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/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at c1 | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : ¬i' < j
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
⊢ i < j | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
c1 : j ≤ i'
⊢ i < j | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : ¬i' < j
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).leng... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at a3_right | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
c1 : j ≤ i'
⊢ i < j | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
a3_right : (if h : i' - j < zs.length then free_ (zs.get ⟨i' - j, ⋯⟩) else bound_ (i' - j - zs.length + j)) = bound_ i
⊢ i < j | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | split at a3_right | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
a3_right : (if h : i' - j < zs.length then free_ (zs.get ⟨i' - j, ⋯⟩) else bound_ (i' - j - zs.length + j)) = bound_ i
⊢ i < j | case inl
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
h✝ : i' - j < zs.length
a3_right : False
⊢ i < j
case inr
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
h✝ : ¬i' - j < z... | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
a3_right : (if h : i' - j < zs.length then free_ (zs.get ⟨i' - j, ⋯⟩) else bound_ (i' - j - zs.length + j)) = bound_ i
⊢ i < j
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c2 =>
contradiction | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : i' - j < zs.length
a3_right : False
⊢ i < j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : i' - j < zs.length
a3_right : False
⊢ i < j
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c2 =>
exfalso
apply c2
exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i < j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i < j
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | contradiction | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : i' - j < zs.length
a3_right : False
⊢ i < j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : i' - j < zs.length
a3_right : False
⊢ i < j
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | exfalso | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i < j | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i < j
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply c2 | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ False | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i' - j < zs.length | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ False
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i' - j < zs.length | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : String
vs : List Var
j : ℕ
zs : List String
i i' : ℕ
a3_left : bound_ i' ∈ vs
a1 : i' < j + zs.length
c1 : j ≤ i'
c2 : ¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i' - j < zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.openList] | phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi.not_) ↔ Formula.lc_at (j + zs.length) phi.not_ | phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi).not_ ↔ Formula.lc_at (j + zs.length) phi.not_ | Please generate a tactic in lean4 to solve the state.
STATE:
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi.not_) ↔ Formula.lc... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.lc_at] | phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi).not_ ↔ Formula.lc_at (j + zs.length) phi.not_ | phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi | Please generate a tactic in lean4 to solve the state.
STATE:
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi).not_ ↔ Formula.lc... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply phi_ih | phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.openList] | phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j : ℕ
zs : List St... | phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j : ℕ
zs : List St... | Please generate a tactic in lean4 to solve the state.
STATE:
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.lc_at] | phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j : ℕ
zs : List St... | phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j : ℕ
zs : List St... | Please generate a tactic in lean4 to solve the state.
STATE:
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | congr! 1 | phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j : ℕ
zs : List St... | case a.h.e'_1.a
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j ... | Please generate a tactic in lean4 to solve the state.
STATE:
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply phi_ih | case a.h.e'_1.a
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (L... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply psi_ih | case a.h.e'_2.a
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi
j ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
phi psi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
psi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (L... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.openList] | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (forall_ a✝ phi)) ↔
Formula.lc_at (j + zs.length) (forall_ a✝ phi) | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (forall_ a✝ (Formula.openList (j + 1) (List.map free_ zs) phi)) ↔
Formula.lc_at (j + zs.length) (forall_ a✝... | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (forall_ a... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.lc_at] | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (forall_ a✝ (Formula.openList (j + 1) (List.map free_ zs) phi)) ↔
Formula.lc_at (j + zs.length) (forall_ a✝... | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at (j + 1) (Formula.openList (j + 1) (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length + 1) phi | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (forall_ a✝ (Formula.openList (j + 1) (List.map fr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [phi_ih] | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at (j + 1) (Formula.openList (j + 1) (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length + 1) phi | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at (j + 1 + zs.length) phi ↔ Formula.lc_at (j + zs.length + 1) phi | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at (j + 1) (Formula.openList (j + 1) (List.map free_ zs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | have s1 : j + 1 + List.length zs = j + List.length zs + 1 | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at (j + 1 + zs.length) phi ↔ Formula.lc_at (j + zs.length + 1) phi | case s1
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ j + 1 + zs.length = j + zs.length + 1
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.l... | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at (j + 1 + zs.length) phi ↔ Formula.lc_at (j + zs.leng... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | linarith | case s1
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ j + 1 + zs.length = j + zs.length + 1
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.l... | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
s1 : j + 1 + zs.length = j + zs.length + 1
⊢ Formula.lc_at (j + 1 + zs.length) phi ↔ Formula.lc_at (j + zs.length + 1) phi | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ j + 1 + zs.length = j + zs.length + 1
a✝ : String
phi : F... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [s1] | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
s1 : j + 1 + zs.length = j + zs.length + 1
⊢ Formula.lc_at (j + 1 + zs.length) phi ↔ Formula.lc_at (j + zs.length + 1) phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
s1 : j + 1 + zs.length = j + zs.length + 1
⊢ Formula.lc_at (j + 1 + ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | induction E generalizing F V V' | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
h1 : ∀ (v : VarName), isFreeIn v F → V v = V' v
⊢ Holds D I V E F ↔ Holds D I V' E F | case nil
D : Type
I : Interpretation D
V V' : VarAssignment D
F : Formula
h1 : ∀ (v : VarName), isFreeIn v F → V v = V' v
⊢ Holds D I V [] F ↔ Holds D I V' [] F
case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName)... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
h1 : ∀ (v : VarName), isFreeIn v F → V v = V' v
⊢ Holds D I V E F ↔ Holds D I V' E F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case cons.def_ hd tl ih X xs =>
split_ifs
case pos c1 =>
apply ih
intro v a1
simp only [isFreeIn_iff_mem_freeVarSet v hd.q] at a1
have s1 : v ∈ List.toFinset hd.args
apply Finset.mem_of_subset hd.h1 a1
simp only [List.mem_toFinset] at s1
apply Function.updateListITE_fun_coincide_mem_eq_l... | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ (if X = hd.name ∧ xs.length = hd.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | induction F generalizing V V' | case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
V V' : VarAssignment D
F : Formula
h1 : ∀ (v : VarName), isFreeIn v F → V v = V' v
⊢ H... | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ (v : VarNam... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
V V' : VarAssignment D
F ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | all_goals
simp only [isFreeIn] at h1
simp only [Holds] | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ (v : VarNam... | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ a✝, V v... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredNam... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case pred_const_ X xs | pred_var_ X xs =>
congr! 1
simp only [List.map_eq_map_iff]
exact h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ I.pred_var_ X ... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case eq_ x y =>
simp at h1
cases h1
case intro h1_left h1_right =>
congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), v = x ∨ v = y → V v = V' v
⊢ V x = V ... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V V' h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v phi → V v ... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp at h1
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih
intro v a1
simp only [Function.updateITE]
split_ifs <;> tauto | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn ... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp only [isFreeIn] at h1 | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFr... | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ a✝, V v = V' v
... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : L... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp only [Holds] | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ a✝, V v = V' v
... | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ a✝, V v = V' v
... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : L... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ I.pred_var_ X ... | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ ... | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : Lis... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | exact h1 | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
X : PredName
xs : Lis... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), v = x ∨ v = y → V v = V' v
⊢ V x = V ... | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment D
h1 : V x = V' x ∧ V y = V' y
⊢ V x = V y ↔ V' x = V' y | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | cases h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment D
h1 : V x = V' x ∧ V y = V' y
⊢ V x = V y ↔ V' x = V' y | case intro
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment D
left✝ : V x = V' x
right✝ : V y = V' y
⊢ V x = V... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case intro h1_left h1_right =>
congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment D
h1_left : V x = V' x
h1_right : V y = V' y
⊢ V x = V y ↔ V'... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment D
h1_left : V x = V' x
h1_right : V y = V' y
⊢ V x = V y ↔ V'... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x y : VarName
V V' : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v phi → V v ... | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFre... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | exact phi_ih V V' h1 | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFre... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi : Formula
phi_i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v phi → ... | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply phi_ih V V' | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | intro v a1 | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply h1 | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | case a.h.e'_1.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | left | case a.h.e'_1.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | case a.h.e'_1.a.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | exact a1 | case a.h.e'_1.a.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formu... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply psi_ih V V' | case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | intro v a1 | case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply h1 | case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), i... | case a.h.e'_2.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | right | case a.h.e'_2.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | case a.h.e'_2.a.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | exact a1 | case a.h.e'_2.a.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.a.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
phi psi : Formu... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp at h1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn ... | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn ... | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), is... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | intro d | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), is... | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), is... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
ph... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply phi_ih | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), is... | case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | 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 V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
ph... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | intro v a1 | case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp only [Function.updateITE] | case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | split_ifs <;> tauto | case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName),... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply forall_congr' | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn ... | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), is... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply exists_congr | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn ... | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), is... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
x : VarName
phi : Formula
phi_ih :
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | split_ifs | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
⊢ (if X = hd.name ∧ xs.length = hd.... | case pos
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
h✝ : X = hd.name ∧ xs.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case pos c1 =>
apply ih
intro v a1
simp only [isFreeIn_iff_mem_freeVarSet v hd.q] at a1
have s1 : v ∈ List.toFinset hd.args
apply Finset.mem_of_subset hd.h1 a1
simp only [List.mem_toFinset] at s1
apply Function.updateListITE_fun_coincide_mem_eq_len V V' hd.args xs v h1 s1
tauto | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.length = hd.a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | case neg c1 =>
apply ih
tauto | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : ¬(X = hd.name ∧ xs.length = hd... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply ih | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.length = hd.a... | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | intro v a1 | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp only [isFreeIn_iff_mem_freeVarSet v hd.q] at a1 | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | have s1 : v ∈ List.toFinset hd.args | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply Finset.mem_of_subset hd.h1 a1 | case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | simp only [List.mem_toFinset] at s1 | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply Function.updateListITE_fun_coincide_mem_eq_len V V' hd.args xs v h1 s1 | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | tauto | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : X = hd.name ∧ xs.lengt... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | apply ih | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : ¬(X = hd.name ∧ xs.length = hd... | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : ¬(X = hd.name ∧ xs.len... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_Var | [101, 1] | [172, 12] | tauto | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ xs, V v = V' v
c1 : ¬(X = hd.name ∧ xs.len... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F : Formula),
(∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tl F ↔ Holds D I V' tl F)
X : DefName
xs : List VarName
V V' : VarAssig... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | induction E generalizing F V | D : Type
I I' : Interpretation D
V : VarAssignment D
E : Env
F : Formula
h1 : I.pred_const_ = I'.pred_const_
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V E F ↔ Holds D I' V E F | case nil
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
V : VarAssignment D
F : Formula
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V [] F ↔ Holds D I' V [] F
case cons
D : Type
I I' : Interpretation D
h1 : I.pred_const_ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
V : VarAssignment D
E : Env
F : Formula
h1 : I.pred_const_ = I'.pred_const_
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V E F ↔ Holds D I' V E F
TACTI... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | case cons.def_ hd tl ih X xs =>
split_ifs
case pos c1 =>
apply ih
intro P ds a1
simp only [predVarOccursIn_iff_mem_predVarSet P ds.length] at a1
simp only [hd.h2] at a1
simp at a1
case neg c1 =>
apply ih
intro P ds a1
simp only [predVarOccursIn] at a1 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | induction F generalizing V | case cons
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds ... | case cons.pred_const_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P d... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | all_goals
simp only [predVarOccursIn] at h2
simp only [Holds] | case cons.pred_const_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail... | case cons.pred_const_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.pred_const_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | case pred_const_ X xs =>
simp only [h1] | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | case pred_var_ X xs =>
simp at h2
specialize h2 X (List.map V xs)
simp at h2
exact h2 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V h2 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
first | apply forall_congr' | apply exists_congr
intro d
exact phi_ih (Function.updateITE V x d) h2 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp only [predVarOccursIn] at h2 | case cons.def_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ H... | case cons.def_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ H... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.def_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp only [Holds] | case cons.def_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ H... | case cons.def_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ H... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.def_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp only [h1] | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp at h2 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | specialize h2 X (List.map V xs) | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | simp at h2 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | exact h2 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | congr! 1 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | case a.h.e'_1.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | exact phi_ih V h2 | case a.h.e'_1.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_va... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | congr! 1 | D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tai... | case a.h.e'_1.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pre... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply phi_ih | case a.h.e'_1.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ ... | case a.h.e'_1.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_va... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | intro P ds a1 | case a.h.e'_1.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F... | case a.h.e'_1.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply h2 | case a.h.e'_1.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F... | case a.h.e'_1.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | left | case a.h.e'_1.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝... | case a.h.e'_1.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tai... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | exact a1 | case a.h.e'_1.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tai... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | apply psi_ih | case a.h.e'_2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ ... | case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_va... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | intro P ds a1 | case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F... | case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred... |
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