaInstantiation h2 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id X : DefName xs : List VarName V : Valuation D E_hd : Definition E_tl : List Definition s1 : E_hd.F.metaVarSet = ∅ c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : ∃ a ∈ E_tl, X = a.name ∧ xs.length = a.args.length E_ih : (∃ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length) → (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ σ') M E_tl (τ X')) E_tl (def_ X xs) ↔ Holds D I V M E_tl (def_ X (List.map (↑σ) xs))) ⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, Holds D I (V' ∘ σ') M (E_hd :: E_tl) (τ X) ↔ Holds D I (V' ∘ σ') M E_tl (τ X)	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 F	D : Type I : Interpretation D M : MetaValuation D E : Env F : Formula Γ : List (VarName × MetaVarName) v : VarName h1 : NotFree Γ v F h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v ⊢ IsNotFree D I M E F v	case meta_var_ 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 a✝ : MetaVarName h1 : NotFree Γ v (meta_var_ a✝) ⊢ IsNotFree D I M E (meta_var_ a✝) v case pred_ 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 a✝¹ : PredName a✝ : List VarName h1 : NotFree Γ v (pred_ a✝¹ a✝) ⊢ IsNotFree D I M E (pred_ a✝¹ a✝) v case eq_ 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 a✝¹ a✝ : VarName h1 : NotFree Γ v (eq_ a✝¹ a✝) ⊢ IsNotFree D I M E (eq_ a✝¹ a✝) v case true_ 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 h1 : NotFree Γ v true_ ⊢ IsNotFree D I M E true_ v case not_ 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 a✝ : Formula a_ih✝ : NotFree Γ v a✝ → IsNotFree D I M E a✝ v h1 : NotFree Γ v a✝.not_ ⊢ IsNotFree D I M E a✝.not_ v case imp_ 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 a✝¹ a✝ : Formula a_ih✝¹ : NotFree Γ v a✝¹ → IsNotFree D I M E a✝¹ v a_ih✝ : NotFree Γ v a✝ → IsNotFree D I M E a✝ v h1 : NotFree Γ v (a✝¹.imp_ a✝) ⊢ IsNotFree D I M E (a✝¹.imp_ a✝) v case forall_ 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 a✝¹ : VarName a✝ : Formula a_ih✝ : NotFree Γ v a✝ → IsNotFree D I M E a✝ v h1 : NotFree Γ v (forall_ a✝¹ a✝) ⊢ IsNotFree D I M E (forall_ a✝¹ a✝) v case def_ 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 a✝¹ : DefName a✝ : List VarName h1 : NotFree Γ v (def_ a✝¹ a✝) ⊢ IsNotFree D I M E (def_ a✝¹ a✝) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case meta_var_ X => unfold NotFree at h1 exact h2 X h1	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 : MetaVarName h1 : NotFree Γ v (meta_var_ X) ⊢ IsNotFree D I M E (meta_var_ X) 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]	case pred_ X xs => unfold NotFree at h1 unfold IsNotFree simp only [Holds] intro V d congr! 1 apply List.map_congr intro x a1 have s1 : ¬ x = v intro contra subst contra contradiction unfold Function.updateITE simp only [if_neg s1]	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 : PredName xs : List VarName h1 : NotFree Γ v (pred_ X xs) ⊢ IsNotFree D I M E (pred_ 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]	case eq_ x y => unfold NotFree at h1 unfold IsNotFree simp only [Holds] intro V d cases h1 case intro h1_left h1_right => simp only [Function.updateITE] simp only [if_neg h1_left] simp only [if_neg h1_right]	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 y : VarName h1 : NotFree Γ v (eq_ x y) ⊢ IsNotFree D I M E (eq_ x y) 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]	case true_ => unfold IsNotFree intro V d simp only [Holds]	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 h1 : NotFree Γ v true_ ⊢ IsNotFree D I M E true_ 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]	case not_ phi phi_ih => unfold NotFree at h1 unfold IsNotFree at phi_ih unfold IsNotFree intro V d simp only [Holds] congr! 1 exact phi_ih h1 V 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 phi : Formula phi_ih : NotFree Γ v phi → IsNotFree D I M E phi v h1 : NotFree Γ v phi.not_ ⊢ IsNotFree D I M E phi.not_ 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 NotFree at h1	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 : MetaVarName h1 : NotFree Γ v (meta_var_ X) ⊢ IsNotFree D I M E (meta_var_ X) v	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 : MetaVarName h1 : (v, X) ∈ Γ ⊢ IsNotFree D I M E (meta_var_ X) v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	exact h2 X h1	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 : MetaVarName h1 : (v, X) ∈ Γ ⊢ IsNotFree D I M E (meta_var_ X) 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 NotFree at h1	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 : PredName xs : List VarName h1 : NotFree Γ v (pred_ X xs) ⊢ IsNotFree D I M E (pred_ X xs) v	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 : PredName xs : List VarName h1 : v ∉ xs ⊢ IsNotFree D I M E (pred_ 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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v X : PredName xs : List VarName h1 : v ∉ xs ⊢ IsNotFree D I M E (pred_ X xs) v	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 : PredName xs : List VarName h1 : v ∉ xs ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E (pred_ X xs) ↔ Holds D I (Function.updateITE V v d) M E (pred_ 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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v X : PredName xs : List VarName h1 : v ∉ xs ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E (pred_ X xs) ↔ Holds D I (Function.updateITE V v d) M E (pred_ X xs)	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 : PredName xs : List VarName h1 : v ∉ xs ⊢ ∀ (V : Valuation D) (d : D), I X (List.map V xs) ↔ I X (List.map (Function.updateITE V v d) 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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v X : PredName xs : List VarName h1 : v ∉ xs ⊢ ∀ (V : Valuation D) (d : D), I X (List.map V xs) ↔ I X (List.map (Function.updateITE V v d) xs)	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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D ⊢ I X (List.map V xs) ↔ I X (List.map (Function.updateITE V v d) xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	congr! 1	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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D ⊢ I X (List.map V xs) ↔ I X (List.map (Function.updateITE V v d) xs)	case a.h.e'_2 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D ⊢ List.map V xs = List.map (Function.updateITE V v d) xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	apply List.map_congr	case a.h.e'_2 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D ⊢ List.map V xs = List.map (Function.updateITE V v d) xs	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D ⊢ ∀ x ∈ xs, V x = Function.updateITE V v d x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	intro x a1	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D ⊢ ∀ x ∈ xs, V x = Function.updateITE V v d x	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs ⊢ V x = Function.updateITE V v d x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	have s1 : ¬ x = v	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs ⊢ V x = Function.updateITE V v d x	case s1 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs ⊢ ¬x = v case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	intro contra	case s1 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs ⊢ ¬x = v case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x	case s1 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs contra : x = v ⊢ False case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x
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 s1 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs contra : x = v ⊢ False case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x	case s1 D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) X : PredName xs : List VarName V : Valuation D d : D x : VarName a1 : x ∈ xs h2 : ∀ (X : MetaVarName), (x, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) x h1 : x ∉ xs ⊢ False case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	contradiction	case s1 D : Type I : Interpretation D M : MetaValuation D E : Env Γ : List (VarName × MetaVarName) X : PredName xs : List VarName V : Valuation D d : D x : VarName a1 : x ∈ xs h2 : ∀ (X : MetaVarName), (x, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) x h1 : x ∉ xs ⊢ False case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	unfold Function.updateITE	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = Function.updateITE V v d x	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = if x = v then d else V x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [if_neg s1]	case a.h.e'_2.a 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 : PredName xs : List VarName h1 : v ∉ xs V : Valuation D d : D x : VarName a1 : x ∈ xs s1 : ¬x = v ⊢ V x = if x = v then d else V x	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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x y : VarName h1 : NotFree Γ v (eq_ x y) ⊢ IsNotFree D I M E (eq_ x y) v	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 y : VarName h1 : ¬x = v ∧ ¬y = v ⊢ IsNotFree D I M E (eq_ x y) 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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x y : VarName h1 : ¬x = v ∧ ¬y = v ⊢ IsNotFree D I M E (eq_ x y) v	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 y : VarName h1 : ¬x = v ∧ ¬y = v ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E (eq_ x y) ↔ Holds D I (Function.updateITE V v d) M E (eq_ x y)
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x y : VarName h1 : ¬x = v ∧ ¬y = v ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E (eq_ x y) ↔ Holds D I (Function.updateITE V v d) M E (eq_ x y)	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 y : VarName h1 : ¬x = v ∧ ¬y = v ⊢ ∀ (V : Valuation D) (d : D), V x = V y ↔ Function.updateITE V v d x = Function.updateITE V v d y
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v x y : VarName h1 : ¬x = v ∧ ¬y = v ⊢ ∀ (V : Valuation D) (d : D), V x = V y ↔ Function.updateITE V v d x = Function.updateITE V v d y	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 y : VarName h1 : ¬x = v ∧ ¬y = v V : Valuation D d : D ⊢ V x = V y ↔ Function.updateITE V v d x = Function.updateITE V v d y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	cases h1	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 y : VarName h1 : ¬x = v ∧ ¬y = v V : Valuation D d : D ⊢ V x = V y ↔ Function.updateITE V v d x = Function.updateITE V v d y	case intro 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 y : VarName V : Valuation D d : D left✝ : ¬x = v right✝ : ¬y = v ⊢ V x = V y ↔ Function.updateITE V v d x = Function.updateITE V v d y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	case intro h1_left h1_right => simp only [Function.updateITE] simp only [if_neg h1_left] simp only [if_neg h1_right]	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 y : VarName V : Valuation D d : D h1_left : ¬x = v h1_right : ¬y = v ⊢ V x = V y ↔ Function.updateITE V v d x = Function.updateITE V v d y	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]	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 y : VarName V : Valuation D d : D h1_left : ¬x = v h1_right : ¬y = v ⊢ V x = V y ↔ Function.updateITE V v d x = Function.updateITE V v d y	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 y : VarName V : Valuation D d : D h1_left : ¬x = v h1_right : ¬y = v ⊢ V x = V y ↔ (if x = v then d else V x) = if y = v then d else V y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [if_neg h1_left]	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 y : VarName V : Valuation D d : D h1_left : ¬x = v h1_right : ¬y = v ⊢ V x = V y ↔ (if x = v then d else V x) = if y = v then d else V y	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 y : VarName V : Valuation D d : D h1_left : ¬x = v h1_right : ¬y = v ⊢ V x = V y ↔ V x = if y = v then d else V y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	simp only [if_neg h1_right]	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 y : VarName V : Valuation D d : D h1_left : ¬x = v h1_right : ¬y = v ⊢ V x = V y ↔ V x = if y = v then d else V y	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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v h1 : NotFree Γ v true_ ⊢ IsNotFree D I M E true_ v	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 h1 : NotFree Γ v true_ ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E true_ ↔ Holds D I (Function.updateITE V v d) M E true_
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v h1 : NotFree Γ v true_ ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E true_ ↔ Holds D I (Function.updateITE V v d) M E true_	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 h1 : NotFree Γ v true_ V : Valuation D d : D ⊢ Holds D I V M E true_ ↔ Holds D I (Function.updateITE V v d) M E true_
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v h1 : NotFree Γ v true_ V : Valuation D d : D ⊢ Holds D I V M E true_ ↔ Holds D I (Function.updateITE V v d) M E true_	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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v phi : Formula phi_ih : NotFree Γ v phi → IsNotFree D I M E phi v h1 : NotFree Γ v phi.not_ ⊢ IsNotFree D I M E phi.not_ v	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 phi : Formula phi_ih : NotFree Γ v phi → IsNotFree D I M E phi v h1 : NotFree Γ v phi ⊢ IsNotFree D I M E phi.not_ 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 phi_ih	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 phi : Formula phi_ih : NotFree Γ v phi → IsNotFree D I M E phi v h1 : NotFree Γ v phi ⊢ IsNotFree D I M E phi.not_ v	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 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 : NotFree Γ v phi ⊢ IsNotFree D I M E phi.not_ 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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v 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 : NotFree Γ v phi ⊢ IsNotFree D I M E phi.not_ v	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 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 : NotFree Γ v phi ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E phi.not_ ↔ Holds D I (Function.updateITE V v d) M E phi.not_
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v 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 : NotFree Γ v phi ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E phi.not_ ↔ Holds D I (Function.updateITE V v d) M E phi.not_	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 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 : NotFree Γ v phi V : Valuation D d : D ⊢ Holds D I V M E phi.not_ ↔ Holds D I (Function.updateITE V v d) M E phi.not_
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v 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 : NotFree Γ v phi V : Valuation D d : D ⊢ Holds D I V M E phi.not_ ↔ Holds D I (Function.updateITE V v d) M E phi.not_	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 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 : 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
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	congr! 1	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 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 : 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	case a.h.e'_1.a 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 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 : 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
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 h1 V d	case a.h.e'_1.a 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 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 : 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	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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v phi psi : Formula phi_ih : NotFree Γ v phi → IsNotFree D I M E phi v psi_ih : NotFree Γ v psi → IsNotFree D I M E psi v h1 : NotFree Γ v (phi.imp_ psi) ⊢ IsNotFree D I M E (phi.imp_ psi) v	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 phi psi : Formula phi_ih : NotFree Γ v phi → IsNotFree D I M E phi v psi_ih : NotFree Γ v psi → IsNotFree D I M E psi v h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ IsNotFree D I M E (phi.imp_ psi) 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 phi_ih	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 phi psi : Formula phi_ih : NotFree Γ v phi → IsNotFree D I M E phi v psi_ih : NotFree Γ v psi → IsNotFree D I M E psi v h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ IsNotFree D I M E (phi.imp_ psi) v	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 phi psi : 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 psi_ih : NotFree Γ v psi → IsNotFree D I M E psi v h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ IsNotFree D I M E (phi.imp_ psi) 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 psi_ih	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 phi psi : 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 psi_ih : NotFree Γ v psi → IsNotFree D I M E psi v h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ IsNotFree D I M E (phi.imp_ psi) v	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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ IsNotFree D I M E (phi.imp_ psi) 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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ IsNotFree D I M E (phi.imp_ psi) v	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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E (phi.imp_ psi) ↔ Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi ⊢ ∀ (V : Valuation D) (d : D), Holds D I V M E (phi.imp_ psi) ↔ Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)	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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi V : Valuation D d : D ⊢ Holds D I V M E (phi.imp_ psi) ↔ Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)
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 E : Env Γ : List (VarName × MetaVarName) v : VarName h2 : ∀ (X : MetaVarName), (v, X) ∈ Γ → IsNotFree D I M E (meta_var_ X) v phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi V : Valuation D d : D ⊢ Holds D I V M E (phi.imp_ psi) ↔ Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)	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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi V : Valuation D d : D ⊢ Holds D I V M E phi → Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E phi → Holds D I (Function.updateITE V v d) M E psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	cases h1	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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi h1 : NotFree Γ v phi ∧ NotFree Γ v psi V : Valuation D d : D ⊢ Holds D I V M E phi → Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E phi → Holds D I (Function.updateITE V v d) M E psi	case intro 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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi V : Valuation D d : D left✝ : NotFree Γ v phi right✝ : NotFree Γ v psi ⊢ Holds D I V M E phi → Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E phi → Holds D I (Function.updateITE V v d) M E psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.not_free_imp_is_not_free	[1604, 1]	[1734, 26]	congr! 1	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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi V : Valuation D d : D h1_left : NotFree Γ v phi h1_right : NotFree Γ v psi ⊢ Holds D I V M E phi → Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E phi → Holds D I (Function.updateITE V v d) M E psi	case a.h.a 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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi V : Valuation D d : D h1_left : NotFree Γ v phi h1_right : NotFree Γ v psi ⊢ Holds D I V M E phi ↔ Holds D I (Function.updateITE V v d) M E phi case a.h'.a 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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi V : Valuation D d : D h1_left : NotFree Γ v phi h1_right : NotFree Γ v psi a✝ : Holds D I (Function.updateITE V v d) M E phi ⊢ Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi
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 h1_left V d	case a.h.a 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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi V : Valuation D d : D h1_left : NotFree Γ v phi h1_right : NotFree Γ v psi ⊢ Holds D I V M E phi ↔ Holds D I (Function.updateITE V 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]	exact psi_ih h1_right V d	case a.h'.a 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 phi psi : 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 psi_ih : NotFree Γ v psi → ∀ (V : Valuation D) (d : D), Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi V : Valuation D d : D h1_left : NotFree Γ v phi h1_right : NotFree Γ v psi a✝ : Holds D I (Function.updateITE V v d) M E phi ⊢ Holds D I V M E psi ↔ Holds D I (Function.updateITE V v d) M E psi	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 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 → IsNotFree D I M E phi v h1 : NotFree Γ v (forall_ x phi) ⊢ IsNotFree D I M E (forall_ x phi) v	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 → IsNotFree D I M E phi v h1 : x = v ∨ NotFree Γ v phi ⊢ IsNotFree D I M E (forall_ x phi) 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 phi_ih	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 → IsNotFree D I M E phi v h1 : x = v ∨ NotFree Γ v phi ⊢ IsNotFree D I M E (forall_ x phi) v	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 ⊢ IsNotFree D I M E (forall_ x phi) 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 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 ⊢ IsNotFree D I M E (forall_ x phi) v	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), Holds D I V M E (forall_ x phi) ↔ Holds D I (Function.updateITE V v d) M E (forall_ x phi)
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 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), Holds D I V M E (forall_ x phi) ↔ Holds D I (Function.updateITE V v d) M E (forall_ x 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 h1 : x = v ∨ NotFree Γ v phi V : Valuation D d : D ⊢ Holds D I V M E (forall_ x phi) ↔ Holds D I (Function.updateITE V v d) M E (forall_ x 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 [Holds]	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 ⊢ Holds D I V M E (forall_ x phi) ↔ Holds D I (Function.updateITE V v d) M E (forall_ x 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 h1 : x = v ∨ NotFree Γ v phi V : Valuation D d : D ⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) M E phi) ↔ ∀ (d_1 : D), Holds D I (Function.updateITE (Function.updateITE V v d) x d_1) 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]	apply forall_congr'	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 : D), Holds D I (Function.updateITE V x d) M E phi) ↔ ∀ (d_1 : D), Holds D I (Function.updateITE (Function.updateITE V v d) x d_1) M E phi	case h 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 ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x a) 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]	intro d'	case h 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 ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) M E phi ↔ Holds D I (Function.updateITE (Function.updateITE V v d) x a) M E phi	case h 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 ⊢ 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]	by_cases c1 : x = v	case h 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 ⊢ 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 pos 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 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
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 pos 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 pos.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 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 pos.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

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