url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
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 ... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ta... |
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' : Valuat... | 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' : Valuat... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' : Valuat... | 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' : Valuat... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' : Valuat... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' : Valuat... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' ... | 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' ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ... | 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' ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ... | 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 :
∀... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :
∀... | 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 :
∀... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ↔ H... |
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 :
∀... | 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 :
... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ↔ H... |
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 :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :
... | 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 :
... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :
... | 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 :
... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :
... | 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 :
... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ... | 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' ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:... |
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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :... | 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' : Valuati... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 :... | 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' : Valuatio... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' : Valuatio... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : Valuatio... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : Valuatio... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 :... | 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' : Valuatio... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' : Valuatio... | 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' : Valuatio... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : Valuatio... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : Valuatio... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' ... | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 : Formu... | 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' ... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : Meta... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : Valuat... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : MetaV... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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) (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : Meta... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : Valuat... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : MetaV... | 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' : MetaV... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : MetaV... | 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' : MetaV... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : MetaV... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : M... | 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' : M... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : M... | 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' : M... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : M... | 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' : M... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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' : M... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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... | 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 : Valuatio... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 : Valuatio... | 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 : Valuatio... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : Valuatio... | 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 : Valuat... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : Valuat... | 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 : Valu... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : Valu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ↔ Hol... |
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 ... | 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 : Valuati... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : Valuati... | 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 : Valuati... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 : Valuati... | 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 : Valua... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 : Valua... | 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 : Val... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ↔ Hold... |
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 : Val... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ↔ Ho... |
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) (... | 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) (... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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) (... | 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) (... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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) (... | 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 : Valuati... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝ ... |
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 : Valuati... | 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 : Valuati... | Please generate a tactic in lean4 to solve the state.
STATE:
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'... |
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 : Valuati... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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'... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : D... |
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... | 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' : Meta... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : D... |
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' : Meta... | 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' : M... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : M... | 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' : M... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : M... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : Meta... | 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' : M... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : M... | 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' : M... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' : M... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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' | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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' | Please generate a tactic in lean4 to solve the state.
STATE:
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'
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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'
TACTIC:
|
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
hea... | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 : 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 =... |
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 (Functi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 (Functi... | 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 (Functi... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 (Functi... | 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 (Functi... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 (Functi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 ... | 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 :: ... | Please generate a tactic in lean4 to solve the state.
STATE:
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_... |
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 :: ... | 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 : Definit... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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 : Definit... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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... | 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 : D... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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