Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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stringclasses
147 values
commit
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147 values
file_path
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101
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2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact a2 d
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi β†’ Holds D I (Function.updateITE V h1_x d) M E h1_psi a2 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi d : D ⊒ Holds D I (Function.updateITE V h1_x d) M E h1_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
have s1 : IsNotFree D I M E h1_phi h1_x
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
case s1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ IsNotFree D I M E h1_phi h1_x D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
apply not_free_imp_is_not_free D I M E h1_phi h1_Ξ“ h1_x h1_2
case s1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ IsNotFree D I M E h1_phi h1_x D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
case s1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (X : MetaVarName), (h1_x, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) h1_x D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact nf h1_x
case s1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (X : MetaVarName), (h1_x, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) h1_x D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V a1 a
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x V : Valuation D a1 : Holds D I V M E h1_phi a : D ⊒ Holds D I (Function.updateITE V h1_x a) M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
unfold IsNotFree at s1
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : IsNotFree D I M E h1_phi h1_x V : Valuation D a1 : Holds D I V M E h1_phi a : D ⊒ Holds D I (Function.updateITE V h1_x a) M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : βˆ€ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi V : Valuation D a1 : Holds D I V M E h1_phi a : D ⊒ Holds D I (Function.updateITE V h1_x a) M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [← s1 V a]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : βˆ€ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi V : Valuation D a1 : Holds D I V M E h1_phi a : D ⊒ Holds D I (Function.updateITE V h1_x a) M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : βˆ€ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi V : Valuation D a1 : Holds D I V M E h1_phi a : D ⊒ Holds D I V M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact a1
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F s1 : βˆ€ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi V : Valuation D a1 : Holds D I V M E h1_phi a : D ⊒ Holds D I V M E h1_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
unfold exists_
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (exists_ h1_x (eq_ h1_x h1_y))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (forall_ h1_x (eq_ h1_x h1_y).not_).not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (forall_ h1_x (eq_ h1_x h1_y).not_).not_
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Β¬βˆ€ (d : D), Β¬Function.updateITE V h1_x d h1_x = Function.updateITE V h1_x d h1_y
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Β¬βˆ€ (d : D), Β¬Function.updateITE V h1_x d h1_x = Function.updateITE V h1_x d h1_y
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), βˆƒ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), βˆƒ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ βˆƒ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
apply Exists.intro (V h1_y)
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ βˆƒ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ Function.updateITE V h1_x (V h1_y) h1_x = Function.updateITE V h1_x (V h1_y) h1_y
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
unfold Function.updateITE
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ Function.updateITE V h1_x (V h1_y) h1_x = Function.updateITE V h1_x (V h1_y) h1_y
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ (if h1_x = h1_x then V h1_y else V h1_x) = if h1_y = h1_x then V h1_y else V h1_y
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ (if h1_x = h1_x then V h1_y else V h1_x) = if h1_y = h1_x then V h1_y else V h1_y
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((eq_ h1_x h1_y).imp_ ((eq_ h1_x h1_z).imp_ (eq_ h1_y h1_z)))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), V h1_x = V h1_y β†’ V h1_x = V h1_z β†’ V h1_y = V h1_z
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V a1 a2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), V h1_x = V h1_y β†’ V h1_x = V h1_z β†’ V h1_y = V h1_z
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : V h1_x = V h1_y a2 : V h1_x = V h1_z ⊒ V h1_y = V h1_z
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
trans V h1_x
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : V h1_x = V h1_y a2 : V h1_x = V h1_z ⊒ V h1_y = V h1_z
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : V h1_x = V h1_y a2 : V h1_x = V h1_z ⊒ V h1_y = V h1_x D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : V h1_x = V h1_y a2 : V h1_x = V h1_z ⊒ V h1_x = V h1_z
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [a1]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : V h1_x = V h1_y a2 : V h1_x = V h1_z ⊒ V h1_y = V h1_x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact a2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y h1_z : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : V h1_x = V h1_y a2 : V h1_x = V h1_z ⊒ V h1_x = V h1_z
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
sorry
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_n : β„• h1_name : PredName h1_xs h1_ys : Fin h1_n β†’ VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (eqSubPred h1_name h1_n h1_xs h1_ys)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
obtain βŸ¨Οƒ', a1⟩ := h1_Οƒ.2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ h1_phi)
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id ⊒ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ h1_phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
have s1 : IsMetaVarOrAllDefInEnv E h1_phi := is_proof_imp_is_meta_var_or_all_def_in_env E h1_Ξ“ h1_Ξ” h1_phi h1_4
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id ⊒ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ h1_phi)
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi ⊒ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ h1_phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi ⊒ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ h1_phi)
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ Holds D I V M E (sub h1_Οƒ h1_Ο„ h1_phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [← holds_sub D I V M E h1_Οƒ Οƒ' h1_Ο„ h1_phi s1 a1]
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ Holds D I V M E (sub h1_Οƒ h1_Ο„ h1_phi)
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ Holds D I (V ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
apply h1_ih_2
case intro D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ Holds D I (V ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E h1_phi
case intro.nf D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E (meta_var_ X) v case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro v X a2
case intro.nf D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E (meta_var_ X) v
case intro.nf D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D v : VarName X : MetaVarName a2 : (v, X) ∈ h1_Ξ“ ⊒ IsNotFree D I (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E (meta_var_ X) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact lem_1 D I M E h1_Ξ“ h1_Ξ“' h1_Οƒ Οƒ' h1_Ο„ a1 nf h1_2 v X a2
case intro.nf D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D v : VarName X : MetaVarName a2 : (v, X) ∈ h1_Ξ“ ⊒ IsNotFree D I (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E (meta_var_ X) v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro psi V' a2
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D ⊒ βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E F
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
have s2 : IsMetaVarOrAllDefInEnv E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case s2 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ IsMetaVarOrAllDefInEnv E psi case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
apply lem_2_b E h1_Οƒ h1_Ο„
case s2 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ IsMetaVarOrAllDefInEnv E psi case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case s2.h1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ IsMetaVarOrAllDefInEnv E (sub h1_Οƒ h1_Ο„ psi) case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
apply is_proof_imp_is_meta_var_or_all_def_in_env E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi)
case s2.h1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ IsMetaVarOrAllDefInEnv E (sub h1_Οƒ h1_Ο„ psi) case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case s2.h1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact h1_3 psi a2
case s2.h1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” ⊒ IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
have s3 : βˆ€ V'' : Valuation D, Holds D I (V'' ∘ h1_Οƒ.val) (fun (X' : MetaVarName) (V' : Valuation D) => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case s3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V''
case s3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi ⊒ βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case s3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi V'' : Valuation D ⊒ Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [holds_sub D I V'' M E h1_Οƒ Οƒ' h1_Ο„ psi s2 a1]
case s3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi V'' : Valuation D ⊒ Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case s3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi V'' : Valuation D ⊒ Holds D I V'' M E (sub h1_Οƒ h1_Ο„ psi) case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact h1_ih_1 psi a2 M nf hyp V''
case s3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi V'' : Valuation D ⊒ Holds D I V'' M E (sub h1_Οƒ h1_Ο„ psi) case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
specialize s3 (V' ∘ Οƒ')
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : βˆ€ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I ((V' ∘ Οƒ') ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Function.comp.assoc] at s3
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I ((V' ∘ Οƒ') ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I (V' ∘ Οƒ' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [a1.right] at s3
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I (V' ∘ Οƒ' ∘ ↑h1_Οƒ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I (V' ∘ id) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Function.comp_id] at s3
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I (V' ∘ id) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact s3
case intro.hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ h1_Ξ“' : List (VarName Γ— MetaVarName) h1_Ξ” h1_Ξ”' : List Formula h1_phi : Formula h1_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“' β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ”' β†’ Holds D I V M E F Οƒ' : VarName β†’ VarName a1 : ↑h1_Οƒ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑h1_Οƒ = id s1 : IsMetaVarOrAllDefInEnv E h1_phi V : Valuation D psi : Formula V' : Valuation D a2 : psi ∈ h1_Ξ” s2 : IsMetaVarOrAllDefInEnv E psi s3 : Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi ⊒ Holds D I V' (fun X' V' => Holds D I (V' ∘ Οƒ') M E (h1_Ο„ X')) E psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi'
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ Holds D I V M E h1_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
have s1 : Holds D I V M E h1_phi := h1_ih M nf hyp V
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ Holds D I V M E h1_phi'
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D s1 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [← holds_conv D I V M E h1_phi h1_phi' h2 h1_3]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D s1 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_phi'
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D s1 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact s1
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D s1 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
induction F generalizing vs
F : Formula vs : List VarName v : VarName Ξ“ : List (VarName Γ— MetaVarName) h1 : NoMetaVarAndAllFreeInList vs F h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v F
case meta_var_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) a✝ : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (meta_var_ a✝) case pred_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) a✝¹ : PredName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (pred_ a✝¹ a✝) case eq_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) a✝¹ a✝ : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (eq_ a✝¹ a✝) case true_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v true_ case not_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) a✝ : Formula a_ih✝ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ β†’ v βˆ‰ vs β†’ NotFree Ξ“ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs a✝.not_ h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v a✝.not_ case imp_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝¹ β†’ v βˆ‰ vs β†’ NotFree Ξ“ v a✝¹ a_ih✝ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ β†’ v βˆ‰ vs β†’ NotFree Ξ“ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (a✝¹.imp_ a✝) case forall_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ β†’ v βˆ‰ vs β†’ NotFree Ξ“ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (forall_ a✝¹ a✝) case def_ v : VarName Ξ“ : List (VarName Γ— MetaVarName) a✝¹ : DefName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
case meta_var_ X => unfold NoMetaVarAndAllFreeInList at h1 contradiction
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (meta_var_ X)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
case pred_ X xs => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree tauto
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (pred_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
case true_ => unfold NotFree simp
v : VarName Ξ“ : List (VarName Γ— MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v true_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
case not_ phi phi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree exact phi_ih vs h1 h2
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v phi.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
case def_ X xs => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree tauto
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NoMetaVarAndAllFreeInList at h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (meta_var_ X)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : MetaVarName vs : List VarName h2 : v βˆ‰ vs h1 : False ⊒ NotFree Ξ“ v (meta_var_ X)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
contradiction
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : MetaVarName vs : List VarName h2 : v βˆ‰ vs h1 : False ⊒ NotFree Ξ“ v (meta_var_ X)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NoMetaVarAndAllFreeInList at h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (pred_ X xs)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : PredName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ NotFree Ξ“ v (pred_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NotFree
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : PredName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ NotFree Ξ“ v (pred_ X xs)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : PredName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ v βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
tauto
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : PredName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ v βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NoMetaVarAndAllFreeInList at h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (eq_ x y)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1 : x ∈ vs ∧ y ∈ vs ⊒ NotFree Ξ“ v (eq_ x y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NotFree
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1 : x ∈ vs ∧ y ∈ vs ⊒ NotFree Ξ“ v (eq_ x y)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1 : x ∈ vs ∧ y ∈ vs ⊒ Β¬x = v ∧ Β¬y = v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
cases h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1 : x ∈ vs ∧ y ∈ vs ⊒ Β¬x = v ∧ Β¬y = v
case intro v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs left✝ : x ∈ vs right✝ : y ∈ vs ⊒ Β¬x = v ∧ Β¬y = v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
constructor
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊒ Β¬x = v ∧ Β¬y = v
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊒ Β¬x = v case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊒ Β¬y = v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
intro contra
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊒ Β¬x = v
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
apply h2
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊒ False
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊒ v ∈ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
subst contra
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊒ v ∈ vs
case left Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : x βˆ‰ vs ⊒ x ∈ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
exact h1_left
case left Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : x βˆ‰ vs ⊒ x ∈ vs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
intro contra
case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊒ Β¬y = v
case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
apply h2
case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊒ False
case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊒ v ∈ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
subst contra
case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h2 : v βˆ‰ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊒ v ∈ vs
case right Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : y βˆ‰ vs ⊒ y ∈ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
exact h1_right
case right Ξ“ : List (VarName Γ— MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : y βˆ‰ vs ⊒ y ∈ vs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NotFree
v : VarName Ξ“ : List (VarName Γ— MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v true_
v : VarName Ξ“ : List (VarName Γ— MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v βˆ‰ vs ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
simp
v : VarName Ξ“ : List (VarName Γ— MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v βˆ‰ vs ⊒ True
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NoMetaVarAndAllFreeInList at h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v phi.not_
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊒ NotFree Ξ“ v phi.not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NotFree
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊒ NotFree Ξ“ v phi.not_
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊒ NotFree Ξ“ v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
exact phi_ih vs h1 h2
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊒ NotFree Ξ“ v phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NoMetaVarAndAllFreeInList at h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (phi.imp_ psi)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v (phi.imp_ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NotFree
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v (phi.imp_ psi)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v phi ∧ NotFree Ξ“ v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
cases h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v phi ∧ NotFree Ξ“ v psi
case intro v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs left✝ : NoMetaVarAndAllFreeInList vs phi right✝ : NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v phi ∧ NotFree Ξ“ v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
constructor
v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v phi ∧ NotFree Ξ“ v psi
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v phi case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
exact phi_ih vs h1_left h2
case left v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
exact psi_ih vs h1_right h2
case right v : VarName Ξ“ : List (VarName Γ— MetaVarName) phi psi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi psi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v psi vs : List VarName h2 : v βˆ‰ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊒ NotFree Ξ“ v psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NoMetaVarAndAllFreeInList at h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (forall_ x phi)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊒ NotFree Ξ“ v (forall_ x phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NotFree
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊒ NotFree Ξ“ v (forall_ x phi)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊒ x = v ∨ NotFree Ξ“ v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
by_cases c1 : x = v
v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊒ x = v ∨ NotFree Ξ“ v phi
case pos v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊒ x = v ∨ NotFree Ξ“ v phi case neg v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ x = v ∨ NotFree Ξ“ v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
left
case pos v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊒ x = v ∨ NotFree Ξ“ v phi
case pos.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊒ x = v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
exact c1
case pos.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊒ x = v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
right
case neg v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ x = v ∨ NotFree Ξ“ v phi
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ NotFree Ξ“ v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
apply phi_ih (x :: vs) h1
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ NotFree Ξ“ v phi
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v βˆ‰ x :: vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
simp
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v βˆ‰ x :: vs
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ Β¬v = x ∧ v βˆ‰ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
push_neg
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ Β¬v = x ∧ v βˆ‰ vs
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v β‰  x ∧ v βˆ‰ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
constructor
case neg.h v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v β‰  x ∧ v βˆ‰ vs
case neg.h.left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v β‰  x case neg.h.right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v βˆ‰ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
tauto
case neg.h.left v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v β‰  x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
exact h2
case neg.h.right v : VarName Ξ“ : List (VarName Γ— MetaVarName) x : VarName phi : Formula phi_ih : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi β†’ v βˆ‰ vs β†’ NotFree Ξ“ v phi vs : List VarName h2 : v βˆ‰ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : Β¬x = v ⊒ v βˆ‰ vs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NoMetaVarAndAllFreeInList at h1
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) h2 : v βˆ‰ vs ⊒ NotFree Ξ“ v (def_ X xs)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : DefName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ NotFree Ξ“ v (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
unfold NotFree
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : DefName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ NotFree Ξ“ v (def_ X xs)
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : DefName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ v βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.all_free_in_list_and_not_in_list_imp_not_free
[572, 1]
[641, 10]
tauto
v : VarName Ξ“ : List (VarName Γ— MetaVarName) X : DefName xs vs : List VarName h2 : v βˆ‰ vs h1 : xs βŠ† vs ⊒ v βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.no_meta_var_imp_meta_var_set_is_empty
[644, 1]
[681, 8]
induction F generalizing vs
F : Formula vs : List VarName h1 : NoMetaVarAndAllFreeInList vs F ⊒ F.metaVarSet = βˆ…
case meta_var_ a✝ : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝) ⊒ (meta_var_ a✝).metaVarSet = βˆ… case pred_ a✝¹ : PredName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝) ⊒ (pred_ a✝¹ a✝).metaVarSet = βˆ… case eq_ a✝¹ a✝ : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝) ⊒ (eq_ a✝¹ a✝).metaVarSet = βˆ… case true_ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ ⊒ true_.metaVarSet = βˆ… case not_ a✝ : Formula a_ih✝ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ β†’ a✝.metaVarSet = βˆ… vs : List VarName h1 : NoMetaVarAndAllFreeInList vs a✝.not_ ⊒ a✝.not_.metaVarSet = βˆ… case imp_ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝¹ β†’ a✝¹.metaVarSet = βˆ… a_ih✝ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ β†’ a✝.metaVarSet = βˆ… vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝) ⊒ (a✝¹.imp_ a✝).metaVarSet = βˆ… case forall_ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ β†’ a✝.metaVarSet = βˆ… vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝) ⊒ (forall_ a✝¹ a✝).metaVarSet = βˆ… case def_ a✝¹ : DefName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝) ⊒ (def_ a✝¹ a✝).metaVarSet = βˆ…
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.no_meta_var_imp_meta_var_set_is_empty
[644, 1]
[681, 8]
case meta_var_ X => unfold NoMetaVarAndAllFreeInList at h1 contradiction
X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) ⊒ (meta_var_ X).metaVarSet = βˆ…
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.no_meta_var_imp_meta_var_set_is_empty
[644, 1]
[681, 8]
case pred_ X xs => rfl
X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) ⊒ (pred_ X xs).metaVarSet = βˆ…
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.no_meta_var_imp_meta_var_set_is_empty
[644, 1]
[681, 8]
case eq_ x y => rfl
x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) ⊒ (eq_ x y).metaVarSet = βˆ…
no goals