Datasets:

internlm
/

Lean-Github

Dataset card Data Studio Files Files and versions Community

Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/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')
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 x' : VarName V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' y y' : VarName h2_left : isAlphaEqvVar binders x y h2_right : isAlphaEqvVar binders x' y' ⊢ Holds D I V (head✝ :: tail✝) (eq_ x x') ↔ Holds D I V' (head✝ :: tail✝) (eq_ y y')	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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_left : isAlphaEqvVar binders x y h2_right : isAlphaEqvVar binders x' y' ⊢ V x = V x' ↔ V' y = V' y'
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 x' : VarName V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' y y' : VarName h2_left : isAlphaEqvVar binders x y h2_right : isAlphaEqvVar binders x' y' ⊢ V x = V x' ↔ V' y = V' y'	case a.h.e'_2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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_left : isAlphaEqvVar binders x y h2_right : isAlphaEqvVar binders x' y' ⊢ V x = V' y case a.h.e'_3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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_left : isAlphaEqvVar binders x y h2_right : isAlphaEqvVar binders x' y' ⊢ V x' = V' y'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	exact aux_1 D binders x y V V' h1 h2_left	case a.h.e'_2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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_left : isAlphaEqvVar binders x y h2_right : isAlphaEqvVar binders x' y' ⊢ V x = V' y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	exact aux_1 D binders x' y' V V' h1 h2_right	case a.h.e'_3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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_left : isAlphaEqvVar binders x y h2_right : isAlphaEqvVar binders x' y' ⊢ V x' = V' y'	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') 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]	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') 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_	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 ↔ ¬Holds D I V' (head✝ :: tail✝) phi'
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') 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 ↔ ¬Holds D I V' (head✝ :: tail✝) phi'	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 ↔ Holds D I V' (head✝ :: tail✝) phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	exact phi_ih V V' phi' binders h1 h2	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 ↔ Holds D I V' (head✝ :: tail✝) phi'	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') phi psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2 : isAlphaEqvAux binders phi phi' ∧ isAlphaEqvAux binders psi psi' ⊢ Holds D I V (head✝ :: tail✝) (phi.iff_ psi) ↔ Holds D I V' (head✝ :: tail✝) (phi'.iff_ psi')	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') phi psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula left✝ : isAlphaEqvAux binders phi phi' right✝ : isAlphaEqvAux binders psi psi' ⊢ Holds D I V (head✝ :: tail✝) (phi.iff_ psi) ↔ Holds D I V' (head✝ :: tail✝) (phi'.iff_ psi')
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') phi psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2_left : isAlphaEqvAux binders phi phi' h2_right : isAlphaEqvAux binders psi psi' ⊢ Holds D I V (head✝ :: tail✝) (phi.iff_ psi) ↔ Holds D I V' (head✝ :: tail✝) (phi'.iff_ psi')	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2_left : isAlphaEqvAux binders phi phi' h2_right : isAlphaEqvAux binders psi psi' ⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔ (Holds D I V' (head✝ :: tail✝) phi' ↔ Holds D I V' (head✝ :: tail✝) psi')
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') phi psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2_left : isAlphaEqvAux binders phi phi' h2_right : isAlphaEqvAux binders psi psi' ⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔ (Holds D I V' (head✝ :: tail✝) phi' ↔ Holds D I V' (head✝ :: tail✝) psi')	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2_left : isAlphaEqvAux binders phi phi' h2_right : isAlphaEqvAux binders psi psi' ⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi' case a.h.e'_2.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2_left : isAlphaEqvAux binders phi phi' h2_right : isAlphaEqvAux binders psi psi' ⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	exact phi_ih V V' phi' binders h1 h2_left	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2_left : isAlphaEqvAux binders phi phi' h2_right : isAlphaEqvAux binders psi psi' ⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	exact psi_ih V V' psi' binders h1 h2_right	case a.h.e'_2.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 psi : 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') psi_ih : ∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)), AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F') V V' : VarAssignment D binders : List (VarName × VarName) h1 : AlphaEqvVarAssignment D binders V V' phi' psi' : Formula h2_left : isAlphaEqvAux binders phi phi' h2_right : isAlphaEqvAux binders psi psi' ⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi'	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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ Holds D I V (head✝ :: tail✝) (exists_ x phi) ↔ Holds D I V' (head✝ :: tail✝) (exists_ y phi')	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔ ∃ d, Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	first \| apply forall_congr' \| apply exists_congr	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔ ∃ d, Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi'	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	intro d	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi'	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' d : D ⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	induction h1	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' d : D ⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi'	case h.nil D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D V✝ : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ Holds D I (Function.updateITE V✝ x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V✝ y d) (head✝ :: tail✝) phi' case h.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') x : VarName 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) y : VarName phi' : Formula d : D binders✝ : List (VarName × VarName) x✝ y✝ : VarName V✝ V'✝ : VarAssignment D d✝ : D a✝ : AlphaEqvVarAssignment D binders✝ V✝ V'✝ a_ih✝ : isAlphaEqvAux ((x, y) :: binders✝) phi phi' → (Holds D I (Function.updateITE V✝ x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V'✝ y d) (head✝ :: tail✝) phi') h2 : isAlphaEqvAux ((x, y) :: (x✝, y✝) :: binders✝) phi phi' ⊢ Holds D I (Function.updateITE (Function.updateITE V✝ x✝ d✝) x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE (Function.updateITE V'✝ y✝ d✝) y d) (head✝ :: tail✝) phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	apply forall_congr'	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi'	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	apply exists_congr	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔ ∃ d, Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi'	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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' y : VarName phi' : Formula h2 : isAlphaEqvAux ((x, y) :: binders) phi phi' ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	apply phi_ih	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE h1_V y d) (head✝ :: tail✝) phi'	case h1 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ AlphaEqvVarAssignment D ?binders (Function.updateITE h1_V x d) (Function.updateITE h1_V y d) case 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ isAlphaEqvAux ?binders phi phi' case binders D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ List (VarName × VarName)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	apply AlphaEqvVarAssignment.cons	case h1 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ AlphaEqvVarAssignment D ?binders (Function.updateITE h1_V x d) (Function.updateITE h1_V y d)	case h1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ AlphaEqvVarAssignment D ?h1.binders h1_V h1_V case h1.binders D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ List (VarName × VarName)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	apply AlphaEqvVarAssignment.nil	case h1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ AlphaEqvVarAssignment D ?h1.binders h1_V h1_V case h1.binders D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (F F' : Formula) (binders : 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ List (VarName × VarName)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Alpha.lean	FOL.NV.isAlphaEqv_Holds_aux	[624, 1]	[734, 58]	exact h2	case 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 : VarName 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) y : VarName phi' : Formula d : D h1_V : VarAssignment D h2 : isAlphaEqvAux [(x, y)] phi phi' ⊢ isAlphaEqvAux [(x, y)] phi phi'	no goals

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.