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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply h2 | case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds | case a.h.e'_2.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | right | case a.h.e'_2.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi | case a.h.e'_2.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | exact a1 | case a.h.e'_2.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | first | apply forall_congr' | apply exists_congr | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | intro d | case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi | case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | exact phi_ih (Function.updateITE V x d) h2 | case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply forall_congr' | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply exists_congr | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | split_ifs | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I' V tl (def_ X xs) | case pos
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
h✝ : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
case neg
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
h✝ : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | case pos c1 =>
apply ih
intro P ds a1
simp only [predVarOccursIn_iff_mem_predVarSet P ds.length] at a1
simp only [hd.h2] at a1
simp at a1 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | case neg c1 =>
apply ih
intro P ds a1
simp only [predVarOccursIn] at a1 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply ih | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | intro P ds a1 | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length hd.q
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp only [predVarOccursIn_iff_mem_predVarSet P ds.length] at a1 | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length hd.q
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ hd.q.predVarSet
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp only [hd.h2] at a1 | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ hd.q.predVarSet
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ ∅
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp at a1 | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ ∅
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply ih | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs) | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | intro P ds a1 | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp only [predVarOccursIn] at a1 | case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | induction F generalizing V | D : Type
I : Interpretation D
V : VarAssignment D
E E' : Env
F : Formula
h1 : ∃ E1, E' = E1 ++ E
h2 : all_def_in_env E F
h3 : E'.nodup_
⊢ Holds D I V E' F ↔ Holds D I V E F | case pred_const_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_const_ a✝¹ a✝)
⊢ Holds D I V E' (pred_const_ a✝¹ a✝) ↔ Holds D I V E (pred_const_ a✝¹ a✝)
case pred_var_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_var_ a✝¹ a✝)
⊢ Holds D I V E' (pred_var_ a✝¹ a✝) ↔ Holds D I V E (pred_var_ a✝¹ a✝)
case eq_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : VarName
V : VarAssignment D
h2 : all_def_in_env E (eq_ a✝¹ a✝)
⊢ Holds D I V E' (eq_ a✝¹ a✝) ↔ Holds D I V E (eq_ a✝¹ a✝)
case true_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E true_
⊢ Holds D I V E' true_ ↔ Holds D I V E true_
case false_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E false_
⊢ Holds D I V E' false_ ↔ Holds D I V E false_
case not_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝.not_
⊢ Holds D I V E' a✝.not_ ↔ Holds D I V E a✝.not_
case imp_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.imp_ a✝)
⊢ Holds D I V E' (a✝¹.imp_ a✝) ↔ Holds D I V E (a✝¹.imp_ a✝)
case and_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.and_ a✝)
⊢ Holds D I V E' (a✝¹.and_ a✝) ↔ Holds D I V E (a✝¹.and_ a✝)
case or_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.or_ a✝)
⊢ Holds D I V E' (a✝¹.or_ a✝) ↔ Holds D I V E (a✝¹.or_ a✝)
case iff_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.iff_ a✝)
⊢ Holds D I V E' (a✝¹.iff_ a✝) ↔ Holds D I V E (a✝¹.iff_ a✝)
case forall_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (forall_ a✝¹ a✝)
⊢ Holds D I V E' (forall_ a✝¹ a✝) ↔ Holds D I V E (forall_ a✝¹ a✝)
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (exists_ a✝¹ a✝)
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝)
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | any_goals
simp only [all_def_in_env] at h2
simp only [Holds] | case pred_const_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_const_ a✝¹ a✝)
⊢ Holds D I V E' (pred_const_ a✝¹ a✝) ↔ Holds D I V E (pred_const_ a✝¹ a✝)
case pred_var_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_var_ a✝¹ a✝)
⊢ Holds D I V E' (pred_var_ a✝¹ a✝) ↔ Holds D I V E (pred_var_ a✝¹ a✝)
case eq_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : VarName
V : VarAssignment D
h2 : all_def_in_env E (eq_ a✝¹ a✝)
⊢ Holds D I V E' (eq_ a✝¹ a✝) ↔ Holds D I V E (eq_ a✝¹ a✝)
case true_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E true_
⊢ Holds D I V E' true_ ↔ Holds D I V E true_
case false_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E false_
⊢ Holds D I V E' false_ ↔ Holds D I V E false_
case not_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝.not_
⊢ Holds D I V E' a✝.not_ ↔ Holds D I V E a✝.not_
case imp_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.imp_ a✝)
⊢ Holds D I V E' (a✝¹.imp_ a✝) ↔ Holds D I V E (a✝¹.imp_ a✝)
case and_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.and_ a✝)
⊢ Holds D I V E' (a✝¹.and_ a✝) ↔ Holds D I V E (a✝¹.and_ a✝)
case or_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.or_ a✝)
⊢ Holds D I V E' (a✝¹.or_ a✝) ↔ Holds D I V E (a✝¹.or_ a✝)
case iff_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.iff_ a✝)
⊢ Holds D I V E' (a✝¹.iff_ a✝) ↔ Holds D I V E (a✝¹.iff_ a✝)
case forall_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (forall_ a✝¹ a✝)
⊢ Holds D I V E' (forall_ a✝¹ a✝) ↔ Holds D I V E (forall_ a✝¹ a✝)
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (exists_ a✝¹ a✝)
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝)
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝) | case not_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ ¬Holds D I V E' a✝ ↔ ¬Holds D I V E a✝
case imp_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ Holds D I V E' a✝¹ → Holds D I V E' a✝ ↔ Holds D I V E a✝¹ → Holds D I V E a✝
case and_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ Holds D I V E' a✝¹ ∧ Holds D I V E' a✝ ↔ Holds D I V E a✝¹ ∧ Holds D I V E a✝
case or_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ Holds D I V E' a✝¹ ∨ Holds D I V E' a✝ ↔ Holds D I V E a✝¹ ∨ Holds D I V E a✝
case iff_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ (Holds D I V E' a✝¹ ↔ Holds D I V E' a✝) ↔ (Holds D I V E a✝¹ ↔ Holds D I V E a✝)
case forall_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ (∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) E' a✝) ↔ ∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) E a✝
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) E' a✝) ↔ ∃ d, Holds D I (Function.updateITE V a✝¹ d) E a✝
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V h2 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ¬Holds D I V E' phi ↔ ¬Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih
exact h2 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp only [all_def_in_env] at h2 | case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝) | case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, a✝¹ = d.name ∧ a✝.length = d.args.length
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp only [Holds] | case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝) | case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) E' a✝) ↔ ∃ d, Holds D I (Function.updateITE V a✝¹ d) E a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | congr! 1 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ¬Holds D I V E' phi ↔ ¬Holds D I V E phi | case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ Holds D I V E' phi ↔ Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact phi_ih V h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ Holds D I V E' phi ↔ Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | cases h2 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2 : all_def_in_env E phi ∧ all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi) | case intro
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
left✝ : all_def_in_env E phi
right✝ : all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | congr! 1 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi) | case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' phi ↔ Holds D I V E phi
case a.h.e'_2.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' psi ↔ Holds D I V E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact phi_ih V h2_left | case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' phi ↔ Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact psi_ih V h2_right | case a.h.e'_2.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' psi ↔ Holds D I V E psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi | case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | intro d | case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi | case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ Holds D I (Function.updateITE V x d) E' phi ↔ Holds D I (Function.updateITE V x d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | apply phi_ih | case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ Holds D I (Function.updateITE V x d) E' phi ↔ Holds D I (Function.updateITE V x d) E phi | case h.h2
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ all_def_in_env E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact h2 | case h.h2
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ all_def_in_env E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | apply forall_congr' | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) E' phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V x d) E phi | case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | apply exists_congr | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi | case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | apply Exists.elim h1 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ ∀ (a : Env), E' = a ++ E → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | intro E1 h1_1 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ ∀ (a : Env), E' = a ++ E → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)) | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | clear h1 | D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp only [all_def_in_env] at h2 | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | apply Exists.elim h2 | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ ∀ (a : Definition),
a ∈ E ∧ X = a.name ∧ xs.length = a.args.length → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | intro a h2_1 | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ ∀ (a : Definition),
a ∈ E ∧ X = a.name ∧ xs.length = a.args.length → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)) | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | clear h2 | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp only [Env.nodup_] at h3 | D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E E' : Env
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) E'
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | subst h1_1 | D : Type
I : Interpretation D
E E' : Env
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) E'
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1 ++ E)
⊢ Holds D I V (E1 ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | induction E1 | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1 ++ E)
⊢ Holds D I V (E1 ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | case nil
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
case cons
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (tail✝ ++ E) →
(Holds D I V (tail✝ ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (head✝ :: tail✝ ++ E)
⊢ Holds D I V (head✝ :: tail✝ ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | case nil =>
simp | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp at h3 | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_hd :: E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | cases h2_1 | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
left✝ : a ∈ E
right✝ : X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | cases h2_1_right | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
h2_1_right : X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
left✝ : X = a.name
right✝ : xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | cases h3 | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
left✝ : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
right✝ : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: (E1_tl ++ E)) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: (E1_tl ++ E)) (def_ X xs) ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ (if X = E1_hd.name ∧ xs.length = E1_hd.args.length then
Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q
else Holds D I V (E1_tl ++ E) (def_ X xs)) ↔
Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | split_ifs | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ (if X = E1_hd.name ∧ xs.length = E1_hd.args.length then
Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q
else Holds D I V (E1_tl ++ E) (def_ X xs)) ↔
Holds D I V E (def_ X xs) | case pos
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h✝ : X = E1_hd.name ∧ xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs)
case neg
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h✝ : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | case _ c1 =>
exact E1_ih h3_right | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | cases c1 | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : X = E1_hd.name ∧ xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs) | case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
left✝ : X = E1_hd.name
right✝ : xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exfalso | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs) | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | apply h3_left a | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ False | case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E1_tl ∨ a ∈ E
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.name = a.name
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = a.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | right | case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E1_tl ∨ a ∈ E | case a.h
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact h2_1_left | case a.h
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | subst c1_left | case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.name = a.name | case a
D : Type
I : Interpretation D
E : Env
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
h2_1_left : a ∈ E
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_right : xs.length = E1_hd.args.length
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ E1_hd.name xs) ↔ Holds D I V E (def_ E1_hd.name xs))
h2_1_right_left : E1_hd.name = a.name
⊢ E1_hd.name = a.name |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact h2_1_right_left | case a
D : Type
I : Interpretation D
E : Env
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
h2_1_left : a ∈ E
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_right : xs.length = E1_hd.args.length
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ E1_hd.name xs) ↔ Holds D I V E (def_ E1_hd.name xs))
h2_1_right_left : E1_hd.name = a.name
⊢ E1_hd.name = a.name | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | trans List.length xs | case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = a.args.length | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = xs.length
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = a.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | simp only [eq_comm] | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = xs.length | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = E1_hd.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact c1_right | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = E1_hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact h2_1_right_right | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = a.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Env | [239, 1] | [316, 35] | exact E1_ih h3_right | D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | apply Finset.union_subset_iff.mpr | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ∪ B ⊆ C ∪ D | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | constructor | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D | case left
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D
case right
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ C ∪ D |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | trans C | case left
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ C ⊆ C ∪ D |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | exact h1 | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | exact Finset.subset_union_left C D | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ C ⊆ C ∪ D | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | trans D | case right
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ C ∪ D | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ D
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ D ⊆ C ∪ D |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | exact h2 | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ D | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | exact Finset.subset_union_right C D | α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ D ⊆ C ∪ D | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | apply Finset.union_subset_iff.mpr | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ∪ B ⊆ C ∪ D ∪ E | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E ∧ B ⊆ C ∪ D ∪ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | constructor | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E ∧ B ⊆ C ∪ D ∪ E | case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ C ∪ D ∪ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | trans C ∪ E | case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ∪ E ⊆ C ∪ D ∪ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | exact h1 | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ E | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | apply Finset.union_subset_union_left | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ∪ E ⊆ C ∪ D ∪ E | case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ⊆ C ∪ D |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | exact Finset.subset_union_left C D | case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ⊆ C ∪ D | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | trans D ∪ E | case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ C ∪ D ∪ E | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ D ∪ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ∪ E ⊆ C ∪ D ∪ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | exact h2 | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ D ∪ E | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | apply Finset.union_subset_union_left | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ∪ E ⊆ C ∪ D ∪ E | case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ⊆ C ∪ D |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_union_left_right | [26, 1] | [43, 42] | exact Finset.subset_union_right C D | case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ⊆ C ∪ D | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | apply Finset.union_subset_iff.mpr | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ∪ B ⊆ (C ∪ D) \ E | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E ∧ B ⊆ (C ∪ D) \ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | constructor | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E ∧ B ⊆ (C ∪ D) \ E | case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ (C ∪ D) \ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | trans C \ E | case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ C \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C \ E ⊆ (C ∪ D) \ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | exact h1 | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ C \ E | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | apply Finset.sdiff_subset_sdiff | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C \ E ⊆ (C ∪ D) \ E | case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C ⊆ C ∪ D
case hvu
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ E ⊆ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | exact Finset.subset_union_left C D | case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C ⊆ C ∪ D | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | rfl | case hvu
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ E ⊆ E | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | trans D \ E | case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ (C ∪ D) \ E | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ D \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D \ E ⊆ (C ∪ D) \ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | exact h2 | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ D \ E | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | apply Finset.sdiff_subset_sdiff | α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D \ E ⊆ (C ∪ D) \ E | case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D ⊆ C ∪ D
case hvu
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ E ⊆ E |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_diff | [46, 1] | [65, 12] | exact Finset.subset_union_right C D | case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D ⊆ C ∪ D | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right_diff | [68, 1] | [89, 12] | apply Finset.union_subset_iff.mpr | α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ∪ B ⊆ E ∪ (C ∪ D) \ F | α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F ∧ B ⊆ E ∪ (C ∪ D) \ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right_diff | [68, 1] | [89, 12] | constructor | α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F ∧ B ⊆ E ∪ (C ∪ D) \ F | case left
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F
case right
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ B ⊆ E ∪ (C ∪ D) \ F |
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