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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi.not_ ↔ Holds D I V E h1_phi'.not_ | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I V E h1_phi ↔ ¬Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | congr! 1 | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I V E h1_phi ↔ ¬Holds D I V E h1_phi' | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E (h1_phi.iff_ h1_psi) ↔ Holds D I V E (h1_phi'.iff_ h1_psi') | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I V E h1_phi ↔ Holds D I V E h1_psi) ↔ (Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | congr! 1 | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I V E h1_phi ↔ Holds D I V E h1_psi) ↔ (Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_psi ↔ Holds D I V E h1_psi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_1 V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_2 V | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_psi ↔ Holds D I V E h1_psi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ Holds D I V E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_psi) | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔ ∃ d, Holds D I (Function.updateITE V h1_x d) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔ ∃ d, Holds D I (Function.updateITE V h1_x d) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | intro d | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih (Function.updateITE V h1_x d) | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) E h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔
∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | apply exists_congr | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔ ∃ d, Holds D I (Function.updateITE V h1_x d) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | rfl | D : Type
I : Interpretation D
E : Env
F F' h1 : Formula
V : VarAssignment D
⊢ Holds D I V E h1 ↔ Holds D I V E h1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | symm | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi' ↔ Holds D I V E h1_phi | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | trans Holds D I V E h1_phi' | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi'' | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi' ↔ Holds D I V E h1_phi'' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_1 V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_2 V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi' ↔ Holds D I V E h1_phi'' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | induction xs | xs : List VarName
⊢ isAlphaEqvVarList [] xs xs | case nil
⊢ isAlphaEqvVarList [] [] []
case cons
head✝ : VarName
tail✝ : List VarName
tail_ih✝ : isAlphaEqvVarList [] tail✝ tail✝
⊢ isAlphaEqvVarList [] (head✝ :: tail✝) (head✝ :: tail✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | case nil =>
simp only [isAlphaEqvVarList] | ⊢ isAlphaEqvVarList [] [] [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | simp only [isAlphaEqvVarList] | ⊢ isAlphaEqvVarList [] [] [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | simp only [isAlphaEqvVarList] | hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVarList [] (hd :: tl) (hd :: tl) | hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVar [] hd hd ∧ isAlphaEqvVarList [] tl tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | constructor | hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVar [] hd hd ∧ isAlphaEqvVarList [] tl tl | case left
hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVar [] hd hd
case right
hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVarList [] tl tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | simp only [isAlphaEqvVar] | case left
hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVar [] hd hd | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | exact ih | case right
hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVarList [] tl tl | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | induction h1 | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvVar binders x y
⊢ V x = V' y | case nil
D : Type
binders : List (VarName × VarName)
x y : VarName
V V' V✝ : VarAssignment D
h2 : isAlphaEqvVar [] x y
⊢ V✝ x = V✝ y
case cons
D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
binders✝ : List (VarName × VarName)
x✝ y✝ : VarName
V✝ V'✝ : VarAssignment D
d✝ : D
a✝ : AlphaEqvVarAssignment D binders✝ V✝ V'✝
a_ih✝ : isAlphaEqvVar binders✝ x y → V✝ x = V'✝ y
h2 : isAlphaEqvVar ((x✝, y✝) :: binders✝) x y
⊢ Function.updateITE V✝ x✝ d✝ x = Function.updateITE V'✝ y✝ d✝ y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | case nil h1_V =>
simp only [isAlphaEqvVar] at h2
subst h2
rfl | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' h1_V : VarAssignment D
h2 : isAlphaEqvVar [] x y
⊢ h1_V x = h1_V y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | case cons h1_l h1_x h1_y h1_V h1_V' h1_d _ h1_ih =>
simp only [isAlphaEqvVar] at h2
simp only [Function.updateITE]
cases h2
case inl h2 =>
cases h2
case intro h2_left h2_right =>
simp only [if_pos h2_left, if_pos h2_right]
case inr h2 =>
cases h2
case intro h2_left h2_right =>
cases h2_left
case intro h2_left_left h2_left_right =>
simp only [if_neg h2_left_left, if_neg h2_left_right]
exact h1_ih h2_right | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : isAlphaEqvVar ((h1_x, h1_y) :: h1_l) x y
⊢ Function.updateITE h1_V h1_x h1_d x = Function.updateITE h1_V' h1_y h1_d y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | simp only [isAlphaEqvVar] at h2 | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' h1_V : VarAssignment D
h2 : isAlphaEqvVar [] x y
⊢ h1_V x = h1_V y | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' h1_V : VarAssignment D
h2 : x = y
⊢ h1_V x = h1_V y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | subst h2 | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' h1_V : VarAssignment D
h2 : x = y
⊢ h1_V x = h1_V y | D : Type
binders : List (VarName × VarName)
x : VarName
V V' h1_V : VarAssignment D
⊢ h1_V x = h1_V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | rfl | D : Type
binders : List (VarName × VarName)
x : VarName
V V' h1_V : VarAssignment D
⊢ h1_V x = h1_V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | simp only [isAlphaEqvVar] at h2 | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : isAlphaEqvVar ((h1_x, h1_y) :: h1_l) x y
⊢ Function.updateITE h1_V h1_x h1_d x = Function.updateITE h1_V' h1_y h1_d y | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : x = h1_x ∧ y = h1_y ∨ (¬x = h1_x ∧ ¬y = h1_y) ∧ isAlphaEqvVar h1_l x y
⊢ Function.updateITE h1_V h1_x h1_d x = Function.updateITE h1_V' h1_y h1_d y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | simp only [Function.updateITE] | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : x = h1_x ∧ y = h1_y ∨ (¬x = h1_x ∧ ¬y = h1_y) ∧ isAlphaEqvVar h1_l x y
⊢ Function.updateITE h1_V h1_x h1_d x = Function.updateITE h1_V' h1_y h1_d y | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : x = h1_x ∧ y = h1_y ∨ (¬x = h1_x ∧ ¬y = h1_y) ∧ isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | cases h2 | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : x = h1_x ∧ y = h1_y ∨ (¬x = h1_x ∧ ¬y = h1_y) ∧ isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | case inl
D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h✝ : x = h1_x ∧ y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y
case inr
D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h✝ : (¬x = h1_x ∧ ¬y = h1_y) ∧ isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | case inl h2 =>
cases h2
case intro h2_left h2_right =>
simp only [if_pos h2_left, if_pos h2_right] | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : x = h1_x ∧ y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | case inr h2 =>
cases h2
case intro h2_left h2_right =>
cases h2_left
case intro h2_left_left h2_left_right =>
simp only [if_neg h2_left_left, if_neg h2_left_right]
exact h1_ih h2_right | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : (¬x = h1_x ∧ ¬y = h1_y) ∧ isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | cases h2 | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : x = h1_x ∧ y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | case intro
D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
left✝ : x = h1_x
right✝ : y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | case intro h2_left h2_right =>
simp only [if_pos h2_left, if_pos h2_right] | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_left : x = h1_x
h2_right : y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | simp only [if_pos h2_left, if_pos h2_right] | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_left : x = h1_x
h2_right : y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | cases h2 | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2 : (¬x = h1_x ∧ ¬y = h1_y) ∧ isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | case intro
D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
left✝ : ¬x = h1_x ∧ ¬y = h1_y
right✝ : isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | case intro h2_left h2_right =>
cases h2_left
case intro h2_left_left h2_left_right =>
simp only [if_neg h2_left_left, if_neg h2_left_right]
exact h1_ih h2_right | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_left : ¬x = h1_x ∧ ¬y = h1_y
h2_right : isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | cases h2_left | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_left : ¬x = h1_x ∧ ¬y = h1_y
h2_right : isAlphaEqvVar h1_l x y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | case intro
D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_right : isAlphaEqvVar h1_l x y
left✝ : ¬x = h1_x
right✝ : ¬y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | case intro h2_left_left h2_left_right =>
simp only [if_neg h2_left_left, if_neg h2_left_right]
exact h1_ih h2_right | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_right : isAlphaEqvVar h1_l x y
h2_left_left : ¬x = h1_x
h2_left_right : ¬y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | simp only [if_neg h2_left_left, if_neg h2_left_right] | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_right : isAlphaEqvVar h1_l x y
h2_left_left : ¬x = h1_x
h2_left_right : ¬y = h1_y
⊢ (if x = h1_x then h1_d else h1_V x) = if y = h1_y then h1_d else h1_V' y | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_right : isAlphaEqvVar h1_l x y
h2_left_left : ¬x = h1_x
h2_left_right : ¬y = h1_y
⊢ h1_V x = h1_V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_1 | [531, 1] | [561, 31] | exact h1_ih h2_right | D : Type
binders : List (VarName × VarName)
x y : VarName
V V' : VarAssignment D
h1_l : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
a✝ : AlphaEqvVarAssignment D h1_l h1_V h1_V'
h1_ih : isAlphaEqvVar h1_l x y → h1_V x = h1_V' y
h2_right : isAlphaEqvVar h1_l x y
h2_left_left : ¬x = h1_x
h2_left_right : ¬y = h1_y
⊢ h1_V x = h1_V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | induction xs generalizing ys | D : Type
binders : List (VarName × VarName)
xs ys : List VarName
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvVarList binders xs ys
⊢ List.map V xs = List.map V' ys | case nil
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2 : isAlphaEqvVarList binders [] ys
⊢ List.map V [] = List.map V' ys
case cons
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
head✝ : VarName
tail✝ : List VarName
tail_ih✝ : ∀ (ys : List VarName), isAlphaEqvVarList binders tail✝ ys → List.map V tail✝ = List.map V' ys
ys : List VarName
h2 : isAlphaEqvVarList binders (head✝ :: tail✝) ys
⊢ List.map V (head✝ :: tail✝) = List.map V' ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | case nil =>
cases ys
case nil =>
simp
case cons ys_hd ys_tl =>
simp only [isAlphaEqvVarList] at h2 | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2 : isAlphaEqvVarList binders [] ys
⊢ List.map V [] = List.map V' ys | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | cases ys | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2 : isAlphaEqvVarList binders [] ys
⊢ List.map V [] = List.map V' ys | case nil
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvVarList binders [] []
⊢ List.map V [] = List.map V' []
case cons
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
head✝ : VarName
tail✝ : List VarName
h2 : isAlphaEqvVarList binders [] (head✝ :: tail✝)
⊢ List.map V [] = List.map V' (head✝ :: tail✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | case nil =>
simp | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvVarList binders [] []
⊢ List.map V [] = List.map V' [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | case cons ys_hd ys_tl =>
simp only [isAlphaEqvVarList] at h2 | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVarList binders [] (ys_hd :: ys_tl)
⊢ List.map V [] = List.map V' (ys_hd :: ys_tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | simp | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvVarList binders [] []
⊢ List.map V [] = List.map V' [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | simp only [isAlphaEqvVarList] at h2 | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVarList binders [] (ys_hd :: ys_tl)
⊢ List.map V [] = List.map V' (ys_hd :: ys_tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | cases ys | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys : List VarName
h2 : isAlphaEqvVarList binders (xs_hd :: xs_tl) ys
⊢ List.map V (xs_hd :: xs_tl) = List.map V' ys | case nil
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
h2 : isAlphaEqvVarList binders (xs_hd :: xs_tl) []
⊢ List.map V (xs_hd :: xs_tl) = List.map V' []
case cons
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
head✝ : VarName
tail✝ : List VarName
h2 : isAlphaEqvVarList binders (xs_hd :: xs_tl) (head✝ :: tail✝)
⊢ List.map V (xs_hd :: xs_tl) = List.map V' (head✝ :: tail✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | case nil =>
simp only [isAlphaEqvVarList] at h2 | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
h2 : isAlphaEqvVarList binders (xs_hd :: xs_tl) []
⊢ List.map V (xs_hd :: xs_tl) = List.map V' [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | simp only [isAlphaEqvVarList] at h2 | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
h2 : isAlphaEqvVarList binders (xs_hd :: xs_tl) []
⊢ List.map V (xs_hd :: xs_tl) = List.map V' [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | simp only [isAlphaEqvVarList] at h2 | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVarList binders (xs_hd :: xs_tl) (ys_hd :: ys_tl)
⊢ List.map V (xs_hd :: xs_tl) = List.map V' (ys_hd :: ys_tl) | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ List.map V (xs_hd :: xs_tl) = List.map V' (ys_hd :: ys_tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | simp | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ List.map V (xs_hd :: xs_tl) = List.map V' (ys_hd :: ys_tl) | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ V xs_hd = V' ys_hd ∧ List.map V xs_tl = List.map V' ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | constructor | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ V xs_hd = V' ys_hd ∧ List.map V xs_tl = List.map V' ys_tl | case left
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ V xs_hd = V' ys_hd
case right
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ List.map V xs_tl = List.map V' ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | cases h2 | case left
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ V xs_hd = V' ys_hd | case left.intro
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
left✝ : isAlphaEqvVar binders xs_hd ys_hd
right✝ : isAlphaEqvVarList binders xs_tl ys_tl
⊢ V xs_hd = V' ys_hd |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | case left.intro h2_left h2_right =>
exact aux_1 D binders xs_hd ys_hd V V' h1 h2_left | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2_left : isAlphaEqvVar binders xs_hd ys_hd
h2_right : isAlphaEqvVarList binders xs_tl ys_tl
⊢ V xs_hd = V' ys_hd | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | exact aux_1 D binders xs_hd ys_hd V V' h1 h2_left | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2_left : isAlphaEqvVar binders xs_hd ys_hd
h2_right : isAlphaEqvVarList binders xs_tl ys_tl
⊢ V xs_hd = V' ys_hd | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | apply xs_ih ys_tl | case right
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ List.map V xs_tl = List.map V' ys_tl | case right
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ isAlphaEqvVarList binders xs_tl ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | cases h2 | case right
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ isAlphaEqvVarList binders xs_tl ys_tl | case right.intro
D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
left✝ : isAlphaEqvVar binders xs_hd ys_hd
right✝ : isAlphaEqvVarList binders xs_tl ys_tl
⊢ isAlphaEqvVarList binders xs_tl ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | case right.intro h2_left h2_right =>
exact h2_right | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2_left : isAlphaEqvVar binders xs_hd ys_hd
h2_right : isAlphaEqvVarList binders xs_tl ys_tl
⊢ isAlphaEqvVarList binders xs_tl ys_tl | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.aux_2 | [564, 1] | [595, 25] | exact h2_right | D : Type
binders : List (VarName × VarName)
V V' : VarAssignment D
h1 : AlphaEqvVarAssignment D binders V V'
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → List.map V xs_tl = List.map V' ys
ys_hd : VarName
ys_tl : List VarName
h2_left : isAlphaEqvVar binders xs_hd ys_hd
h2_right : isAlphaEqvVarList binders xs_tl ys_tl
⊢ isAlphaEqvVarList binders xs_tl ys_tl | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | induction xs generalizing ys | binders : List (VarName × VarName)
xs ys : List VarName
h1 : isAlphaEqvVarList binders xs ys
⊢ xs.length = ys.length | case nil
binders : List (VarName × VarName)
ys : List VarName
h1 : isAlphaEqvVarList binders [] ys
⊢ [].length = ys.length
case cons
binders : List (VarName × VarName)
head✝ : VarName
tail✝ : List VarName
tail_ih✝ : ∀ (ys : List VarName), isAlphaEqvVarList binders tail✝ ys → tail✝.length = ys.length
ys : List VarName
h1 : isAlphaEqvVarList binders (head✝ :: tail✝) ys
⊢ (head✝ :: tail✝).length = ys.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | case nil =>
cases ys
case nil =>
rfl
case cons ys_hd ys_tl =>
simp only [isAlphaEqvVarList] at h1 | binders : List (VarName × VarName)
ys : List VarName
h1 : isAlphaEqvVarList binders [] ys
⊢ [].length = ys.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | case cons xs_hd xs_tl xs_ih =>
cases ys
case nil =>
simp only [isAlphaEqvVarList] at h1
case cons ys_hd ys_tl =>
simp only [isAlphaEqvVarList] at h1
simp
cases h1
case intro h1_left h1_right =>
exact xs_ih ys_tl h1_right | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys : List VarName
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) ys
⊢ (xs_hd :: xs_tl).length = ys.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | cases ys | binders : List (VarName × VarName)
ys : List VarName
h1 : isAlphaEqvVarList binders [] ys
⊢ [].length = ys.length | case nil
binders : List (VarName × VarName)
h1 : isAlphaEqvVarList binders [] []
⊢ [].length = [].length
case cons
binders : List (VarName × VarName)
head✝ : VarName
tail✝ : List VarName
h1 : isAlphaEqvVarList binders [] (head✝ :: tail✝)
⊢ [].length = (head✝ :: tail✝).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | case nil =>
rfl | binders : List (VarName × VarName)
h1 : isAlphaEqvVarList binders [] []
⊢ [].length = [].length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | case cons ys_hd ys_tl =>
simp only [isAlphaEqvVarList] at h1 | binders : List (VarName × VarName)
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVarList binders [] (ys_hd :: ys_tl)
⊢ [].length = (ys_hd :: ys_tl).length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | rfl | binders : List (VarName × VarName)
h1 : isAlphaEqvVarList binders [] []
⊢ [].length = [].length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | simp only [isAlphaEqvVarList] at h1 | binders : List (VarName × VarName)
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVarList binders [] (ys_hd :: ys_tl)
⊢ [].length = (ys_hd :: ys_tl).length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | cases ys | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys : List VarName
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) ys
⊢ (xs_hd :: xs_tl).length = ys.length | case nil
binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) []
⊢ (xs_hd :: xs_tl).length = [].length
case cons
binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
head✝ : VarName
tail✝ : List VarName
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) (head✝ :: tail✝)
⊢ (xs_hd :: xs_tl).length = (head✝ :: tail✝).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | case nil =>
simp only [isAlphaEqvVarList] at h1 | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) []
⊢ (xs_hd :: xs_tl).length = [].length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | case cons ys_hd ys_tl =>
simp only [isAlphaEqvVarList] at h1
simp
cases h1
case intro h1_left h1_right =>
exact xs_ih ys_tl h1_right | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) (ys_hd :: ys_tl)
⊢ (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | simp only [isAlphaEqvVarList] at h1 | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) []
⊢ (xs_hd :: xs_tl).length = [].length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | simp only [isAlphaEqvVarList] at h1 | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVarList binders (xs_hd :: xs_tl) (ys_hd :: ys_tl)
⊢ (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | simp | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ xs_tl.length = ys_tl.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | cases h1 | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1 : isAlphaEqvVar binders xs_hd ys_hd ∧ isAlphaEqvVarList binders xs_tl ys_tl
⊢ xs_tl.length = ys_tl.length | case intro
binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
left✝ : isAlphaEqvVar binders xs_hd ys_hd
right✝ : isAlphaEqvVarList binders xs_tl ys_tl
⊢ xs_tl.length = ys_tl.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | case intro h1_left h1_right =>
exact xs_ih ys_tl h1_right | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1_left : isAlphaEqvVar binders xs_hd ys_hd
h1_right : isAlphaEqvVarList binders xs_tl ys_tl
⊢ xs_tl.length = ys_tl.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarList_length | [598, 1] | [621, 35] | exact xs_ih ys_tl h1_right | binders : List (VarName × VarName)
xs_hd : VarName
xs_tl : List VarName
xs_ih : ∀ (ys : List VarName), isAlphaEqvVarList binders xs_tl ys → xs_tl.length = ys.length
ys_hd : VarName
ys_tl : List VarName
h1_left : isAlphaEqvVar binders xs_hd ys_hd
h1_right : isAlphaEqvVarList binders xs_tl ys_tl
⊢ xs_tl.length = ys_tl.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | induction E generalizing F F' binders V V' | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders F F'
⊢ Holds D I V E F ↔ Holds D I V' E F' | case nil
D : Type
I : Interpretation D
V V' : VarAssignment D
F F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders F F'
⊢ Holds D I V [] F ↔ Holds D I V' [] F'
case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
F F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders F F'
⊢ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) F' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case nil.def_.def_ =>
simp only [Holds] | D : Type
I : Interpretation D
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : a✝³ = a✝¹ ∧ isAlphaEqvVarList binders a✝² a✝
⊢ Holds D I V [] (def_ a✝³ a✝²) ↔ Holds D I V' [] (def_ a✝¹ a✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | induction F generalizing F' binders V V' | case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
F F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders F F'
⊢ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) F' | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_const_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_var_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (eq_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders true_ F'
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders false_ F'
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders a✝.not_ F'
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.imp_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.and_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.or_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.iff_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (forall_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (exists_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | all_goals
cases F' | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_const_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_var_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (eq_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders true_ F'
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders false_ F'
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders a✝.not_ F'
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.imp_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.and_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.or_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.iff_ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (forall_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (exists_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F'
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F' | case cons.pred_const_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_const_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.pred_const_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.pred_const_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_const_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.pred_const_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_const_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.pred_const_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : PredName
a✝¹ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.pred_const_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.pred_const_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.pred_const_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.pred_const_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.pred_const_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.pred_const_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.pred_const_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.pred_var_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.pred_var_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.pred_var_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_var_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.pred_var_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_var_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.pred_var_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : PredName
a✝¹ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.pred_var_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.pred_var_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.pred_var_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.pred_var_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.pred_var_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.pred_var_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.pred_var_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.eq_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.eq_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.eq_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (eq_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.eq_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (eq_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.eq_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.eq_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.eq_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.eq_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.eq_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.eq_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.eq_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.eq_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.true_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders true_ (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.true_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders true_ (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.true_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders true_ (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders true_ true_
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.true_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders true_ false_
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.true_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders true_ a✝.not_
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.true_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.true_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.true_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.true_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.true_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders true_ (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.true_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders true_ (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.true_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders true_ (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.false_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders false_ (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.false_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders false_ (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.false_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders false_ (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.false_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders false_ true_
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.false_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders false_ false_
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.false_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders false_ a✝.not_
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.false_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.false_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.false_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.false_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.false_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders false_ (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.false_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders false_ (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.false_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders false_ (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.not_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders a✝².not_ (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.not_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders a✝².not_ (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.not_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders a✝².not_ (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.not_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders a✝.not_ true_
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.not_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders a✝.not_ false_
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.not_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders a✝¹.not_ a✝.not_
⊢ Holds D I V (head✝ :: tail✝) a✝¹.not_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.not_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.not_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.not_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.not_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.not_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.not_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.not_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders a✝².not_ (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.imp_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.imp_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.imp_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.imp_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.imp_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.imp_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.imp_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.imp_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².imp_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².imp_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.imp_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.imp_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.imp_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.imp_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.imp_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.imp_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.imp_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.and_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.and_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.and_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.and_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.and_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.and_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.and_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.and_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².and_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².and_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.and_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.and_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.and_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.and_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.and_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.and_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.or_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.or_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.or_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.or_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.or_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.or_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.or_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.or_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².or_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².or_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.or_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.or_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.or_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.or_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.or_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.or_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.iff_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.iff_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.iff_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.iff_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.iff_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.iff_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.iff_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.iff_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².iff_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².iff_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.iff_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.iff_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.iff_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.iff_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.iff_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.iff_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.forall_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.forall_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.forall_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.forall_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (forall_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.forall_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (forall_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.forall_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : VarName
a✝¹ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.forall_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.forall_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.forall_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.forall_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.forall_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.forall_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.exists_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.exists_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.exists_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.exists_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (exists_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.exists_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (exists_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.exists_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : VarName
a✝¹ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.exists_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.exists_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.exists_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.exists_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.exists_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.exists_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.def_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.def_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.def_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.def_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.def_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.def_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : DefName
a✝¹ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.def_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.def_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.def_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.def_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.def_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.def_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.def_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | any_goals
simp only [isAlphaEqvAux] at h2 | case cons.pred_const_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_const_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.pred_const_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.pred_const_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_const_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.pred_const_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_const_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.pred_const_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : PredName
a✝¹ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.pred_const_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.pred_const_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.pred_const_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.pred_const_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.pred_const_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.pred_const_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.pred_const_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_const_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.pred_var_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.pred_var_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.pred_var_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_var_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.pred_var_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (pred_var_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.pred_var_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : PredName
a✝¹ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.pred_var_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.pred_var_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.pred_var_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.pred_var_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.pred_var_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.pred_var_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.pred_var_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (pred_var_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.eq_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.eq_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.eq_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (eq_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.eq_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (eq_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.eq_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.eq_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.eq_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.eq_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.eq_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.eq_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.eq_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.eq_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (eq_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.true_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders true_ (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.true_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders true_ (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.true_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders true_ (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders true_ true_
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.true_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders true_ false_
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.true_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders true_ a✝.not_
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.true_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.true_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.true_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.true_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders true_ (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.true_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders true_ (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.true_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders true_ (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.true_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders true_ (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.false_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders false_ (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.false_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders false_ (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.false_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders false_ (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.false_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders false_ true_
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.false_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders false_ false_
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.false_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders false_ a✝.not_
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.false_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.false_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.false_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.false_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders false_ (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.false_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders false_ (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.false_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders false_ (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.false_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders false_ (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.not_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders a✝².not_ (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.not_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders a✝².not_ (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.not_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders a✝².not_ (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.not_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders a✝.not_ true_
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.not_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders a✝.not_ false_
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.not_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders a✝¹.not_ a✝.not_
⊢ Holds D I V (head✝ :: tail✝) a✝¹.not_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.not_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.not_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.not_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.not_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.not_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.not_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders a✝².not_ (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.not_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders a✝².not_ (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) a✝².not_ ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.imp_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.imp_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.imp_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.imp_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.imp_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.imp_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.imp_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.imp_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².imp_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².imp_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.imp_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.imp_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.imp_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.imp_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.imp_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.imp_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.imp_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.imp_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.and_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.and_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.and_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.and_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.and_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.and_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.and_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.and_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².and_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².and_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.and_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.and_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.and_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.and_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.and_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.and_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.and_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.or_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.or_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.or_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.or_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.or_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.or_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.or_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.or_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².or_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².or_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.or_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.or_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.or_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.or_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.or_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.or_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.or_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.iff_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.iff_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.iff_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.iff_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.iff_ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.iff_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (a✝¹.iff_ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.iff_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² a✝¹ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝².iff_ a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (a✝².iff_ a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.iff_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.iff_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.iff_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.iff_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.iff_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.iff_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (a✝³.iff_ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.forall_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.forall_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.forall_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.forall_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (forall_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.forall_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (forall_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.forall_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : VarName
a✝¹ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.forall_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.forall_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.forall_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.forall_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.forall_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.forall_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (forall_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.exists_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.exists_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.exists_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.exists_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (exists_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.exists_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝ F' → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (exists_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.exists_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : VarName
a✝¹ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.exists_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.exists_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.exists_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.exists_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.exists_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.exists_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (exists_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
case cons.def_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.def_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.def_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.def_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.def_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.def_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : DefName
a✝¹ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.def_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.def_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.def_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.def_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.def_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.def_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.def_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) | case cons.pred_const_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : a✝³ = a✝¹ ∧ isAlphaEqvVarList binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : PredName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : a✝³ = a✝¹ ∧ isAlphaEqvVarList binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvVar binders a✝³ a✝¹ ∧ isAlphaEqvVar binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : True
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.false_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : True
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.not_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝¹ F' → (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders a✝¹ a✝
⊢ Holds D I V (head✝ :: tail✝) a✝¹.not_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.imp_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝³ a✝¹ ∧ isAlphaEqvAux binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (a✝³.imp_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝³ a✝¹ ∧ isAlphaEqvAux binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (a✝³.and_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝³ a✝¹ ∧ isAlphaEqvAux binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (a✝³.or_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ a✝² : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝³ F' → (Holds D I V (head✝ :: tail✝) a✝³ ↔ Holds D I V' (head✝ :: tail✝) F')
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders a✝³ a✝¹ ∧ isAlphaEqvAux binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (a✝³.iff_ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux ((a✝³, a✝¹) :: binders) a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : VarName
a✝² : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders a✝² F' → (Holds D I V (head✝ :: tail✝) a✝² ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux ((a✝³, a✝¹) :: binders) a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.def_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : a✝³ = a✝¹ ∧ isAlphaEqvVarList binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case
pred_const_.pred_const_ X xs Y ys
| pred_var_.pred_var_ X xs Y ys =>
cases h2
case intro h2_left h2_right =>
simp only [Holds]
subst h2_left
congr! 1
exact aux_2 D binders xs ys V V' h1 h2_right | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : PredName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ Y ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case true_.true_ | false_.false_ =>
simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : True
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case not_.not_ phi phi_ih phi' =>
simp only [Holds]
congr! 1
exact phi_ih V V' phi' binders h1 h2 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' : Formula
h2 : isAlphaEqvAux binders phi phi'
⊢ Holds D I V (head✝ :: tail✝) phi.not_ ↔ Holds D I V' (head✝ :: tail✝) phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases F' | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
F' : Formula
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) F'
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) F' | case cons.def_.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (pred_const_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.def_.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : PredName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (pred_var_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.def_.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (eq_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.def_.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) true_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.def_.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : isAlphaEqvAux binders (def_ a✝¹ a✝) false_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.def_.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝² : DefName
a✝¹ : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝² a✝¹) a✝.not_
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝² a✝¹) ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.def_.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.imp_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.def_.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.and_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.def_.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.or_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.def_.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (a✝¹.iff_ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.def_.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (forall_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.def_.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : VarName
a✝ : Formula
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (exists_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.def_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [isAlphaEqvAux] at h2 | case cons.def_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : isAlphaEqvAux binders (def_ a✝³ a✝²) (def_ a✝¹ a✝)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) | case cons.def_.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : a✝³ = a✝¹ ∧ isAlphaEqvVarList binders a✝² a✝
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝³ a✝²) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases h2 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : PredName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ Y ys) | case intro
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : PredName
ys : List VarName
left✝ : X = Y
right✝ : isAlphaEqvVarList binders xs ys
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ Y ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case intro h2_left h2_right =>
simp only [Holds]
subst h2_left
congr! 1
exact aux_2 D binders xs ys V V' h1 h2_right | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : PredName
ys : List VarName
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ Y ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : PredName
ys : List VarName
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ Y ys) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : PredName
ys : List VarName
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ Y (List.map V' ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | subst h2_left | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : PredName
ys : List VarName
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ Y (List.map V' ys) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2_right : isAlphaEqvVarList binders xs ys
⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V' ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2_right : isAlphaEqvVarList binders xs ys
⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V' ys) | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2_right : isAlphaEqvVarList binders xs ys
⊢ List.map V xs = List.map V' ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact aux_2 D binders xs ys V V' h1 h2_right | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
X : PredName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2_right : isAlphaEqvVarList binders xs ys
⊢ List.map V xs = List.map V' ys | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases h2 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2 : isAlphaEqvVar binders x y ∧ isAlphaEqvVar binders x' y'
⊢ Holds D I V (head✝ :: tail✝) (eq_ x x') ↔ Holds D I V' (head✝ :: tail✝) (eq_ y y') | case intro
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
left✝ : isAlphaEqvVar binders x y
right✝ : isAlphaEqvVar binders x' y'
⊢ Holds D I V (head✝ :: tail✝) (eq_ x x') ↔ Holds D I V' (head✝ :: tail✝) (eq_ y y') |
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