Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
start
stringlengths
6
10
end
stringlengths
6
11
tactic
stringlengths
1
11.2k
state_before
stringlengths
3
2.09M
state_after
stringlengths
6
2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp only [Holds]
Ξ” : List Formula F : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E (false_.iff_ true_.not_)
Ξ” : List Formula F : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ False ↔ Β¬True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
tauto
Ξ” : List Formula F : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ False ↔ Β¬True
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
intro D I V E _
Ξ” : List Formula F phi psi : Formula ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ [], Holds D I V E H) β†’ Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_)
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp only [Holds]
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_)
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E phi ∧ Holds D I V E psi ↔ Β¬(Holds D I V E phi β†’ Β¬Holds D I V E psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
tauto
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E phi ∧ Holds D I V E psi ↔ Β¬(Holds D I V E phi β†’ Β¬Holds D I V E psi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
intro D I V E _
Ξ” : List Formula F phi psi : Formula ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ [], Holds D I V E H) β†’ Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi))
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp only [Holds]
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi))
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E phi ∨ Holds D I V E psi ↔ Β¬Holds D I V E phi β†’ Holds D I V E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
tauto
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E phi ∨ Holds D I V E psi ↔ Β¬Holds D I V E phi β†’ Holds D I V E psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
intro D I V E _
Ξ” : List Formula F phi psi : Formula ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ [], Holds D I V E H) β†’ Holds D I V E (((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_ (((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ psi)).not_).not_
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E (((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_ (((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ psi)).not_).not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp only [Holds]
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E (((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_ (((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ psi)).not_).not_
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Β¬(((Holds D I V E phi ↔ Holds D I V E psi) β†’ Β¬((Holds D I V E phi β†’ Holds D I V E psi) β†’ Β¬(Holds D I V E psi β†’ Holds D I V E phi))) β†’ Β¬(Β¬((Holds D I V E phi β†’ Holds D I V E psi) β†’ Β¬(Holds D I V E psi β†’ Holds D I V E phi)) β†’ (Holds D I V E phi ↔ Holds D I V E psi)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
tauto
Ξ” : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Β¬(((Holds D I V E phi ↔ Holds D I V E psi) β†’ Β¬((Holds D I V E phi β†’ Holds D I V E psi) β†’ Β¬(Holds D I V E psi β†’ Holds D I V E phi))) β†’ Β¬(Β¬((Holds D I V E phi β†’ Holds D I V E psi) β†’ Β¬(Holds D I V E psi β†’ Holds D I V E phi)) β†’ (Holds D I V E phi ↔ Holds D I V E psi)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
intro D I V E _
Ξ” : List Formula F : Formula v : VarName phi : Formula ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ [], Holds D I V E H) β†’ Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_)
Ξ” : List Formula F : Formula v : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp only [Holds]
Ξ” : List Formula F : Formula v : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_)
Ξ” : List Formula F : Formula v : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ (βˆƒ d, Holds D I (Function.updateITE V v d) E phi) ↔ Β¬βˆ€ (d : D), Β¬Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp
Ξ” : List Formula F : Formula v : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : βˆ€ H ∈ [], Holds D I V E H ⊒ (βˆƒ d, Holds D I (Function.updateITE V v d) E phi) ↔ Β¬βˆ€ (d : D), Β¬Holds D I (Function.updateITE V v d) E phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
intro D I V E a1
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ List.map (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„) Ξ”', Holds D I V E H) β†’ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : βˆ€ H ∈ List.map (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„) Ξ”', Holds D I V E H ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp at a1
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : βˆ€ H ∈ List.map (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„) Ξ”', Holds D I V E H ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : βˆ€ a ∈ Ξ”', Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ a) ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
obtain s1 := Sub.Pred.All.Rec.Option.Fresh.substitution_theorem D I V E freshChar Ο„
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : βˆ€ a ∈ Ξ”', Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ a) ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : βˆ€ a ∈ Ξ”', Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ a) s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp only [← s1] at a1
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : βˆ€ a ∈ Ξ”', Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ a) s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) a1 : βˆ€ a ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E a ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
simp only [← s1]
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) a1 : βˆ€ a ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E a ⊒ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ phi)
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) a1 : βˆ€ a ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E a ⊒ Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
apply ih_2
Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) a1 : βˆ€ a ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E a ⊒ Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E phi
case a Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) a1 : βˆ€ a ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E a ⊒ βˆ€ H ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.soundness
[793, 1]
[979, 13]
exact a1
case a Ξ” : List Formula F : Formula Ξ”' : List Formula phi : Formula Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : IsDeduct Ξ”' phi ih_2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (βˆ€ H ∈ Ξ”', Holds D I V E H) β†’ Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env s1 : βˆ€ (F : Formula), Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E F ↔ Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο„ F) a1 : βˆ€ a ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E a ⊒ βˆ€ H ∈ Ξ”', Holds D (Sub.Pred.All.Rec.Option.Fresh.I' D I V E Ο„) V E H
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.not_IsDeduct_false
[982, 1]
[988, 26]
intro contra
⊒ ¬IsDeduct [] false_
contra : IsDeduct [] false_ ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.not_IsDeduct_false
[982, 1]
[988, 26]
obtain s1 := soundness [] false_ contra Unit default default default
contra : IsDeduct [] false_ ⊒ False
contra : IsDeduct [] false_ s1 : (βˆ€ H ∈ [], Holds Unit default default default H) β†’ Holds Unit default default default false_ ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.not_IsDeduct_false
[982, 1]
[988, 26]
simp at s1
contra : IsDeduct [] false_ s1 : (βˆ€ H ∈ [], Holds Unit default default default H) β†’ Holds Unit default default default false_ ⊒ False
contra : IsDeduct [] false_ s1 : Holds Unit default default default false_ ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Program/Backend.lean
FOL.NV.Program.Backend.not_IsDeduct_false
[982, 1]
[988, 26]
simp only [Holds] at s1
contra : IsDeduct [] false_ s1 : Holds Unit default default default false_ ⊒ False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
induction F generalizing V
D : Type I : Interpretation D V : VarAssignment D E : Env F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ ⊒ Holds D I (V ∘ Οƒ) E F ↔ Holds D I V E (replaceAll Οƒ F)
case pred_const_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_const_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (pred_const_ a✝¹ a✝)) case pred_var_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_var_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (pred_var_ a✝¹ a✝)) case eq_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (eq_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (eq_ a✝¹ a✝)) case true_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E true_ ↔ Holds D I V E (replaceAll Οƒ true_) case false_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E false_ ↔ Holds D I V E (replaceAll Οƒ false_) case not_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E a✝.not_ ↔ Holds D I V E (replaceAll Οƒ a✝.not_) case imp_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.imp_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.imp_ a✝)) case and_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.and_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.and_ a✝)) case or_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.or_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.or_ a✝)) case iff_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.iff_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.iff_ a✝)) case forall_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (forall_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (forall_ a✝¹ a✝)) case exists_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (exists_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (exists_ a✝¹ a✝)) case def_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : DefName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
all_goals simp only [replaceAll]
case pred_const_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_const_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (pred_const_ a✝¹ a✝)) case pred_var_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_var_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (pred_var_ a✝¹ a✝)) case eq_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (eq_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (eq_ a✝¹ a✝)) case true_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E true_ ↔ Holds D I V E (replaceAll Οƒ true_) case false_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E false_ ↔ Holds D I V E (replaceAll Οƒ false_) case not_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E a✝.not_ ↔ Holds D I V E (replaceAll Οƒ a✝.not_) case imp_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.imp_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.imp_ a✝)) case and_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.and_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.and_ a✝)) case or_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.or_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.or_ a✝)) case iff_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.iff_ a✝) ↔ Holds D I V E (replaceAll Οƒ (a✝¹.iff_ a✝)) case forall_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (forall_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (forall_ a✝¹ a✝)) case exists_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (exists_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (exists_ a✝¹ a✝)) case def_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : DefName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (def_ a✝¹ a✝))
case pred_const_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_const_ a✝¹ a✝) ↔ Holds D I V E (pred_const_ a✝¹ (List.map Οƒ a✝)) case pred_var_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_var_ a✝¹ a✝) ↔ Holds D I V E (pred_var_ a✝¹ (List.map Οƒ a✝)) case eq_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (eq_ a✝¹ a✝) ↔ Holds D I V E (eq_ (Οƒ a✝¹) (Οƒ a✝)) case true_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E true_ ↔ Holds D I V E true_ case false_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E false_ ↔ Holds D I V E false_ case not_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E a✝.not_ ↔ Holds D I V E (replaceAll Οƒ a✝).not_ case imp_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.imp_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).imp_ (replaceAll Οƒ a✝)) case and_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.and_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).and_ (replaceAll Οƒ a✝)) case or_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.or_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).or_ (replaceAll Οƒ a✝)) case iff_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.iff_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).iff_ (replaceAll Οƒ a✝)) case forall_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (forall_ a✝¹ a✝) ↔ Holds D I V E (forall_ (Οƒ a✝¹) (replaceAll Οƒ a✝)) case exists_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ (Οƒ a✝¹) (replaceAll Οƒ a✝)) case def_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : DefName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ (List.map Οƒ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
any_goals simp only [Holds]
case pred_const_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_const_ a✝¹ a✝) ↔ Holds D I V E (pred_const_ a✝¹ (List.map Οƒ a✝)) case pred_var_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (pred_var_ a✝¹ a✝) ↔ Holds D I V E (pred_var_ a✝¹ (List.map Οƒ a✝)) case eq_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (eq_ a✝¹ a✝) ↔ Holds D I V E (eq_ (Οƒ a✝¹) (Οƒ a✝)) case true_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E true_ ↔ Holds D I V E true_ case false_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E false_ ↔ Holds D I V E false_ case not_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E a✝.not_ ↔ Holds D I V E (replaceAll Οƒ a✝).not_ case imp_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.imp_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).imp_ (replaceAll Οƒ a✝)) case and_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.and_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).and_ (replaceAll Οƒ a✝)) case or_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.or_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).or_ (replaceAll Οƒ a✝)) case iff_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (a✝¹.iff_ a✝) ↔ Holds D I V E ((replaceAll Οƒ a✝¹).iff_ (replaceAll Οƒ a✝)) case forall_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (forall_ a✝¹ a✝) ↔ Holds D I V E (forall_ (Οƒ a✝¹) (replaceAll Οƒ a✝)) case exists_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ (Οƒ a✝¹) (replaceAll Οƒ a✝)) case def_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : DefName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ (List.map Οƒ a✝))
case pred_const_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ I.pred_const_ a✝¹ (List.map (V ∘ Οƒ) a✝) ↔ I.pred_const_ a✝¹ (List.map V (List.map Οƒ a✝)) case pred_var_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : PredName a✝ : List VarName V : VarAssignment D ⊒ I.pred_var_ a✝¹ (List.map (V ∘ Οƒ) a✝) ↔ I.pred_var_ a✝¹ (List.map V (List.map Οƒ a✝)) case eq_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : VarName V : VarAssignment D ⊒ (V ∘ Οƒ) a✝¹ = (V ∘ Οƒ) a✝ ↔ V (Οƒ a✝¹) = V (Οƒ a✝) case not_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Β¬Holds D I (V ∘ Οƒ) E a✝ ↔ Β¬Holds D I V E (replaceAll Οƒ a✝) case imp_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E a✝¹ β†’ Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝¹) β†’ Holds D I V E (replaceAll Οƒ a✝) case and_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E a✝¹ ∧ Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝¹) ∧ Holds D I V E (replaceAll Οƒ a✝) case or_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E a✝¹ ∨ Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝¹) ∨ Holds D I V E (replaceAll Οƒ a✝) case iff_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I V E (replaceAll Οƒ a✝¹) a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ (Holds D I (V ∘ Οƒ) E a✝¹ ↔ Holds D I (V ∘ Οƒ) E a✝) ↔ (Holds D I V E (replaceAll Οƒ a✝¹) ↔ Holds D I V E (replaceAll Οƒ a✝)) case forall_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE (V ∘ Οƒ) a✝¹ d) E a✝) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V (Οƒ a✝¹) d) E (replaceAll Οƒ a✝) case exists_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ (βˆƒ d, Holds D I (Function.updateITE (V ∘ Οƒ) a✝¹ d) E a✝) ↔ βˆƒ d, Holds D I (Function.updateITE V (Οƒ a✝¹) d) E (replaceAll Οƒ a✝) case def_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : DefName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ (List.map Οƒ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
case pred_const_ X xs | pred_var_ X xs => simp
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : PredName xs : List VarName V : VarAssignment D ⊒ I.pred_var_ X (List.map (V ∘ Οƒ) xs) ↔ I.pred_var_ X (List.map V (List.map Οƒ xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
case eq_ x y => simp
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x y : VarName V : VarAssignment D ⊒ (V ∘ Οƒ) x = (V ∘ Οƒ) y ↔ V (Οƒ x) = V (Οƒ y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
case not_ phi phi_ih => congr! 1 exact phi_ih V
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ Β¬Holds D I (V ∘ Οƒ) E phi ↔ Β¬Holds D I V E (replaceAll Οƒ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
case forall_ x phi phi_ih | exists_ x phi phi_ih => first | apply forall_congr' | apply exists_congr intro a have s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a apply Function.updateITE_comp_right_injective apply h1 simp only [← s1] exact phi_ih (Function.updateITE V (Οƒ x) a)
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ (βˆƒ d, Holds D I (Function.updateITE (V ∘ Οƒ) x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V (Οƒ x) d) E (replaceAll Οƒ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [replaceAll]
case def_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : DefName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ a✝¹ a✝) ↔ Holds D I V E (replaceAll Οƒ (def_ a✝¹ a✝))
case def_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : DefName a✝ : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ (List.map Οƒ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [Holds]
case exists_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ (Οƒ a✝¹) (replaceAll Οƒ a✝))
case exists_ D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E a✝ ↔ Holds D I V E (replaceAll Οƒ a✝) V : VarAssignment D ⊒ (βˆƒ d, Holds D I (Function.updateITE (V ∘ Οƒ) a✝¹ d) E a✝) ↔ βˆƒ d, Holds D I (Function.updateITE V (Οƒ a✝¹) d) E (replaceAll Οƒ a✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : PredName xs : List VarName V : VarAssignment D ⊒ I.pred_var_ X (List.map (V ∘ Οƒ) xs) ↔ I.pred_var_ X (List.map V (List.map Οƒ xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x y : VarName V : VarAssignment D ⊒ (V ∘ Οƒ) x = (V ∘ Οƒ) y ↔ V (Οƒ x) = V (Οƒ y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
congr! 1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ Β¬Holds D I (V ∘ Οƒ) E phi ↔ Β¬Holds D I V E (replaceAll Οƒ phi)
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
exact phi_ih V
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
congr! 1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) psi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E psi ↔ Holds D I V E (replaceAll Οƒ psi) V : VarAssignment D ⊒ (Holds D I (V ∘ Οƒ) E phi ↔ Holds D I (V ∘ Οƒ) E psi) ↔ (Holds D I V E (replaceAll Οƒ phi) ↔ Holds D I V E (replaceAll Οƒ psi))
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) psi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E psi ↔ Holds D I V E (replaceAll Οƒ psi) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) case a.h.e'_2.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) psi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E psi ↔ Holds D I V E (replaceAll Οƒ psi) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E psi ↔ Holds D I V E (replaceAll Οƒ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
exact phi_ih V
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) psi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E psi ↔ Holds D I V E (replaceAll Οƒ psi) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
exact psi_ih V
case a.h.e'_2.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) psi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E psi ↔ Holds D I V E (replaceAll Οƒ psi) V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E psi ↔ Holds D I V E (replaceAll Οƒ psi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
first | apply forall_congr' | apply exists_congr
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ (βˆƒ d, Holds D I (Function.updateITE (V ∘ Οƒ) x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V (Οƒ x) d) E (replaceAll Οƒ phi)
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ βˆ€ (a : D), Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
intro a
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ βˆ€ (a : D), Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
have s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D ⊒ Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply Function.updateITE_comp_right_injective
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D ⊒ Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
case s1.h1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D ⊒ Function.Injective Οƒ case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply h1
case s1.h1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D ⊒ Function.Injective Οƒ case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [← s1]
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE V (Οƒ x) a ∘ Οƒ) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
exact phi_ih (Function.updateITE V (Οƒ x) a)
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D a : D s1 : Function.updateITE V (Οƒ x) a ∘ Οƒ = Function.updateITE (V ∘ Οƒ) x a ⊒ Holds D I (Function.updateITE V (Οƒ x) a ∘ Οƒ) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply forall_congr'
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE (V ∘ Οƒ) x d) E phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V (Οƒ x) d) E (replaceAll Οƒ phi)
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ βˆ€ (a : D), Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply exists_congr
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ (βˆƒ d, Holds D I (Function.updateITE (V ∘ Οƒ) x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V (Οƒ x) d) E (replaceAll Οƒ phi)
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D), Holds D I (V ∘ Οƒ) E phi ↔ Holds D I V E (replaceAll Οƒ phi) V : VarAssignment D ⊒ βˆ€ (a : D), Holds D I (Function.updateITE (V ∘ Οƒ) x a) E phi ↔ Holds D I (Function.updateITE V (Οƒ x) a) E (replaceAll Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
induction E
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) E (def_ X xs) ↔ Holds D I V E (def_ X (List.map Οƒ xs))
case nil D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) [] (def_ X xs) ↔ Holds D I V [] (def_ X (List.map Οƒ xs)) case cons D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D head✝ : Definition tail✝ : List Definition tail_ih✝ : Holds D I (V ∘ Οƒ) tail✝ (def_ X xs) ↔ Holds D I V tail✝ (def_ X (List.map Οƒ xs)) ⊒ Holds D I (V ∘ Οƒ) (head✝ :: tail✝) (def_ X xs) ↔ Holds D I V (head✝ :: tail✝) (def_ X (List.map Οƒ xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
case nil => simp only [Holds]
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) [] (def_ X xs) ↔ Holds D I V [] (def_ X (List.map Οƒ xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [Holds]
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D ⊒ Holds D I (V ∘ Οƒ) [] (def_ X xs) ↔ Holds D I V [] (def_ X (List.map Οƒ xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [Holds]
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ Holds D I (V ∘ Οƒ) (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V (E_hd :: E_tl) (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D I (V ∘ Οƒ) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D I (V ∘ Οƒ) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D I (V ∘ Οƒ) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
split_ifs
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D I (V ∘ Οƒ) E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
case pos D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length ⊒ Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q case neg D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) h✝ : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊒ Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
case _ c1 => apply E_ih
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊒ Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
cases c1
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊒ Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
case intro D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) left✝ : X = E_hd.name right✝ : xs.length = E_hd.args.length ⊒ Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply Holds_coincide_Var
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊒ Holds D I (Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊒ βˆ€ (v : VarName), isFreeIn v E_hd.q β†’ Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
intro v a1
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length ⊒ βˆ€ (v : VarName), isFreeIn v E_hd.q β†’ Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : isFreeIn v E_hd.q ⊒ Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [isFreeIn_iff_mem_freeVarSet v E_hd.q] at a1
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : isFreeIn v E_hd.q ⊒ Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply Function.updateListITE_mem_eq_len
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ Function.updateListITE (V ∘ Οƒ) E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ v ∈ E_hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ E_hd.args.length = (List.map (V ∘ Οƒ) xs).length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [<- List.mem_toFinset]
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ v ∈ E_hd.args
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ v ∈ E_hd.args.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply Finset.mem_of_subset E_hd.h1 a1
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ v ∈ E_hd.args.toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ E_hd.args.length = (List.map (V ∘ Οƒ) xs).length
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ E_hd.args.length = xs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
simp only [c1_right]
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1_left : X = E_hd.name c1_right : xs.length = E_hd.args.length v : VarName a1 : v ∈ E_hd.q.freeVarSet ⊒ E_hd.args.length = xs.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem
[33, 1]
[96, 19]
apply E_ih
D : Type I : Interpretation D Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ X : DefName xs : List VarName V : VarAssignment D E_hd : Definition E_tl : List Definition E_ih : Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊒ Holds D I (V ∘ Οƒ) E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid
[99, 1]
[111, 13]
simp only [IsValid] at h2
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : F.IsValid ⊒ (replaceAll Οƒ F).IsValid
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ (replaceAll Οƒ F).IsValid
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid
[99, 1]
[111, 13]
simp only [IsValid]
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ (replaceAll Οƒ F).IsValid
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replaceAll Οƒ F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid
[99, 1]
[111, 13]
intro D I V E
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replaceAll Οƒ F)
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊒ Holds D I V E (replaceAll Οƒ F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid
[99, 1]
[111, 13]
simp only [← substitution_theorem D I V E F Οƒ h1]
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊒ Holds D I V E (replaceAll Οƒ F)
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊒ Holds D I (V ∘ Οƒ) E F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean
FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid
[99, 1]
[111, 13]
apply h2
F : Formula Οƒ : VarName β†’ VarName h1 : Function.Injective Οƒ h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊒ Holds D I (V ∘ Οƒ) E F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
induction E generalizing F binders V V' Οƒ Οƒ'
D : Type I : Interpretation D V V' : VarAssignment D E : Env Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName F : Formula h1 : admitsAux Οƒ binders F h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V E F ↔ Holds D I V' E (fastReplaceFree Οƒ' F)
case nil D : Type I : Interpretation D V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName F : Formula h1 : admitsAux Οƒ binders F h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V [] F ↔ Holds D I V' [] (fastReplaceFree Οƒ' F) case cons D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName F : Formula h1 : admitsAux Οƒ binders F h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
induction F generalizing binders V V' Οƒ Οƒ'
case cons D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName F : Formula h1 : admitsAux Οƒ binders F h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' F)
case cons.pred_const_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (pred_const_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (pred_const_ a✝¹ a✝)) case cons.pred_var_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (pred_var_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (pred_var_ a✝¹ a✝)) case cons.eq_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (eq_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (eq_ a✝¹ a✝)) case cons.true_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders true_ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' true_) case cons.false_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders false_ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' false_) case cons.not_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders a✝.not_ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝.not_) case cons.imp_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.imp_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.imp_ a✝)) case cons.and_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.and_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.and_ a✝)) case cons.or_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.or_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.or_ a✝)) case cons.iff_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.iff_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.iff_ a✝)) case cons.forall_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (forall_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (forall_ a✝¹ a✝)) case cons.exists_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (exists_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (exists_ a✝¹ a✝)) case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (def_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
all_goals simp only [admitsAux] at h1 simp only [fastReplaceFree] simp only [Holds]
case cons.pred_const_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (pred_const_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (pred_const_ a✝¹ a✝)) case cons.pred_var_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (pred_var_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (pred_var_ a✝¹ a✝)) case cons.eq_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (eq_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (eq_ a✝¹ a✝)) case cons.true_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders true_ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' true_) case cons.false_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders false_ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' false_) case cons.not_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders a✝.not_ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝.not_) case cons.imp_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.imp_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.imp_ a✝)) case cons.and_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.and_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.and_ a✝)) case cons.or_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.or_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.or_ a✝)) case cons.iff_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (a✝¹.iff_ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (a✝¹.iff_ a✝)) case cons.forall_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (forall_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (forall_ a✝¹ a✝)) case cons.exists_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (exists_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (exists_ a✝¹ a✝)) case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (def_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (def_ a✝¹ a✝))
case cons.pred_const_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ I.pred_const_ a✝¹ (List.map V a✝) ↔ I.pred_const_ a✝¹ (List.map V' (List.map Οƒ' a✝)) case cons.pred_var_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ I.pred_var_ a✝¹ (List.map V a✝) ↔ I.pred_var_ a✝¹ (List.map V' (List.map Οƒ' a✝)) case cons.eq_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : (a✝¹ βˆ‰ binders β†’ Οƒ a✝¹ βˆ‰ binders) ∧ (a✝ βˆ‰ binders β†’ Οƒ a✝ βˆ‰ binders) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ V a✝¹ = V a✝ ↔ V' (Οƒ' a✝¹) = V' (Οƒ' a✝) case cons.not_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders a✝ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Β¬Holds D I V (head✝ :: tail✝) a✝ ↔ Β¬Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝) case cons.imp_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders a✝¹ ∧ admitsAux Οƒ binders a✝ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) a✝¹ β†’ Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹) β†’ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝) case cons.and_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders a✝¹ ∧ admitsAux Οƒ binders a✝ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) a✝¹ ∧ Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹) ∧ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝) case cons.or_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders a✝¹ ∧ admitsAux Οƒ binders a✝ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) a✝¹ ∨ Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹) ∨ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝) case cons.iff_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝¹ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders a✝¹ ∧ admitsAux Οƒ binders a✝ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V (head✝ :: tail✝) a✝) ↔ (Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝¹) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) case cons.forall_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ (binders βˆͺ {a✝¹}) a✝ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE Οƒ' a✝¹ a✝¹) a✝) case cons.exists_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders a✝ β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' a✝)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ (binders βˆͺ {a✝¹}) a✝ h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ (βˆƒ d, Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔ βˆƒ d, Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE Οƒ' a✝¹ a✝¹) a✝) case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔ if a✝¹ = head✝.name ∧ (List.map Οƒ' a✝).length = head✝.args.length then Holds D I (Function.updateListITE V' head✝.args (List.map V' (List.map Οƒ' a✝))) tail✝ head✝.q else Holds D I V' tail✝ (def_ a✝¹ (List.map Οƒ' a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
case not_ phi phi_ih => congr! 1 exact phi_ih V V' Οƒ Οƒ' binders h1 h2 h2' h3
D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName), admitsAux Οƒ binders phi β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' phi)) V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders phi h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Β¬Holds D I V (head✝ :: tail✝) phi ↔ Β¬Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
simp only [admitsAux] at h1
case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : admitsAux Οƒ binders (def_ a✝¹ a✝) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (def_ a✝¹ a✝))
case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
simp only [fastReplaceFree]
case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree Οƒ' (def_ a✝¹ a✝))
case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ (List.map Οƒ' a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
simp only [Holds]
case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ (List.map Οƒ' a✝))
case cons.def_ D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ a✝, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔ if a✝¹ = head✝.name ∧ (List.map Οƒ' a✝).length = head✝.args.length then Holds D I (Function.updateListITE V' head✝.args (List.map V' (List.map Οƒ' a✝))) tail✝ head✝.q else Holds D I V' tail✝ (def_ a✝¹ (List.map Οƒ' a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
congr! 1
D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V' (List.map Οƒ' xs))
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ List.map V xs = List.map V' (List.map Οƒ' xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
simp
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ List.map V xs = List.map V' (List.map Οƒ' xs)
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ List.map V xs = List.map (V' ∘ Οƒ') xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
simp only [List.map_eq_map_iff]
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ List.map V xs = List.map (V' ∘ Οƒ') xs
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ βˆ€ x ∈ xs, V x = (V' ∘ Οƒ') x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
intro v a1
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ βˆ€ x ∈ xs, V x = (V' ∘ Οƒ') x
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs ⊒ V v = (V' ∘ Οƒ') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
apply h2
case a.h.e'_4 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs ⊒ V v = (V' ∘ Οƒ') v
case a.h.e'_4.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs ⊒ v ∈ binders ∨ Οƒ' v βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
by_cases c1 : v ∈ binders
case a.h.e'_4.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs ⊒ v ∈ binders ∨ Οƒ' v βˆ‰ binders
case pos D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v ∈ binders ⊒ v ∈ binders ∨ Οƒ' v βˆ‰ binders case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v βˆ‰ binders ⊒ v ∈ binders ∨ Οƒ' v βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
left
case pos D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v ∈ binders ⊒ v ∈ binders ∨ Οƒ' v βˆ‰ binders
case pos.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v ∈ binders ⊒ v ∈ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
exact c1
case pos.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v ∈ binders ⊒ v ∈ binders
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
right
case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v βˆ‰ binders ⊒ v ∈ binders ∨ Οƒ' v βˆ‰ binders
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v βˆ‰ binders ⊒ Οƒ' v βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
simp only [h3 v c1]
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v βˆ‰ binders ⊒ Οƒ' v βˆ‰ binders
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v βˆ‰ binders ⊒ Οƒ v βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
exact h1 v a1 c1
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) X : PredName xs : List VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : βˆ€ v ∈ xs, v βˆ‰ binders β†’ Οƒ v βˆ‰ binders h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v v : VarName a1 : v ∈ xs c1 : v βˆ‰ binders ⊒ Οƒ v βˆ‰ binders
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
cases h1
D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h1 : (x βˆ‰ binders β†’ Οƒ x βˆ‰ binders) ∧ (y βˆ‰ binders β†’ Οƒ y βˆ‰ binders) h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v ⊒ V x = V y ↔ V' (Οƒ' x) = V' (Οƒ' y)
case intro D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v left✝ : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders right✝ : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ V x = V y ↔ V' (Οƒ' x) = V' (Οƒ' y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
congr! 1
D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ V x = V y ↔ V' (Οƒ' x) = V' (Οƒ' y)
case a.h.e'_2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ V x = V' (Οƒ' x) case a.h.e'_3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ V y = V' (Οƒ' y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
apply h2
case a.h.e'_2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ V x = V' (Οƒ' x)
case a.h.e'_2.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ x ∈ binders ∨ Οƒ' x βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
by_cases c1 : x ∈ binders
case a.h.e'_2.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ x ∈ binders ∨ Οƒ' x βˆ‰ binders
case pos D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x ∈ binders ⊒ x ∈ binders ∨ Οƒ' x βˆ‰ binders case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x βˆ‰ binders ⊒ x ∈ binders ∨ Οƒ' x βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
left
case pos D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x ∈ binders ⊒ x ∈ binders ∨ Οƒ' x βˆ‰ binders
case pos.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x ∈ binders ⊒ x ∈ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
exact c1
case pos.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x ∈ binders ⊒ x ∈ binders
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
right
case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x βˆ‰ binders ⊒ x ∈ binders ∨ Οƒ' x βˆ‰ binders
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x βˆ‰ binders ⊒ Οƒ' x βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
simp only [h3 x c1]
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x βˆ‰ binders ⊒ Οƒ' x βˆ‰ binders
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x βˆ‰ binders ⊒ Οƒ x βˆ‰ binders
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
exact h1_left c1
case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders c1 : x βˆ‰ binders ⊒ Οƒ x βˆ‰ binders
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Rec/Admits.lean
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
[74, 1]
[207, 28]
apply h2
case a.h.e'_3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ V y = V' (Οƒ' y)
case a.h.e'_3.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ Οƒ' : VarName β†’ VarName) (binders : Finset VarName) (F : Formula), admitsAux Οƒ binders F β†’ (βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders, v = Οƒ' v) β†’ (βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v) β†’ (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree Οƒ' F)) x y : VarName V V' : VarAssignment D Οƒ Οƒ' : VarName β†’ VarName binders : Finset VarName h2 : βˆ€ (v : VarName), v ∈ binders ∨ Οƒ' v βˆ‰ binders β†’ V v = V' (Οƒ' v) h2' : βˆ€ v ∈ binders, v = Οƒ' v h3 : βˆ€ v βˆ‰ binders, Οƒ' v = Οƒ v h1_left : x βˆ‰ binders β†’ Οƒ x βˆ‰ binders h1_right : y βˆ‰ binders β†’ Οƒ y βˆ‰ binders ⊒ y ∈ binders ∨ Οƒ' y βˆ‰ binders