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stringlengths 1
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stringlengths 6
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stringlengths 6
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stringlengths 1
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stringlengths 3
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| state_after
stringlengths 6
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | congr! 1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ ¬Holds D I V M (head✝ :: tail✝) phi ↔ ¬Holds D I V' M (head✝ :: tail✝) phi | case a.h.e'_1.a
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact phi_ih V V' vs h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList at h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V' M (head✝ :: tail✝) (phi.imp_ psi) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V' M (head✝ :: tail✝) (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V' M (head✝ :: tail✝) (phi.imp_ psi) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) phi → Holds D I V M (head✝ :: tail✝) psi ↔
Holds D I V' M (head✝ :: tail✝) phi → Holds D I V' M (head✝ :: tail✝) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | cases h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) phi → Holds D I V M (head✝ :: tail✝) psi ↔
Holds D I V' M (head✝ :: tail✝) phi → Holds D I V' M (head✝ :: tail✝) psi | case intro
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
left✝ : NoMetaVarAndAllFreeInList vs phi
right✝ : NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) phi → Holds D I V M (head✝ :: tail✝) psi ↔
Holds D I V' M (head✝ :: tail✝) phi → Holds D I V' M (head✝ :: tail✝) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | congr! 1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) phi → Holds D I V M (head✝ :: tail✝) psi ↔
Holds D I V' M (head✝ :: tail✝) phi → Holds D I V' M (head✝ :: tail✝) psi | case a.h.a
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi
case a.h'.a
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
a✝ : Holds D I V' M (head✝ :: tail✝) phi
⊢ Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact phi_ih V V' vs h1_left h2 | case a.h.a
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact psi_ih V V' vs h1_right h2 | case a.h'.a
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs psi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
a✝ : Holds D I V' M (head✝ :: tail✝) phi
⊢ Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V' M (head✝ :: tail✝) psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList at h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V' M (head✝ :: tail✝) (forall_ x phi) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V' M (head✝ :: tail✝) (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V' M (head✝ :: tail✝) (forall_ x phi) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) M (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V' x d) M (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | apply forall_congr' | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) M (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V' x d) M (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) M (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | intro a | case h
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) M (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
⊢ Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) M (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | apply phi_ih | case h
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
⊢ Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) M (head✝ :: tail✝) phi | case h.h1
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
⊢ NoMetaVarAndAllFreeInList ?h.vs phi
case h.h2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
⊢ ∀ v ∈ ?h.vs, Function.updateITE V x a v = Function.updateITE V' x a v
case h.vs
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
⊢ List VarName |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact h1 | case h.h1
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
⊢ NoMetaVarAndAllFreeInList ?h.vs phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | intro v a1 | case h.h2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
⊢ ∀ v ∈ x :: vs, Function.updateITE V x a v = Function.updateITE V' x a v | case h.h2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
⊢ Function.updateITE V x a v = Function.updateITE V' x a v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold Function.updateITE | case h.h2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
⊢ Function.updateITE V x a v = Function.updateITE V' x a v | case h.h2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
⊢ (if v = x then a else V v) = if v = x then a else V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | split_ifs | case h.h2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
⊢ (if v = x then a else V v) = if v = x then a else V' v | case pos
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
h✝ : v = x
⊢ a = a
case neg
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
h✝ : ¬v = x
⊢ V v = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | case _ c1 =>
rfl | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
c1 : v = x
⊢ a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | case _ c1 =>
simp at a1
tauto | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
c1 : ¬v = x
⊢ V v = V' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | rfl | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
c1 : v = x
⊢ a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp at a1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
a1 : v ∈ x :: vs
c1 : ¬v = x
⊢ V v = V' v | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
c1 : ¬v = x
a1 : v = x ∨ v ∈ vs
⊢ V v = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | tauto | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
c1 : ¬v = x
a1 : v = x ∨ v ∈ vs
⊢ V v = V' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I V' M [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList at h1 | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V' M (E_hd :: E_tl) (def_ X xs) | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V' M (E_hd :: E_tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V' M (E_hd :: E_tl) (def_ X xs) | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) ↔
if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F
else Holds D I V' M E_tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | split_ifs | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) ↔
if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F
else Holds D I V' M E_tl (def_ X xs) | case pos
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔
Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F
case neg
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I V' M E_tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | apply E_ih (Function.updateListITE V E_hd.args (List.map V xs)) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_hd.F E_hd.args E_hd.nf | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔
Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ v ∈ E_hd.args,
Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | intro v a1 | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ v ∈ E_hd.args,
Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | apply Function.updateListITE_fun_coincide_mem_eq_len | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v | case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ ∀ v ∈ xs, V v = V' v
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ v ∈ E_hd.args
case h3
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ E_hd.args.length = xs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | tauto | case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ ∀ v ∈ xs, V v = V' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact a1 | case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ v ∈ E_hd.args | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | tauto | case h3
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ E_hd.args.length = xs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | apply E_ih V V' (def_ X xs) vs | D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I V' M E_tl (def_ X xs) | case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ NoMetaVarAndAllFreeInList vs (def_ X xs)
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ v ∈ vs, V v = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList | case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ NoMetaVarAndAllFreeInList vs (def_ X xs) | case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ xs ⊆ vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact h1 | case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ xs ⊆ vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact h2 | case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ v ∈ vs, V v = V' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | induction E generalizing F V M M' | D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
E : Env
F : Formula
h1 : ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M E F ↔ Holds D I V M' E F | case nil
D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
F : Formula
h1 : ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M [] F ↔ Holds D I V M' [] F
case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
V : Valuation D
M M' : MetaValuation D
F : Formula
h1 : ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) F ↔ Holds D I V M' (head✝ :: tail✝) F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | case nil.def_ X xs =>
simp only [Holds] | D : Type
I : Interpretation D
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I V M' [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | induction F generalizing V M M' | case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
V : Valuation D
M M' : MetaValuation D
F : Formula
h1 : ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) F ↔ Holds D I V M' (head✝ :: tail✝) F | case cons.meta_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
a✝ : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (meta_var_ a✝).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ a✝) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ a✝)
case cons.pred_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (pred_ a✝¹ a✝).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (pred_ a✝¹ a✝) ↔ Holds D I V M' (head✝ :: tail✝) (pred_ a✝¹ a✝)
case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
a✝¹ a✝ : VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (eq_ a✝¹ a✝).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V M' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ true_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) true_ ↔ Holds D I V M' (head✝ :: tail✝) true_
case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
a✝ : Formula
a_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ a✝.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) a✝ ↔ Holds D I V M' (head✝ :: tail✝) a✝)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ a✝.not_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) a✝.not_ ↔ Holds D I V M' (head✝ :: tail✝) a✝.not_
case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ a✝¹.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) a✝¹ ↔ Holds D I V M' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ a✝.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) a✝ ↔ Holds D I V M' (head✝ :: tail✝) a✝)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (a✝¹.imp_ a✝).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V M' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ a✝.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) a✝ ↔ Holds D I V M' (head✝ :: tail✝) a✝)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (forall_ a✝¹ a✝).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V M' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (def_ a✝¹ a✝).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V M' (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | case meta_var_ X =>
unfold Formula.metaVarSet at h1
simp at h1
simp only [Holds]
exact h1 V | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (meta_var_ X).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | case pred_ X xs =>
simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : PredName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (pred_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (pred_ X xs) ↔ Holds D I V M' (head✝ :: tail✝) (pred_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | case eq_ x y =>
simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x y : VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (eq_ x y).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V M' (head✝ :: tail✝) (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | case true_ =>
simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ true_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) true_ ↔ Holds D I V M' (head✝ :: tail✝) true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | case not_ phi phi_ih =>
unfold Formula.metaVarSet at h1
simp only [Holds]
congr! 1
exact phi_ih V M M' h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.not_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V M' (head✝ :: tail✝) phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | case forall_ x phi phi_ih =>
unfold Formula.metaVarSet at h1
simp only [Holds]
apply forall_congr'
intro a
exact phi_ih (Function.updateITE V x a) M M' h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (forall_ x phi).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V M' (head✝ :: tail✝) (forall_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | unfold Formula.metaVarSet at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (meta_var_ X).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ {X}, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ {X}, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), M X V' ↔ M' X V'
⊢ M X V ↔ M' X V |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | exact h1 V | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), M X V' ↔ M' X V'
⊢ M X V ↔ M' X V | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : PredName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (pred_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (pred_ X xs) ↔ Holds D I V M' (head✝ :: tail✝) (pred_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x y : VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (eq_ x y).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V M' (head✝ :: tail✝) (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ true_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) true_ ↔ Holds D I V M' (head✝ :: tail✝) true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | unfold Formula.metaVarSet at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.not_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V M' (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V M' (head✝ :: tail✝) phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V M' (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ ¬Holds D I V M (head✝ :: tail✝) phi ↔ ¬Holds D I V M' (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ ¬Holds D I V M (head✝ :: tail✝) phi ↔ ¬Holds D I V M' (head✝ :: tail✝) phi | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | exact phi_ih V M M' h1 | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | unfold Formula.metaVarSet at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (phi.imp_ psi).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V M' (head✝ :: tail✝) (phi.imp_ psi) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet ∪ psi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V M' (head✝ :: tail✝) (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet ∪ psi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V M' (head✝ :: tail✝) (phi.imp_ psi) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V M' (head✝ :: tail✝) (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ Holds D I V M (head✝ :: tail✝) (phi.imp_ psi) ↔ Holds D I V M' (head✝ :: tail✝) (phi.imp_ psi) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ Holds D I V M (head✝ :: tail✝) phi → Holds D I V M (head✝ :: tail✝) psi ↔
Holds D I V M' (head✝ :: tail✝) phi → Holds D I V M' (head✝ :: tail✝) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ Holds D I V M (head✝ :: tail✝) phi → Holds D I V M (head✝ :: tail✝) psi ↔
Holds D I V M' (head✝ :: tail✝) phi → Holds D I V M' (head✝ :: tail✝) psi | case a.h.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi
case a.h'.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
⊢ Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | apply phi_ih | case a.h.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi | case a.h.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | intro V' X a1 | case a.h.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
⊢ ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V' | case a.h.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
V' : Valuation D
X : MetaVarName
a1 : X ∈ phi.metaVarSet
⊢ M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | apply h1 | case a.h.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
V' : Valuation D
X : MetaVarName
a1 : X ∈ phi.metaVarSet
⊢ M X V' ↔ M' X V' | case a.h.a.h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
V' : Valuation D
X : MetaVarName
a1 : X ∈ phi.metaVarSet
⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | left | case a.h.a.h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
V' : Valuation D
X : MetaVarName
a1 : X ∈ phi.metaVarSet
⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet | case a.h.a.h1.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
V' : Valuation D
X : MetaVarName
a1 : X ∈ phi.metaVarSet
⊢ X ∈ phi.metaVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | exact a1 | case a.h.a.h1.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
V' : Valuation D
X : MetaVarName
a1 : X ∈ phi.metaVarSet
⊢ X ∈ phi.metaVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | apply psi_ih | case a.h'.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
⊢ Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi | case a.h'.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
⊢ ∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | intro V' X a1 | case a.h'.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
⊢ ∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V' | case a.h'.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
V' : Valuation D
X : MetaVarName
a1 : X ∈ psi.metaVarSet
⊢ M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | apply h1 | case a.h'.a.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
V' : Valuation D
X : MetaVarName
a1 : X ∈ psi.metaVarSet
⊢ M X V' ↔ M' X V' | case a.h'.a.h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
V' : Valuation D
X : MetaVarName
a1 : X ∈ psi.metaVarSet
⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | right | case a.h'.a.h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
V' : Valuation D
X : MetaVarName
a1 : X ∈ psi.metaVarSet
⊢ X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet | case a.h'.a.h1.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
V' : Valuation D
X : MetaVarName
a1 : X ∈ psi.metaVarSet
⊢ X ∈ psi.metaVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | exact a1 | case a.h'.a.h1.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
psi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ psi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) psi ↔ Holds D I V M' (head✝ :: tail✝) psi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D) (X : MetaVarName), X ∈ phi.metaVarSet ∨ X ∈ psi.metaVarSet → (M X V' ↔ M' X V')
a✝ : Holds D I V M' (head✝ :: tail✝) phi
V' : Valuation D
X : MetaVarName
a1 : X ∈ psi.metaVarSet
⊢ X ∈ psi.metaVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | unfold Formula.metaVarSet at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (forall_ x phi).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V M' (head✝ :: tail✝) (forall_ x phi) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V M' (head✝ :: tail✝) (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V M' (head✝ :: tail✝) (forall_ x phi) | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) M (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V x d) M' (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | apply forall_congr' | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) M (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V x d) M' (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) M' (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | intro a | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) M' (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
a : D
⊢ Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) M' (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | exact phi_ih (Function.updateITE V x a) M M' h1 | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : Valuation D) (M M' : MetaValuation D),
(∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
(Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V'
a : D
⊢ Holds D I (Function.updateITE V x a) M (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) M' (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I V M' [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [Holds] | D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V M' (E_hd :: E_tl) (def_ X xs) | D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) ↔
if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M' E_tl E_hd.F
else Holds D I V M' E_tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | split_ifs | D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) ↔
if X = E_hd.name ∧ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M' E_tl E_hd.F
else Holds D I V M' E_tl (def_ X xs) | case pos
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M' E_tl E_hd.F
case neg
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I V M' E_tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | apply E_ih | case pos
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M' E_tl E_hd.F | case pos.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ (V' : Valuation D), ∀ X ∈ E_hd.F.metaVarSet, M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp only [no_meta_var_imp_meta_var_set_is_empty E_hd.F E_hd.args E_hd.nf] | case pos.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ (V' : Valuation D), ∀ X ∈ E_hd.F.metaVarSet, M X V' ↔ M' X V' | case pos.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp | case pos.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, M X V' ↔ M' X V' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | apply E_ih | case neg
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I V M' E_tl (def_ X xs) | case neg.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | unfold Formula.metaVarSet | case neg.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V' | case neg.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var | [1203, 1] | [1269, 11] | simp | case neg.h1
D : Type
I : Interpretation D
E_hd : Definition
E_tl : List Definition
E_ih :
∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
(∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M E_tl F ↔ Holds D I V M' E_tl F)
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, M X V' ↔ M' X V' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var_no_meta_var | [1272, 1] | [1284, 7] | apply holds_coincide_meta_var | D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
E : Env
F : Formula
h1 : F.metaVarSet = ∅
⊢ Holds D I V M E F ↔ Holds D I V M' E F | case h1
D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
E : Env
F : Formula
h1 : F.metaVarSet = ∅
⊢ ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var_no_meta_var | [1272, 1] | [1284, 7] | simp only [h1] | case h1
D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
E : Env
F : Formula
h1 : F.metaVarSet = ∅
⊢ ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V' | case h1
D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
E : Env
F : Formula
h1 : F.metaVarSet = ∅
⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, M X V' ↔ M' X V' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_meta_var_no_meta_var | [1272, 1] | [1284, 7] | simp | case h1
D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
E : Env
F : Formula
h1 : F.metaVarSet = ∅
⊢ ∀ (V' : Valuation D), ∀ X ∈ ∅, M X V' ↔ M' X V' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | induction E | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
E : Env
name : DefName
args : List VarName
h1 : Holds D I V M E (def_ name args)
⊢ ∃ d ∈ E, name = d.name ∧ args.length = d.args.length | case nil
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
h1 : Holds D I V M [] (def_ name args)
⊢ ∃ d ∈ [], name = d.name ∧ args.length = d.args.length
case cons
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : Holds D I V M tail✝ (def_ name args) → ∃ d ∈ tail✝, name = d.name ∧ args.length = d.args.length
h1 : Holds D I V M (head✝ :: tail✝) (def_ name args)
⊢ ∃ d ∈ head✝ :: tail✝, name = d.name ∧ args.length = d.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | case nil =>
simp only [Holds] at h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
h1 : Holds D I V M [] (def_ name args)
⊢ ∃ d ∈ [], name = d.name ∧ args.length = d.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | simp only [Holds] at h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
h1 : Holds D I V M [] (def_ name args)
⊢ ∃ d ∈ [], name = d.name ∧ args.length = d.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | simp only [Holds] at h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
h1 : Holds D I V M (E_hd :: E_tl) (def_ name args)
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
h1 :
if name = E_hd.name ∧ args.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ name args)
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | split_ifs at h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
h1 :
if name = E_hd.name ∧ args.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ name args)
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | case pos
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
h✝ : name = E_hd.name ∧ args.length = E_hd.args.length
h1 : Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length
case neg
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
h✝ : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | case _ c1 =>
apply Exists.intro E_hd
simp
exact c1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
c1 : name = E_hd.name ∧ args.length = E_hd.args.length
h1 : Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | apply Exists.intro E_hd | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
c1 : name = E_hd.name ∧ args.length = E_hd.args.length
h1 : Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
c1 : name = E_hd.name ∧ args.length = E_hd.args.length
h1 : Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
⊢ E_hd ∈ E_hd :: E_tl ∧ name = E_hd.name ∧ args.length = E_hd.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | simp | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
c1 : name = E_hd.name ∧ args.length = E_hd.args.length
h1 : Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
⊢ E_hd ∈ E_hd :: E_tl ∧ name = E_hd.name ∧ args.length = E_hd.args.length | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
c1 : name = E_hd.name ∧ args.length = E_hd.args.length
h1 : Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
⊢ name = E_hd.name ∧ args.length = E_hd.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | exact c1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
c1 : name = E_hd.name ∧ args.length = E_hd.args.length
h1 : Holds D I (Function.updateListITE V E_hd.args (List.map V args)) M E_tl E_hd.F
⊢ name = E_hd.name ∧ args.length = E_hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | specialize E_ih h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V M E_tl (def_ name args) → ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
E_ih : ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | apply Exists.elim E_ih | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
E_ih : ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
E_ih : ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
⊢ ∀ (a : Definition),
a ∈ E_tl ∧ name = a.name ∧ args.length = a.args.length →
∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | intro d E_ih_1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
E_ih : ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
⊢ ∀ (a : Definition),
a ∈ E_tl ∧ name = a.name ∧ args.length = a.args.length →
∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
E_ih : ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
d : Definition
E_ih_1 : d ∈ E_tl ∧ name = d.name ∧ args.length = d.args.length
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.def_holds_imp_def_in_env | [1287, 1] | [1320, 29] | cases E_ih_1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
E_ih : ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
d : Definition
E_ih_1 : d ∈ E_tl ∧ name = d.name ∧ args.length = d.args.length
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length | case intro
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
name : DefName
args : List VarName
E_hd : Definition
E_tl : List Definition
c1 : ¬(name = E_hd.name ∧ args.length = E_hd.args.length)
h1 : Holds D I V M E_tl (def_ name args)
E_ih : ∃ d ∈ E_tl, name = d.name ∧ args.length = d.args.length
d : Definition
left✝ : d ∈ E_tl
right✝ : name = d.name ∧ args.length = d.args.length
⊢ ∃ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length |
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