url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | by_cases c1 : v = x | case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Function.updateITE] | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [if_pos c1] | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Function.updateITE] | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [if_neg c1] | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | cases a1 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg.inl
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c2 =>
simp only [s1 v c1 c2]
simp
exact h2 v c2 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c2 =>
contradiction | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [s1 v c1 c2] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h2 v c2 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | contradiction | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro v a1 | case h.h3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | case h.h3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | by_cases c1 : v = x | case h.h3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Function.updateITE] | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [if_pos c1] | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Function.updateITE] at a1 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [if_neg c1] at a1 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Function.updateITE] | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [if_neg c1] | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro a2 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [if_neg a2] | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply h3 v a1 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro v a1 | case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Function.updateITE] | case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | split_ifs | case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c1 =>
exact c1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c1 =>
cases a1
case _ c2 =>
exact h4 v c2
case _ c2 =>
contradiction | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact c1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | cases a1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | case inl
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c2 =>
exact h4 v c2 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c2 =>
contradiction | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h4 v c2 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | contradiction | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | induction E | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case nil
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case nil =>
simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | split_ifs | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c1 c2 =>
simp only [List.length_map] at c2
contradiction | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c1 c2 =>
simp at c2
contradiction | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case _ c1 c2 =>
exact ih | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply Holds_coincide_Var | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro v a1 | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | have s1 : List.map V xs' = List.map (V' β Ο') xs' | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [List.map_eq_map_iff] | case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro x a2 | case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | by_cases c3 : x β binders' | case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [s1] | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply Function.updateListITE_mem_eq_len | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β b... | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | obtain s2 := hd.h1 a1 | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp at s2 | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact s2 | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp at c2 | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | tauto | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h2 x c3 | case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply h3 x | case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply ih_1 x a2 c3 | case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [List.length_map] at c2 | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | contradiction | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp at c2 | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | contradiction | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact ih | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | apply substitution_theorem_aux D I (V β Ο) V E Ο β
F F' h1 | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ Holds D I (V β Ο) E F β Holds D I V E F' | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, (V β Ο) v = V (Ο v)
case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β (v : VarName), Ο v β β
β (V β Ο) v = V (Ο ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ Holds D I (V β Ο) E F β Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | simp | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, (V β Ο) v = V (Ο v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, (V β Ο) v = V (Ο v)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | simp | case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β (v : VarName), Ο v β β
β (V β Ο) v = V (Ο v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β (v : VarName), Ο v β β
β (V β Ο) v = V (Ο v)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | simp | case h4
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, v = Ο v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h4
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, v = Ο v
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | simp only [IsValid] at h2 | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : F.IsValid
β’ F'.IsValid | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : F.IsValid
β’ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | simp only [IsValid] | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F' | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | intro D I V E | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F' | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I V E F' | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | simp only [β substitution_theorem D I V E Ο F F' h1] | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I V E F' | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I (V β Ο) E F | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | apply h2 | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I (V β Ο) E F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I (V β Ο) E F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | induction h1 | Ξ : List Formula
F : Formula
h1 : IsDeduct Ξ F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ, Holds D I V E H) β Holds D I V E F | case struct_1_
Ξ : List Formula
F : Formula
Ξβ : List Formula
Hβ phiβ : Formula
aβ : IsDeduct Ξβ phiβ
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξβ, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Hβ :: Ξβ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
h1 : IsDeduct Ξ F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ, Holds D I V E H) β Holds D I V E F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case struct_1_ Ξ' H _ _ ih_2 =>
intro D I V E a1
apply ih_2
intro H' a2
simp at a1
cases a1
case _ a1_left a1_right =>
exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : Var... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case struct_2_ Ξ' H _ _ ih_2 =>
intro D I V E a1
apply ih_2
intro H' a2
simp at a1
cases a1
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 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
β’ β (D : Type) (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case struct_3_ Ξ_1 Ξ_2 H_1 H_2 _ _ ih_2 =>
intro D I V E a1
apply ih_2
intro H' a2
simp at a1
apply a1
simp at a2
tauto | Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_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] | case assume_ phi =>
intro D I V E a1
simp at a1
exact a1 | Ξ : List Formula
F phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [phi], Holds D I V E H) β Holds D I V E phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [phi], Holds D I V E H) β Holds D I V E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_0_ =>
intro D I V E _
simp only [Holds] | Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E true_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E true_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_1_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phi.imp_ (psi.imp_ phi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phi.imp_ (psi.imp_ phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_2_ phi psi chi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi chi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi chi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi)))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_3_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case pred_1_ v phi psi =>
intro D I V E _
simp only [Holds]
intro a2 a3 d
apply a2 d
exact a3 d | Ξ : List Formula
F : Formula
v : VarName
phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi)))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case pred_2_ v t phi =>
intro D I V E _
obtain s1 := FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateITE id v t) freshChar phi
simp only [Holds]
intro a2
simp only [s1]
simp only [Function.updateITE_comp_left]
simp
exact a2 (V t) | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case pred_3_ v phi ih =>
intro D I V E _
simp only [Holds]
intro a2 d
have s1 : Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
{
apply Holds_coincide_Var
intro v' a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
subst c1
contradiction
case _ c1 =>
... | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phi.imp_ (forall_ v phi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phi.imp_ (forall_ v phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case gen_ v phi _ ih_2 =>
intro D I V E _
simp only [Holds]
intro d
apply ih_2
simp | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβ : IsDeduct [] phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβ : IsDeduct [] phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case eq_1_ v =>
intro D I V E _
simp only [Holds] | Ξ : List Formula
F : Formula
v : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (eq_ v v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (eq_ v v)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case eq_2_eq_ x_0 x_1 y_0 y_1 =>
intro D I V E _
simp only [Holds]
intro a2
cases a2
case _ a2_left a2_right =>
simp only [a2_left]
simp only [a2_right] | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
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))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
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)))
TA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_false_ =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (false_.iff_ true_.not_) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (false_.iff_ true_.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_and_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_or_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_iff_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
(((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
((... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_exists_ v phi =>
intro D I V E _
simp only [Holds]
simp | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case sub_ Ξ' phi Ο _ ih_2 =>
intro D I V E a1
simp at a1
obtain s1 := Sub.Pred.All.Rec.Option.Fresh.substitution_theorem D I V E freshChar Ο
simp only [β s1] at a1
simp only [β s1]
apply ih_2
exact a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β 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 D I V E a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E ... | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ Holds D... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : Var... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ Holds D... | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignmen... |
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