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stringclasses 147
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stringclasses 147
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stringlengths 7
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stringlengths 1
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| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
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stringlengths 3
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stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
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 |
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 |
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 < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
⊢ i < j |
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 |
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 |
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 |
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 |
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 |
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 |
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_ |
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 |
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 |
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 String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (phi.imp_ psi)) ↔ Formula.lc_at (j + zs.length) (phi.imp_ psi) | 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 String
⊢ Formula.lc_at j ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)) ↔
Formula.lc_at (j + zs.length) (phi.imp_ psi) |
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 String
⊢ Formula.lc_at j ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)) ↔
Formula.lc_at (j + zs.length) (phi.imp_ psi) | 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 String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ∧
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔
Formula.lc_at (j + zs.length) phi ∧ Formula.lc_at (j + zs.length) psi |
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 String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ∧
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔
Formula.lc_at (j + zs.length) phi ∧ Formula.lc_at (j + zs.length) psi | 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 : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
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 : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi |
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 : 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 |
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 : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) ↔ Formula.lc_at (j + zs.length) psi | no goals |
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✝ phi) |
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✝ 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) (Formula.openList (j + 1) (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length + 1) phi |
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 |
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.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 |
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.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 | 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 |
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 |
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), 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
⊢ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) F |
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_len V V' hd.args xs v h1 s1
tauto
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
⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D I (Function.updateListITE V' hd.args (List.map V' xs)) tl hd.q
else Holds D I V' tl (def_ X xs) | no goals |
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
⊢ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) F | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : 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 : VarName), isFreeIn v (pred_const_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_
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 : VarName), isFreeIn v (pred_var_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_
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 a✝ : VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (eq_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
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
h1 : ∀ (v : VarName), isFreeIn v true_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.false_
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
h1 : ∀ (v : VarName), isFreeIn v false_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.not_
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✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝.not_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.imp_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.imp_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.and_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.or_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.iff_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_
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 : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (forall_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_
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 : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (exists_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
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), isFreeIn v (def_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) |
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 : VarName), isFreeIn v (pred_const_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_
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 : VarName), isFreeIn v (pred_var_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_
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 a✝ : VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (eq_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
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
h1 : ∀ (v : VarName), isFreeIn v true_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) true_
case cons.false_
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
h1 : ∀ (v : VarName), isFreeIn v false_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_
case cons.not_
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✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝.not_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_
case cons.imp_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.imp_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.and_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.or_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.iff_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_
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 : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (forall_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_
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 : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (exists_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)
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), isFreeIn v (def_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) | 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 = V' v
⊢ I.pred_const_ a✝¹ (List.map V a✝) ↔ I.pred_const_ a✝¹ (List.map V' a✝)
case cons.pred_var_
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 = V' v
⊢ I.pred_var_ a✝¹ (List.map V a✝) ↔ I.pred_var_ a✝¹ (List.map V' a✝)
case cons.eq_
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 a✝ : VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), v = a✝¹ ∨ v = a✝ → V v = V' v
⊢ V a✝¹ = V a✝ ↔ V' a✝¹ = V' a✝
case cons.not_
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✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝ → V v = V' v
⊢ ¬Holds D I V (head✝ :: tail✝) a✝ ↔ ¬Holds D I V' (head✝ :: tail✝) a✝
case cons.imp_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ → Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I V' (head✝ :: tail✝) a✝¹ → Holds D I V' (head✝ :: tail✝) a✝
case cons.and_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∧ Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I V' (head✝ :: tail✝) a✝¹ ∧ Holds D I V' (head✝ :: tail✝) a✝
case cons.or_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∨ Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I V' (head✝ :: tail✝) a✝¹ ∨ Holds D I V' (head✝ :: tail✝) a✝
case cons.iff_
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 a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V (head✝ :: tail✝) a✝) ↔
(Holds D I V' (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝)
case cons.forall_
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 : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → V v = V' v
⊢ (∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
∀ (d : D), Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) a✝
case cons.exists_
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 : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → V v = V' v
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
∃ d, Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) a✝
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
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V' head✝.args (List.map V' a✝)) tail✝ head✝.q
else Holds D I V' tail✝ (def_ a✝¹ a✝) |
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 (List.map V xs) ↔ I.pred_var_ X (List.map V' xs) | no goals |
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 y ↔ V' x = V' y | no goals |
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 = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi → V v = V' v
⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I V' (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → V v = V' v
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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), isFreeIn v (def_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) | 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
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) |
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
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝) | 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
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V' head✝.args (List.map V' a✝)) tail✝ head✝.q
else Holds D I V' tail✝ (def_ a✝¹ a✝) |
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 (List.map V xs) ↔ I.pred_var_ X (List.map V' xs) | 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
⊢ List.map V xs = List.map V' xs |
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
⊢ List.map V xs = List.map V' xs | 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
⊢ ∀ x ∈ xs, V x = V' x |
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
⊢ ∀ x ∈ xs, V x = V' x | no goals |
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 y ↔ V' x = V' y | 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 |
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 y ↔ V' x = V' y |
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' x = V' y | no goals |
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' x = V' y | no goals |
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 = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi → V v = V' v
⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I V' (head✝ :: tail✝) phi | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : 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 = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/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 → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔
(Holds D I V' (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) psi) | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : 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 → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi
case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : 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 → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F : 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 → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ ∀ (v : VarName), isFreeIn v phi → V v = V' v |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ ∀ (v : VarName), isFreeIn v phi → V v = 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v phi
⊢ V v = V' v |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v phi
⊢ V v = V' v | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi ∨ isFreeIn v psi |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi ∨ isFreeIn v psi | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi | no goals |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ ∀ (v : VarName), isFreeIn v psi → V v = V' v |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
⊢ ∀ (v : VarName), isFreeIn v psi → V v = 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v psi
⊢ V v = V' v |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v psi
⊢ V v = V' v | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v phi ∨ isFreeIn v psi |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v phi ∨ isFreeIn v psi | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v psi |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
psi_ih :
∀ (V V' : VarAssignment D),
(∀ (v : VarName), isFreeIn v psi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → V v = V' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v psi | no goals |
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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → V v = V' v
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi | 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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi | case h
D : Type
I : 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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) phi | case h
D : Type
I : 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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE V x d v = Function.updateITE V' x d v |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE V x d v = Function.updateITE V' x d v | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
v : VarName
a1 : isFreeIn v phi
⊢ Function.updateITE V x d v = Function.updateITE V' x d v |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
v : VarName
a1 : isFreeIn v phi
⊢ Function.updateITE V x d v = Function.updateITE V' x d v | 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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
v : VarName
a1 : isFreeIn v phi
⊢ (if v = x then d else V v) = if v = x then d else V' v |
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), isFreeIn v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
d : D
v : VarName
a1 : isFreeIn v phi
⊢ (if v = x then d else V v) = if v = x then d else V' v | no goals |
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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi | case h
D : Type
I : 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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) phi | case h
D : Type
I : 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 v phi → V v = V' v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = x → isFreeIn v phi → V v = V' v
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D I (Function.updateListITE V' hd.args (List.map V' xs)) tl hd.q
else Holds D I V' tl (def_ X xs) | case pos
D : Type
I : 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.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V' hd.args (List.map V' xs)) tl hd.q
case neg
D : Type
I : 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.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V' hd.args (List.map V' xs)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_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.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V' hd.args (List.map V' xs)) tl hd.q | case 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.length = hd.args.length
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v |
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.length = hd.args.length
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v | 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.length = hd.args.length
v : VarName
a1 : isFreeIn v hd.q
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v |
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.length = hd.args.length
v : VarName
a1 : isFreeIn v hd.q
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v | 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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v |
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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v | 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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ v ∈ hd.args.toFinset
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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args.toFinset
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v |
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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ v ∈ hd.args.toFinset
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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args.toFinset
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v | 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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args.toFinset
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v |
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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args.toFinset
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v | 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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v |
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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' xs) v | 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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args
⊢ hd.args.length = xs.length |
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.length = hd.args.length
v : VarName
a1 : v ∈ hd.q.freeVarSet
s1 : v ∈ hd.args
⊢ hd.args.length = xs.length | no goals |
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.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ X xs) | 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.length = hd.args.length)
⊢ ∀ (v : VarName), isFreeIn v (def_ X xs) → V v = V' v |
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.length = hd.args.length)
⊢ ∀ (v : VarName), isFreeIn v (def_ X xs) → V v = V' v | no goals |
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_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
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 (head✝ :: tail✝) F ↔ Holds D I' V (head✝ :: tail✝) F |
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
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I' V tl (def_ X xs) | no goals |
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 D I' V tail✝ F)
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 (head✝ :: tail✝) F ↔ Holds D I' V (head✝ :: tail✝) F | 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✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length (pred_const_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length (pred_var_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (eq_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length true_ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I' V (head✝ :: tail✝) true_
case cons.false_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length false_ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I' V (head✝ :: tail✝) false_
case cons.not_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝.not_ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I' V (head✝ :: tail✝) a✝.not_
case cons.imp_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.imp_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.and_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.or_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.iff_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (forall_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (exists_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (exists_ a✝¹ a✝)
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 ↔ Holds D I' V tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (def_ a✝¹ a✝) |
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✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length (pred_const_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length (pred_var_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (eq_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length true_ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I' V (head✝ :: tail✝) true_
case cons.false_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length false_ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I' V (head✝ :: tail✝) false_
case cons.not_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝.not_ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I' V (head✝ :: tail✝) a✝.not_
case cons.imp_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.imp_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.and_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.or_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (a✝¹.iff_ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I' V (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (forall_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (exists_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (exists_ a✝¹ a✝)
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 ↔ Holds D I' V tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (def_ a✝¹ a✝) | 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✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_const_ a✝¹ (List.map V a✝) ↔ I'.pred_const_ a✝¹ (List.map V a✝)
case cons.pred_var_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), P = a✝¹ ∧ ds.length = a✝.length → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_var_ a✝¹ (List.map V a✝) ↔ I'.pred_var_ a✝¹ (List.map V a✝)
case cons.not_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ¬Holds D I V (head✝ :: tail✝) a✝ ↔ ¬Holds D I' V (head✝ :: tail✝) a✝
case cons.imp_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length a✝¹ ∨ predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) a✝¹ → Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I' V (head✝ :: tail✝) a✝¹ → Holds D I' V (head✝ :: tail✝) a✝
case cons.and_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length a✝¹ ∨ predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∧ Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I' V (head✝ :: tail✝) a✝¹ ∧ Holds D I' V (head✝ :: tail✝) a✝
case cons.or_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length a✝¹ ∨ predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∨ Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I' V (head✝ :: tail✝) a✝¹ ∨ Holds D I' V (head✝ :: tail✝) a✝
case cons.iff_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝¹ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length a✝¹ ∨ predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V (head✝ :: tail✝) a✝) ↔
(Holds D I' V (head✝ :: tail✝) a✝¹ ↔ Holds D I' V (head✝ :: tail✝) a✝)
case cons.forall_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
∀ (d : D), Holds D I' (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝
case cons.exists_
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I' V (head✝ :: tail✝) a✝)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length a✝ → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
∃ d, Holds D I' (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝
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 ↔ Holds D I' V tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I' (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I' V tail✝ (def_ a✝¹ a✝) |
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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_const_ X (List.map V xs) ↔ I'.pred_const_ X (List.map V xs) | no goals |
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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), P = X ∧ ds.length = xs.length → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs) | no goals |
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 tail✝ F)
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I' V (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/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 tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi | no goals |
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 ↔ Holds D I' V tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ a✝¹ a✝) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (def_ a✝¹ a✝) | 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 ↔ Holds D I' V tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (def_ a✝¹ a✝) |
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 ↔ Holds D I' V tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I' V (head✝ :: tail✝) (def_ a✝¹ a✝) | 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 ↔ Holds D I' V tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I' (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I' V tail✝ (def_ a✝¹ a✝) |
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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_const_ X (List.map V xs) ↔ I'.pred_const_ X (List.map V xs) | no goals |
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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), P = X ∧ ds.length = xs.length → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ 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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), P = X → ds.length = xs.length → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs) |
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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), P = X → ds.length = xs.length → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ 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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : X = X → (List.map V xs).length = xs.length → (I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs))
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs) |
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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : X = X → (List.map V xs).length = xs.length → (I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs))
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ 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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs)
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs) |
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 tail✝ F)
X : PredName
xs : List VarName
V : VarAssignment D
h2 : I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs)
⊢ I.pred_var_ X (List.map V xs) ↔ I'.pred_var_ X (List.map V xs) | no goals |
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 tail✝ F)
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I' V (head✝ :: tail✝) phi | 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 ↔ Holds D I' V tail✝ F)
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | exact phi_ih 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 ↔ Holds D I' V tail✝ F)
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/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 tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔
(Holds D I' V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) psi) | case a.h.e'_1.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi
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 ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi |
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 ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi | 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 ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | intro P ds a1 | case 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 ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) | 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 ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length phi
⊢ 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] | 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 ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length phi
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds | 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✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length phi
⊢ predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | 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✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length phi
⊢ predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi | 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 tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length phi
⊢ predVarOccursIn P ds.length phi |
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 tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length phi
⊢ predVarOccursIn P ds.length phi | no goals |
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 ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi | case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Semantics.lean | FOL.NV.Holds_coincide_PredVar | [175, 1] | [236, 40] | intro P ds a1 | case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds) | case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds |
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