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/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply h1 | case h.e'_1.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' ... | case h.e'_1.h1.a
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v =... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | left | case h.e'_1.h1.a
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο... | case h.e'_1.h1.a.h
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h1.a
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | exact a1 | case h.e'_1.h1.a.h
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h1.a.h
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply psi_ih | case h.e'_2
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : V... | case h.e'_2.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | intro v a1 | case h.e'_2.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' ... | case h.e'_2.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v =... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply h1 | case h.e'_2.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' ... | case h.e'_2.h1.a
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v =... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | right | case h.e'_2.h1.a
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο... | case h.e'_2.h1.a.h
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1.a
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | exact a1 | case h.e'_2.h1.a.h
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1.a.h
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | congr! 1 | x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ exists_ x (fastReplaceFree (Function.updateITE Ο x x) phi) =
ex... | case h.e'_2
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ fastReplaceFree (Function.updateITE Ο x x) phi = fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ exists... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply phi_ih | case h.e'_2
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ fastReplaceFree (Function.updateITE Ο x x) phi = fastRe... | case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ β (v : VarName), isFreeIn v phi β Function.updateITE... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | intro v a1 | case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ β (v : VarName), isFreeIn v phi β Function.updateITE... | case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
β’ Function.updateITE Ο... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | simp only [Function.updateITE] | case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
β’ Function.updateITE Ο... | case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
β’ (if v = x then x els... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | split_ifs | case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
β’ (if v = x then x els... | case pos
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
hβ : v = x
β’ x = x
case neg... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | case _ c1 =>
rfl | x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
c1 : v = x
β’ x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarN... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | case _ c1 =>
apply h1
tauto | x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
c1 : Β¬v = x
β’ Ο v = Ο' v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarN... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | rfl | x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
c1 : v = x
β’ x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarN... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply h1 | x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
c1 : Β¬v = x
β’ Ο v = Ο' v | case a
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
c1 : Β¬v = x
β’ Β¬v = x β§ isFreeI... | Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarN... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | tauto | case a
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
c1 : Β¬v = x
β’ Β¬v = x β§ isFreeI... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
v... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | induction F generalizing binders | F : Formula
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders F = replaceFreeAux Ο' binders F | case pred_const_
Ο Ο' : VarName β VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders (pred_const_ aβΒΉ aβ) = replaceFreeAux Ο' binders (pred_const_ aβΒΉ aβ)
case pred_var_
Ο Ο' : VarName β VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset V... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders F = replaceFreeAux Ο' binders F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | all_goals
simp only [replaceFreeAux] | case pred_const_
Ο Ο' : VarName β VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders (pred_const_ aβΒΉ aβ) = replaceFreeAux Ο' binders (pred_const_ aβΒΉ aβ)
case pred_var_
Ο Ο' : VarName β VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset V... | case pred_const_
Ο Ο' : VarName β VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ pred_const_ aβΒΉ (List.map (fun x => if x β binders then Ο x else x) aβ) =
pred_const_ aβΒΉ (List.map (fun x => if x β binders then Ο' x else x) aβ)
case pred_var_
Ο Ο' : VarName β Var... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
Ο Ο' : VarName β VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders (pred_const_ aβΒΉ aβ) = replaceFreeAux Ο' binders (pred_const_ aβΒΉ aβ)
case pred_var_
Ο Ο' : VarName ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr! 1
simp only [List.map_eq_map_iff]
intro x _
split_ifs <;> tauto | Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ def_ X (List.map (fun x => if x β binders then Ο x else x) xs) =
def_ X (List.map (fun x => if x β binders then Ο' x else x) xs) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ def_ X (List.map (fun x => if x β binders then Ο x else x) xs) =
def_ X (List.map (fun x => if x β binders then Ο' x else x) xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case not_ phi phi_ih =>
simp
tauto | Ο Ο' : VarName β VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (replaceFreeAux Ο binders phi).not_ = (replaceFreeAux Ο' binders phi).not_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (replaceFreeAux Ο binders ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto | Ο Ο' : VarName β VarName
phi psi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
psi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders psi = replaceFreeAux Ο' binders psi
binde... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
phi psi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
psi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β repla... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr! 1
apply phi_ih
simp
tauto | Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ exists_ x (replaceFreeAux Ο (binders βͺ {x}) phi) = exists_ x (replaceFreeAu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ exists_ x (rep... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp only [replaceFreeAux] | case def_
Ο Ο' : VarName β VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders (def_ aβΒΉ aβ) = replaceFreeAux Ο' binders (def_ aβΒΉ aβ) | case def_
Ο Ο' : VarName β VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ def_ aβΒΉ (List.map (fun x => if x β binders then Ο x else x) aβ) =
def_ aβΒΉ (List.map (fun x => if x β binders then Ο' x else x) aβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
Ο Ο' : VarName β VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders (def_ aβΒΉ aβ) = replaceFreeAux Ο' binders (def_ aβΒΉ aβ)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | congr! 1 | Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ def_ X (List.map (fun x => if x β binders then Ο x else x) xs) =
def_ X (List.map (fun x => if x β binders then Ο' x else x) xs) | case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ List.map (fun x => if x β binders then Ο x else x) xs = List.map (fun x => if x β binders then Ο' x else x) xs | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ def_ X (List.map (fun x => if x β binders then Ο x else x) xs) =
def_ X (List.map (fun x => if x β binders then Ο' x else x) xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp only [List.map_eq_map_iff] | case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ List.map (fun x => if x β binders then Ο x else x) xs = List.map (fun x => if x β binders then Ο' x else x) xs | case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ β x β xs, (if x β binders then Ο x else x) = if x β binders then Ο' x else x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ List.map (fun x => if x β binders then Ο x else x) xs = List.map (fun x => if x β binders then Ο' x else x) xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | intro x _ | case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ β x β xs, (if x β binders then Ο x else x) = if x β binders then Ο' x else x | case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
x : VarName
aβ : x β xs
β’ (if x β binders then Ο x else x) = if x β binders then Ο' x else x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ β x β xs, (if x β binders then Ο x else x) = if x β binders then Ο' x else x
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | split_ifs <;> tauto | case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
x : VarName
aβ : x β xs
β’ (if x β binders then Ο x else x) = if x β binders then Ο' x else x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
Ο Ο' : VarName β VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
x : VarName
aβ : x β xs
β’ (if x β binders then Ο x else x) = if x β binders then Ο' x else x
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | congr! 1 | Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ eq_ (if x β binders then Ο x else x) (if y β binders then Ο y else y) =
eq_ (if x β binders then Ο' x else x) (if y β binders then Ο' y else y) | case h.e'_1
Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (if x β binders then Ο x else x) = if x β binders then Ο' x else x
case h.e'_2
Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (if y β binders then Ο y else y) =... | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ eq_ (if x β binders then Ο x else x) (if y β binders then Ο y else y) =
eq_ (if x β binders then Ο' x else x) (if y β binders then Ο' y else y)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | split_ifs <;> tauto | case h.e'_1
Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (if x β binders then Ο x else x) = if x β binders then Ο' x else x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (if x β binders then Ο x else x) = if x β binders then Ο' x else x
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | split_ifs <;> tauto | case h.e'_2
Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (if y β binders then Ο y else y) = if y β binders then Ο' y else y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
Ο Ο' : VarName β VarName
x y : VarName
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (if y β binders then Ο y else y) = if y β binders then Ο' y else y
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp | Ο Ο' : VarName β VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (replaceFreeAux Ο binders phi).not_ = (replaceFreeAux Ο' binders phi).not_ | Ο Ο' : VarName β VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ (replaceFreeAux Ο binders ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | tauto | Ο Ο' : VarName β VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο binders p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp | Ο Ο' : VarName β VarName
phi psi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
psi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders psi = replaceFreeAux Ο' binders psi
binde... | Ο Ο' : VarName β VarName
phi psi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
psi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders psi = replaceFreeAux Ο' binders psi
binde... | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
phi psi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
psi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β repla... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | tauto | Ο Ο' : VarName β VarName
phi psi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
psi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders psi = replaceFreeAux Ο' binders psi
binde... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
phi psi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
psi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β repla... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | congr! 1 | Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ exists_ x (replaceFreeAux Ο (binders βͺ {x}) phi) = exists_ x (replaceFreeAu... | case h.e'_2
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο (binders βͺ {x}) phi = replaceFreeAux Ο' (binde... | Please generate a tactic in lean4 to solve the state.
STATE:
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ exists_ x (rep... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | apply phi_ih | case h.e'_2
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ replaceFreeAux Ο (binders βͺ {x}) phi = replaceFreeAux Ο' (binde... | case h.e'_2.h1
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ β v β binders βͺ {x}, Ο v = Ο' v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ re... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp | case h.e'_2.h1
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ β v β binders βͺ {x}, Ο v = Ο' v | case h.e'_2.h1
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ β v β binders, Β¬v = x β Ο v = Ο' v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | tauto | case h.e'_2.h1
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’ β v β binders, Β¬v = x β Ο v = Ο' v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
Ο Ο' : VarName β VarName
x : VarName
phi : Formula
phi_ih :
β (binders : Finset VarName),
(β v β binders, Ο v = Ο' v) β replaceFreeAux Ο binders phi = replaceFreeAux Ο' binders phi
binders : Finset VarName
h1 : β v β binders, Ο v = Ο' v
β’... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | induction F generalizing Ο | Ο : VarName β VarName
c : Char
F : Formula
β’ (sub Ο c F).freeVarSet = Finset.image Ο F.freeVarSet | case pred_const_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (sub Ο c (pred_const_ aβΒΉ aβ)).freeVarSet = Finset.image Ο (pred_const_ aβΒΉ aβ).freeVarSet
case pred_var_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (sub Ο c (pred_var_ aβΒΉ aβ)).freeVarSet = Finset.image Ο (pred_v... | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
c : Char
F : Formula
β’ (sub Ο c F).freeVarSet = Finset.image Ο F.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | all_goals
simp only [sub]
simp only [freeVarSet] | case pred_const_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (sub Ο c (pred_const_ aβΒΉ aβ)).freeVarSet = Finset.image Ο (pred_const_ aβΒΉ aβ).freeVarSet
case pred_var_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (sub Ο c (pred_var_ aβΒΉ aβ)).freeVarSet = Finset.image Ο (pred_v... | case pred_const_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (List.map Ο aβ).toFinset = Finset.image Ο aβ.toFinset
case pred_var_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (List.map Ο aβ).toFinset = Finset.image Ο aβ.toFinset
case eq_
c : Char
aβΒΉ aβ : VarName
Ο : VarName... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (sub Ο c (pred_const_ aβΒΉ aβ)).freeVarSet = Finset.image Ο (pred_const_ aβΒΉ aβ).freeVarSet
case pred_var_
c : Char
aβΒΉ : PredName
aβ : List VarName
Ο : VarName β VarName
β’ (su... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case pred_const_ X xs | pred_var_ X xs | eq_ x y | def_ X xs =>
apply Finset.ext
intro a
simp | c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
β’ (List.map Ο xs).toFinset = Finset.image Ο xs.toFinset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
β’ (List.map Ο xs).toFinset = Finset.image Ο xs.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case true_ | false_ =>
simp | c : Char
Ο : VarName β VarName
β’ β
= Finset.image Ο β
| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
Ο : VarName β VarName
β’ β
= Finset.image Ο β
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case not_ phi phi_ih =>
exact phi_ih Ο | c : Char
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [phi_ih]
simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x Ο]
split_ifs
case _ c1 =>
obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE Ο x x) (freeVarSet phi))
generalize (
fresh x c (Finset.image (Fun... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [sub] | case def_
c : Char
aβΒΉ : DefName
aβ : List VarName
Ο : VarName β VarName
β’ (sub Ο c (def_ aβΒΉ aβ)).freeVarSet = Finset.image Ο (def_ aβΒΉ aβ).freeVarSet | case def_
c : Char
aβΒΉ : DefName
aβ : List VarName
Ο : VarName β VarName
β’ (def_ aβΒΉ (List.map Ο aβ)).freeVarSet = Finset.image Ο (def_ aβΒΉ aβ).freeVarSet | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
c : Char
aβΒΉ : DefName
aβ : List VarName
Ο : VarName β VarName
β’ (sub Ο c (def_ aβΒΉ aβ)).freeVarSet = Finset.image Ο (def_ aβΒΉ aβ).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [freeVarSet] | case def_
c : Char
aβΒΉ : DefName
aβ : List VarName
Ο : VarName β VarName
β’ (def_ aβΒΉ (List.map Ο aβ)).freeVarSet = Finset.image Ο (def_ aβΒΉ aβ).freeVarSet | case def_
c : Char
aβΒΉ : DefName
aβ : List VarName
Ο : VarName β VarName
β’ (List.map Ο aβ).toFinset = Finset.image Ο aβ.toFinset | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
c : Char
aβΒΉ : DefName
aβ : List VarName
Ο : VarName β VarName
β’ (def_ aβΒΉ (List.map Ο aβ)).freeVarSet = Finset.image Ο (def_ aβΒΉ aβ).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.ext | c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
β’ (List.map Ο xs).toFinset = Finset.image Ο xs.toFinset | case a
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
β’ β (a : VarName), a β (List.map Ο xs).toFinset β a β Finset.image Ο xs.toFinset | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
β’ (List.map Ο xs).toFinset = Finset.image Ο xs.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | intro a | case a
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
β’ β (a : VarName), a β (List.map Ο xs).toFinset β a β Finset.image Ο xs.toFinset | case a
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
a : VarName
β’ a β (List.map Ο xs).toFinset β a β Finset.image Ο xs.toFinset | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
β’ β (a : VarName), a β (List.map Ο xs).toFinset β a β Finset.image Ο xs.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case a
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
a : VarName
β’ a β (List.map Ο xs).toFinset β a β Finset.image Ο xs.toFinset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : Char
X : DefName
xs : List VarName
Ο : VarName β VarName
a : VarName
β’ a β (List.map Ο xs).toFinset β a β Finset.image Ο xs.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | c : Char
Ο : VarName β VarName
β’ β
= Finset.image Ο β
| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
Ο : VarName β VarName
β’ β
= Finset.image Ο β
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact phi_ih Ο | c : Char
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.image_union] | c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet βͺ (sub Ο c psi).freeVarSet = Finset.image Ο (phi.freeVarSe... | c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet βͺ (sub Ο c psi).freeVarSet = Finset.image Ο phi.freeVarSet... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarS... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | congr! | c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet βͺ (sub Ο c psi).freeVarSet = Finset.image Ο phi.freeVarSet... | case h.e'_3
c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
case h.e'_4
c... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarS... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact phi_ih Ο | case h.e'_3
c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact psi_ih Ο | case h.e'_4
c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
c : Char
phi psi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
psi_ih : β (Ο : VarName β VarName), (sub Ο c psi).freeVarSet = Finset.image Ο psi.freeVarSet
Ο : VarName β VarName
β’ (sub Ο c p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [phi_ih] | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ Finset.image
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then
fresh x c (Finset.image (Function.updateITE Ο x x) phi.f... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ (sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x Ο] | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ Finset.image
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then
fresh x c (Finset.image (Function.updateITE Ο x x) phi.f... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ Finset.image
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then
fresh x c (Finset.image (Function.updateITE Ο x x) phi.f... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ Finset.image
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | split_ifs | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ Finset.image
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then
fresh x c (Finset.image (Function.updateITE Ο x x) phi.f... | case pos
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
hβ : β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x (fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet)))
phi... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
β’ Finset.image
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case _ c1 =>
obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE Ο x x) (freeVarSet phi))
generalize (
fresh x c (Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) ) = x' at *
have s2 : Finset.image (Function.updateITE Ο x x) (freeVarSet phi \ {x}) β Finset.image (Function.updateITE Ο x x) ... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x (fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet)))
phi.freeVarS... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x (fresh x c (Finset.image (Fun... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case _ c1 =>
simp at c1
have s1 : Finset.image (Function.updateITE Ο x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE Ο x x) (freeVarSet phi \ {x}) \ {x}
apply Finset.image_sdiff_singleton
simp only [Function.updateITE]
simp
simp only [s1]
clear s1
have s2 : x β Finset.image (Function.upda... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x (fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet)))
phi.freeVarS... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
s1 :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet) β
Finset.image (Function.updateITE Ο x x) phi.freeV... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x (fresh x c (Finset.image (Fun... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | generalize (
fresh x c (Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) ) = x' at * | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
s1 :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet) β
Finset.image (Function.updateITE Ο x x) phi.freeV... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Finset.image (Function.updateITE Ο x x') phi.free... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
s1 :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVa... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s2 : Finset.image (Function.updateITE Ο x x) (freeVarSet phi \ {x}) β Finset.image (Function.updateITE Ο x x) (freeVarSet phi) | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Finset.image (Function.updateITE Ο x x') phi.free... | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Finset.image (Function.updateITE Ο x x) (... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.image_subset_image | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Finset.image (Function.updateITE Ο x x) (... | case s2.h
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ phi.freeVarSet \ {x} β phi.freeVarSet
... | Please generate a tactic in lean4 to solve the state.
STATE:
case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s2.h
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ phi.freeVarSet \ {x} β phi.freeVarSet
... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s3 : x' β Finset.image (Function.updateITE Ο x x) (freeVarSet phi \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | case s3
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.not_mem_mono s2 s1 | case s3
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | Please generate a tactic in lean4 to solve the state.
STATE:
case s3
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | calc
Finset.image (Function.updateITE Ο x x') (freeVarSet phi) \ {x'}
= Finset.image (Function.updateITE Ο x x') (freeVarSet phi \ {x}) \ {x'} :=
by
{
apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE Ο x x')
simp only [Function.updateITE]
simp
}
_ = Finset.image... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE Ο x x') | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Function.updateITE] | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact Finset.sdiff_singleton_eq_self s3 | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE Ο x x) (phi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s1 : x' β Finset.image (Function.updateITE Ο x x) phi.f... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp at c1 | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.freeV... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s1 : Finset.image (Function.updateITE Ο x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE Ο x x) (freeVarSet phi \ {x}) \ {x} | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.freeV... | case s1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (p... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.image_sdiff_singleton | case s1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (p... | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Function.updateITE Ο x x x = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (su... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) phi.free... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Function.updateITE] | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Function.updateITE Ο x x x = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (su... | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ (if True then x else Ο x) = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Function.updateITE Ο x x x = x
c : Char
x : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ (if True then x else Ο x) = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s1 :
Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ (if True then x else Ο x) = x
c : Char
x : V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [s1] | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s1 :
Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s1 :
Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s1 :
Finset.image (Function.updateITE Ο x x) phi.freeVar... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | clear s1 | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s1 :
Finset.image (Function.updateITE Ο x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE Ο x x) (phi.... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE Ο x x) (p... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s1 :
Finset.image (Function.updateITE Ο x x) phi.freeVar... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s2 : x β Finset.image (Function.updateITE Ο x x) (freeVarSet phi \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE Ο x x) (p... | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ x β Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {x})
c : Char
x : VarName
phi : Formula
phi_ih... | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Finset.image (Function.updateITE Ο x x) (phi.freeVarSet ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.mem_image] | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ x β Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {x})
c : Char
x : VarName
phi : Formula
phi_ih... | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Β¬β a β phi.freeVarSet \ {x}, Function.updateITE Ο x x a = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο... | Please generate a tactic in lean4 to solve the state.
STATE:
case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ x β Finset.image (Function.updateITE Ο x x) (phi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Β¬β a β phi.freeVarSet \ {x}, Function.updateITE Ο x x a = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο... | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Function.updateITE Ο x x x_1 = x
c : Char
x : VarName
phi : Formula
phi_... | Please generate a tactic in lean4 to solve the state.
STATE:
case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ Β¬β a β phi.freeVarSet \ {x}, Function.updateITE ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Function.updateITE] | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Function.updateITE Ο x x x_1 = x
c : Char
x : VarName
phi : Formula
phi_... | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬(if x_1 = x then x else Ο x_1) = x
c : Char
x : VarName
phi : Formula
ph... | Please generate a tactic in lean4 to solve the state.
STATE:
case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬(if x_1 = x then x else Ο x_1) = x
c : Char
x : VarName
phi : Formula
ph... | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬x_1 = x β§ Β¬Ο x_1 = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬(if x_1 = x ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | tauto | case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬x_1 = x β§ Β¬Ο x_1 = x
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : ... | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s2 : x β Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {x})
β’ Finset.image (Function.updateITE Ο x x) (phi.... | Please generate a tactic in lean4 to solve the state.
STATE:
case s2
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
β’ β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬x_1 = x β§ Β¬Ο... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.sdiff_singleton_eq_self s2] | c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s2 : x β Finset.image (Function.updateITE Ο x x) (phi.freeVarSet \ {x})
β’ Finset.image (Function.updateITE Ο x x) (phi.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
x : VarName
phi : Formula
phi_ih : β (Ο : VarName β VarName), (sub Ο c phi).freeVarSet = Finset.image Ο phi.freeVarSet
Ο : VarName β VarName
c1 : β x_1 β phi.freeVarSet, Β¬x_1 = x β Β¬Ο x_1 = x
s2 : x β Finset.image (Function.updateITE Ο x x) (phi.free... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | induction F generalizing Ο V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
c : Char
F : Formula
β’ Holds D I V E (sub Ο c F) β Holds D I (V β Ο) E F | case pred_const_
D : Type
I : Interpretation D
E : Env
c : Char
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (pred_const_ aβΒΉ aβ)) β Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ)
case pred_var_
D : Type
I : Interpretation D
E : Env
c : Char
aβΒΉ : PredName
aβ : List Var... | 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
c : Char
F : Formula
β’ Holds D I V E (sub Ο c F) β Holds D I (V β Ο) E F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case pred_const_ X xs | pred_var_ X xs | eq_ x y =>
simp only [sub]
simp only [Holds]
simp | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (eq_ x y)) β Holds D I (V β Ο) E (eq_ x y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (eq_ x y)) β Holds D I (V β Ο) E (eq_ x y)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case true_ | false_ =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c false_) β Holds D I (V β Ο) E false_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c false_) β Holds D I (V β Ο) E false_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case not_ phi phi_ih =>
simp only [sub]
simp only [Holds]
congr! 1
exact phi_ih V Ο | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi.not_) β Holds D I (V β Ο) E phi.not_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi.not_) β Holds D... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (eq_ x y)) β Holds D I (V β Ο) E (eq_ x y) | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (eq_ (Ο x) (Ο y)) β Holds D I (V β Ο) E (eq_ x y) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (eq_ x y)) β Holds D I (V β Ο) E (eq_ x y)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (eq_ (Ο x) (Ο y)) β Holds D I (V β Ο) E (eq_ x y) | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ V (Ο x) = V (Ο y) β (V β Ο) x = (V β Ο) y | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (eq_ (Ο x) (Ο y)) β Holds D I (V β Ο) E (eq_ x y)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ V (Ο x) = V (Ο y) β (V β Ο) x = (V β Ο) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
Ο : VarName β VarName
β’ V (Ο x) = V (Ο y) β (V β Ο) x = (V β Ο) y
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c false_) β Holds D I (V β Ο) E false_ | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E false_ β Holds D I (V β Ο) E false_ | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c false_) β Holds D I (V β Ο) E false_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E false_ β Holds D I (V β Ο) E false_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E false_ β Holds D I (V β Ο) E false_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi.not_) β Holds D I (V β Ο) E phi.not_ | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi).not_ β Holds D I (V β Ο) E phi.not_ | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi.not_) β Holds D... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi).not_ β Holds D I (V β Ο) E phi.not_ | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Β¬Holds D I V E (sub Ο c phi) β Β¬Holds D I (V β Ο) E phi | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi).not_ β Holds D... |
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