url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
⊢ ∀ H ∈ Δ', Holds D I V E H | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
H' : Formula
a2 : H' ∈ Δ'
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
H' : Formula
a2 : H' ∈ Δ'
⊢ Holds D I V E H' | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ'
a1 : Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a1 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ'
a1 : Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' | case a.intro
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ'
left✝ : Holds D I V E H
right✝ : ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a1_left a1_right =>
exact a1_right H' a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ'
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1_right H' a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct Δ' phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ'
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝ | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
⊢ Holds D I V E phi✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
⊢ Holds D I V E phi✝ | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
⊢ ∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
⊢ ∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
H' : Formula
a2 : H' ∈ H :: H :: Δ'
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
H' : Formula
a2 : H' ∈ H :: H :: Δ'
⊢ Holds D I V E H' | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ H :: H :: Δ'
a1 : Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a1 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ H :: H :: Δ'
a1 : Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' | case a.intro
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ H :: H :: Δ'
left✝ : Holds D I V E H
right✝ : ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a1_left a1_right =>
simp at a2
cases a2
case _ a2 =>
simp only [a2]
exact a1_left
case _ a2 =>
exact a1_right H' a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ H :: H :: Δ'
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ H :: H :: Δ'
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
⊢ Holds D I V E H' | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' = H ∨ H' ∈ Δ'
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' = H ∨ H' ∈ Δ'
⊢ Holds D I V E H' | case inl
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
h✝ : H' = H
⊢ Holds D I V E H'
case inr
Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
h✝ : H' ∈ Δ'
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a2 =>
simp only [a2]
exact a1_left | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' = H
⊢ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a2 =>
exact a1_right H' a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' ∈ Δ'
⊢ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [a2] | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' = H
⊢ Holds D I V E H' | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' = H
⊢ Holds D I V E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1_left | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' = H
⊢ Holds D I V E H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1_right H' a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi✝ : Formula
a✝ : IsDeduct (H :: H :: Δ') phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : ∀ a ∈ Δ', Holds D I V E a
a2 : H' ∈ Δ'
⊢ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ | Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H
⊢ Holds D I V E phi✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H
⊢ Holds D I V E phi✝ | case a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H
⊢ ∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H
⊢ ∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H | case a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H
H' : Formula
a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | case a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H
H' : Formula
a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2
⊢ Holds D I V E H' | case a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2
a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply a1 | case a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2
a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H
⊢ Holds D I V E H' | case a.a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2
a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H
⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a2 | case a.a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2
a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H
⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2 | case a.a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H
a2 : H' ∈ Δ_1 ∨ H' = H_1 ∨ H' = H_2 ∨ H' ∈ Δ_2
⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | case a.a
Δ : List Formula
F : Formula
Δ_1 Δ_2 : List Formula
H_1 H_2 phi✝ : Formula
a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H
a2 : H' ∈ Δ_1 ∨ H' = H_1 ∨ H' = H_2 ∨ H' ∈ Δ_2
⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Δ : List Formula
F phi : Formula
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [phi], Holds D I V E H) → Holds D I V E phi | Δ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ [phi], Holds D I V E H
⊢ Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | Δ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ [phi], Holds D I V E H
⊢ Holds D I V E phi | Δ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : Holds D I V E phi
⊢ Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1 | Δ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : Holds D I V E phi
⊢ Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E true_ | Δ : List Formula
F : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E true_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F phi psi : Formula
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi.imp_ (psi.imp_ phi)) | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (phi.imp_ (psi.imp_ phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (phi.imp_ (psi.imp_ phi)) | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E phi → Holds D I V E psi → Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E phi → Holds D I V E psi → Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F phi psi chi : Formula
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) | Δ : List Formula
F phi psi chi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F phi psi chi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) | Δ : List Formula
F phi psi chi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ (Holds D I V E phi → Holds D I V E psi → Holds D I V E chi) →
(Holds D I V E phi → Holds D I V E psi) → Holds D I V E phi → Holds D I V E chi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | Δ : List Formula
F phi psi chi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ (Holds D I V E phi → Holds D I V E psi → Holds D I V E chi) →
(Holds D I V E phi → Holds D I V E psi) → Holds D I V E phi → Holds D I V E chi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F phi psi : Formula
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ (¬Holds D I V E phi → ¬Holds D I V E psi) → Holds D I V E psi → Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | Δ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ (¬Holds D I V E phi → ¬Holds D I V E psi) → Holds D I V E psi → Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (phi.imp_ psi)
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E psi | Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (phi.imp_ psi)
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
⊢ Holds D I V E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] at ih_major | Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (phi.imp_ psi)
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
⊢ Holds D I V E psi | Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ Holds D I V E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_major | Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ Holds D I V E psi | case a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ ∀ H ∈ Δ', Holds D I V E H
case a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ ∀ H ∈ Δ', Holds D I V E H | case a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
H' : Formula
a2 : H' ∈ Δ'
⊢ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1 H' a2 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
H' : Formula
a2 : H' ∈ Δ'
⊢ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_minor | case a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ Holds D I V E phi | case a.a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ ∀ H ∈ Δ', Holds D I V E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1 | case a.a
Δ : List Formula
F : Formula
Δ' : List Formula
phi psi : Formula
a✝¹ : IsDeduct Δ' (phi.imp_ psi)
a✝ : IsDeduct Δ' phi
ih_minor :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
ih_major :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi
⊢ ∀ H ∈ Δ', Holds D I V E H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (H.imp_ phi) | Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
⊢ Holds D I V E (H.imp_ phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
⊢ Holds D I V E (H.imp_ phi) | Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
⊢ Holds D I V E H → Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
⊢ Holds D I V E H → Holds D I V E phi | Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ Holds D I V E phi | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ ∀ H_1 ∈ H :: Δ', Holds D I V E H_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ ∀ H_1 ∈ H :: Δ', Holds D I V E H_1 | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | constructor | case a
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a | case a.left
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ Holds D I V E H
case a.right
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ ∀ a ∈ Δ', Holds D I V E a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a2 | case a.left
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ Holds D I V E H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1 | case a.right
Δ : List Formula
F : Formula
Δ' : List Formula
H phi : Formula
a✝ : IsDeduct (H :: Δ') phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : ∀ H ∈ Δ', Holds D I V E H
a2 : Holds D I V E H
⊢ ∀ a ∈ Δ', Holds D I V E a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) → Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi) →
(∀ (d : D), Holds D I (Function.updateITE V v d) E phi) → ∀ (d : D), Holds D I (Function.updateITE V v d) E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 a3 d | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi) →
(∀ (d : D), Holds D I (Function.updateITE V v d) E phi) → ∀ (d : D), Holds D I (Function.updateITE V v d) E psi | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi
a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply a2 d | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi
a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E psi | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi
a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a3 d | Δ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi
a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) →
Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | obtain s1 := FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateITE id v t) freshChar phi | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi) →
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi) →
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [s1] | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I (V ∘ Function.updateITE id v t) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Function.updateITE_comp_left] | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I (V ∘ Function.updateITE id v t) E phi | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I (Function.updateITE (V ∘ id) v (V t)) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I (Function.updateITE (V ∘ id) v (V t)) E phi | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I (Function.updateITE V v (V t)) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a2 (V t) | Δ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔
Holds D I (V ∘ Function.updateITE id v t) E phi
a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
⊢ Holds D I (Function.updateITE V v (V t)) E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi.imp_ (forall_ v phi)) | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (phi.imp_ (forall_ v phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (phi.imp_ (forall_ v phi)) | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E phi → ∀ (d : D), Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 d | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E phi → ∀ (d : D), Holds D I (Function.updateITE V v d) E phi | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | have s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E phi | case s1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
⊢ Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | {
apply Holds_coincide_Var
intro v' a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
subst c1
contradiction
case _ c1 =>
rfl
} | case s1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
⊢ Holds D I (Function.updateITE V v d) E phi | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
⊢ Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [s1] | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
⊢ Holds D I (Function.updateITE V v d) E phi | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
⊢ Holds D I V E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a2 | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi
⊢ Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply Holds_coincide_Var | case s1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
⊢ Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi | case s1.h1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
⊢ ∀ (v_1 : VarName), isFreeIn v_1 phi → Function.updateITE V v d v_1 = V v_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro v' a1 | case s1.h1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
⊢ ∀ (v_1 : VarName), isFreeIn v_1 phi → Function.updateITE V v d v_1 = V v_1 | case s1.h1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
⊢ Function.updateITE V v d v' = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Function.updateITE] | case s1.h1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
⊢ Function.updateITE V v d v' = V v' | case s1.h1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
⊢ (if v' = v then d else V v') = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | split_ifs | case s1.h1
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
⊢ (if v' = v then d else V v') = V v' | case pos
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
h✝ : v' = v
⊢ d = V v'
case neg
Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
h✝ : ¬v' = v
⊢ V v' = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ c1 =>
subst c1
contradiction | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : v' = v
⊢ d = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ c1 =>
rfl | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : ¬v' = v
⊢ V v' = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | subst c1 | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : v' = v
⊢ d = V v' | Δ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
ih : ¬isFreeIn v' phi
⊢ d = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | contradiction | Δ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
ih : ¬isFreeIn v' phi
⊢ d = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | rfl | Δ : List Formula
F : Formula
v : VarName
phi : Formula
ih : ¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : ¬v' = v
⊢ V v' = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) → Holds D I V E (forall_ v phi) | Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (forall_ v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (forall_ v phi) | Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ ∀ (d : D), Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro d | Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ ∀ (d : D), Holds D I (Function.updateITE V v d) E phi | Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
d : D
⊢ Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
d : D
⊢ Holds D I (Function.updateITE V v d) E phi | case a
Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
d : D
⊢ ∀ H ∈ [], Holds D I (Function.updateITE V v d) E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp | case a
Δ : List Formula
F : Formula
v : VarName
phi : Formula
a✝¹ : IsDeduct [] phi
ih_2 :
∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
d : D
⊢ ∀ H ∈ [], Holds D I (Function.updateITE V v d) E H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
v : VarName
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) → Holds D I V E (eq_ v v) | Δ : List Formula
F : Formula
v : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (eq_ v v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
v : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (eq_ v v) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) →
Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1))) | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1))) | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ V x_0 = V y_0 ∧ V x_1 = V y_1 → (V x_0 = V x_1 ↔ V y_0 = V y_1) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ V x_0 = V y_0 ∧ V x_1 = V y_1 → (V x_0 = V x_1 ↔ V y_0 = V y_1) | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : V x_0 = V y_0 ∧ V x_1 = V y_1
⊢ V x_0 = V x_1 ↔ V y_0 = V y_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a2 | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2 : V x_0 = V y_0 ∧ V x_1 = V y_1
⊢ V x_0 = V x_1 ↔ V y_0 = V y_1 | case intro
Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
left✝ : V x_0 = V y_0
right✝ : V x_1 = V y_1
⊢ V x_0 = V x_1 ↔ V y_0 = V y_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a2_left a2_right =>
simp only [a2_left]
simp only [a2_right] | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
⊢ V x_0 = V x_1 ↔ V y_0 = V y_1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [a2_left] | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
⊢ V x_0 = V x_1 ↔ V y_0 = V y_1 | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
⊢ V y_0 = V x_1 ↔ V y_0 = V y_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [a2_right] | Δ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
⊢ V y_0 = V x_1 ↔ V y_0 = V y_1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Δ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(∀ H ∈ [], Holds D I V E H) →
Holds D I V E ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys))) | Δ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Δ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys))) | Δ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) →
(I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V ys)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a1 | Δ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
⊢ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) →
(I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V ys)) | Δ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a✝ : ∀ H ∈ [], Holds D I V E H
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys))
⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V ys) |
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