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stringlengths 6
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_a | [1766, 1] | [1837, 13] | exact h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ β d β E, X = d.name β§ xs.length = d.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | induction F | E : Env
Ο : Instantiation
Ο : MetaInstantiation
F : Formula
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο F)
β’ IsMetaVarOrAllDefInEnv E F | case meta_var_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβ : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ aβ))
β’ IsMetaVarOrAllDefInEnv E (meta_var_ aβ)
case pred_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ : PredName
aβ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ aβΒΉ aβ))
β’ IsMetaVarOrAllDefInEnv E (pred_ aβΒΉ aβ)
case eq_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ aβ : VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ aβΒΉ aβ))
β’ IsMetaVarOrAllDefInEnv E (eq_ aβΒΉ aβ)
case true_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
β’ IsMetaVarOrAllDefInEnv E true_
case not_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβ : Formula
a_ihβ : IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ) β IsMetaVarOrAllDefInEnv E aβ
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ.not_)
β’ IsMetaVarOrAllDefInEnv E aβ.not_
case imp_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ aβ : Formula
a_ihβΒΉ : IsMetaVarOrAllDefInEnv E (sub Ο Ο aβΒΉ) β IsMetaVarOrAllDefInEnv E aβΒΉ
a_ihβ : IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ) β IsMetaVarOrAllDefInEnv E aβ
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (aβΒΉ.imp_ aβ))
β’ IsMetaVarOrAllDefInEnv E (aβΒΉ.imp_ aβ)
case forall_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ : VarName
aβ : Formula
a_ihβ : IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ) β IsMetaVarOrAllDefInEnv E aβ
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ aβΒΉ aβ))
β’ IsMetaVarOrAllDefInEnv E (forall_ aβΒΉ aβ)
case def_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ : DefName
aβ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ aβΒΉ aβ))
β’ IsMetaVarOrAllDefInEnv E (def_ aβΒΉ aβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | case meta_var_ X =>
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
β’ IsMetaVarOrAllDefInEnv E (meta_var_ X) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | case pred_ X xs =>
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ X xs))
β’ IsMetaVarOrAllDefInEnv E (pred_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | case eq_ x y =>
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ x y))
β’ IsMetaVarOrAllDefInEnv E (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | case true_ =>
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
β’ IsMetaVarOrAllDefInEnv E true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | case not_ phi phi_ih =>
unfold sub at h1
unfold IsMetaVarOrAllDefInEnv at h1
unfold IsMetaVarOrAllDefInEnv
exact phi_ih h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
β’ IsMetaVarOrAllDefInEnv E phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | case forall_ x phi phi_ih =>
unfold sub at h1
unfold IsMetaVarOrAllDefInEnv at h1
unfold IsMetaVarOrAllDefInEnv
exact phi_ih h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
β’ IsMetaVarOrAllDefInEnv E (forall_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | case def_ X xs =>
unfold sub at h1
unfold IsMetaVarOrAllDefInEnv at h1
simp at h1
unfold IsMetaVarOrAllDefInEnv
exact h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ X xs))
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
β’ IsMetaVarOrAllDefInEnv E (meta_var_ X) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
β’ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
β’ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ X xs))
β’ IsMetaVarOrAllDefInEnv E (pred_ X xs) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ X xs))
β’ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ X xs))
β’ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ x y))
β’ IsMetaVarOrAllDefInEnv E (eq_ x y) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ x y))
β’ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ x y))
β’ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
β’ IsMetaVarOrAllDefInEnv E true_ | E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
β’ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | simp only | E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
β’ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold sub at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
β’ IsMetaVarOrAllDefInEnv E phi.not_ | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi).not_
β’ IsMetaVarOrAllDefInEnv E phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi).not_
β’ IsMetaVarOrAllDefInEnv E phi.not_ | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E phi.not_ | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | exact phi_ih h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold sub at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
β’ IsMetaVarOrAllDefInEnv E (phi.imp_ psi) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1 : IsMetaVarOrAllDefInEnv E ((sub Ο Ο phi).imp_ (sub Ο Ο psi))
β’ IsMetaVarOrAllDefInEnv E (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1 : IsMetaVarOrAllDefInEnv E ((sub Ο Ο phi).imp_ (sub Ο Ο psi))
β’ IsMetaVarOrAllDefInEnv E (phi.imp_ psi) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E (phi.imp_ psi) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | cases h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi | case intro
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
leftβ : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
rightβ : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | constructor | E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1_left : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1_right : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi | case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1_left : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1_right : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E phi
case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1_left : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1_right : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | exact phi_ih h1_left | case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1_left : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1_right : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | exact psi_ih h1_right | case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
psi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi) β IsMetaVarOrAllDefInEnv E psi
h1_left : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1_right : IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
β’ IsMetaVarOrAllDefInEnv E psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold sub at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
β’ IsMetaVarOrAllDefInEnv E (forall_ x phi) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (forall_ (βΟ x) (sub Ο Ο phi))
β’ IsMetaVarOrAllDefInEnv E (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (forall_ (βΟ x) (sub Ο Ο phi))
β’ IsMetaVarOrAllDefInEnv E (forall_ x phi) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E (forall_ x phi) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | exact phi_ih h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β IsMetaVarOrAllDefInEnv E phi
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
β’ IsMetaVarOrAllDefInEnv E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold sub at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ X xs))
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ X (List.map (βΟ) xs))
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ X (List.map (βΟ) xs))
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : β d β E, X = d.name β§ (List.map (βΟ) xs).length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | simp at h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : β d β E, X = d.name β§ (List.map (βΟ) xs).length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (def_ X xs) | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ β d β E, X = d.name β§ xs.length = d.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_2_b | [1840, 1] | [1889, 13] | exact h1 | E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ β d β E, X = d.name β§ xs.length = d.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | induction h1 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1 : IsProof E Ξ Ξ F
β’ IsMetaVarOrAllDefInEnv E F | case hyp
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ : Formula
aβΒΉ : IsMetaVarOrAllDefInEnv E phiβ
aβ : phiβ β Ξβ
β’ IsMetaVarOrAllDefInEnv E phiβ
case mp
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ psiβ : Formula
aβΒΉ : IsProof E Ξβ Ξβ phiβ
aβ : IsProof E Ξβ Ξβ (phiβ.imp_ psiβ)
a_ihβΒΉ : IsMetaVarOrAllDefInEnv E phiβ
a_ihβ : IsMetaVarOrAllDefInEnv E (phiβ.imp_ psiβ)
β’ IsMetaVarOrAllDefInEnv E psiβ
case prop_1
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ psiβ : Formula
aβΒΉ : IsMetaVarOrAllDefInEnv E phiβ
aβ : IsMetaVarOrAllDefInEnv E psiβ
β’ IsMetaVarOrAllDefInEnv E (phiβ.imp_ (psiβ.imp_ phiβ))
case prop_2
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ psiβ chiβ : Formula
aβΒ² : IsMetaVarOrAllDefInEnv E phiβ
aβΒΉ : IsMetaVarOrAllDefInEnv E psiβ
aβ : IsMetaVarOrAllDefInEnv E chiβ
β’ IsMetaVarOrAllDefInEnv E ((phiβ.imp_ (psiβ.imp_ chiβ)).imp_ ((phiβ.imp_ psiβ).imp_ (phiβ.imp_ chiβ)))
case prop_3
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ psiβ : Formula
aβΒΉ : IsMetaVarOrAllDefInEnv E phiβ
aβ : IsMetaVarOrAllDefInEnv E psiβ
β’ IsMetaVarOrAllDefInEnv E ((phiβ.not_.imp_ psiβ.not_).imp_ (psiβ.imp_ phiβ))
case gen
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ : Formula
xβ : VarName
aβ : IsProof E Ξβ Ξβ phiβ
a_ihβ : IsMetaVarOrAllDefInEnv E phiβ
β’ IsMetaVarOrAllDefInEnv E (forall_ xβ phiβ)
case pred_1
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ psiβ : Formula
xβ : VarName
aβΒΉ : IsMetaVarOrAllDefInEnv E phiβ
aβ : IsMetaVarOrAllDefInEnv E psiβ
β’ IsMetaVarOrAllDefInEnv E ((forall_ xβ (phiβ.imp_ psiβ)).imp_ ((forall_ xβ phiβ).imp_ (forall_ xβ psiβ)))
case pred_2
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ : Formula
xβ : VarName
aβΒΉ : IsMetaVarOrAllDefInEnv E phiβ
aβ : NotFree Ξβ xβ phiβ
β’ IsMetaVarOrAllDefInEnv E (phiβ.imp_ (forall_ xβ phiβ))
case eq_1
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
xβ yβ : VarName
aβ : Β¬yβ = xβ
β’ IsMetaVarOrAllDefInEnv E (exists_ xβ (eq_ xβ yβ))
case eq_2
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
xβ yβ zβ : VarName
β’ IsMetaVarOrAllDefInEnv E ((eq_ xβ yβ).imp_ ((eq_ xβ zβ).imp_ (eq_ yβ zβ)))
case eq_3
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
nβ : β
nameβ : PredName
xsβ ysβ : Fin nβ β VarName
β’ IsMetaVarOrAllDefInEnv E (eqSubPred nameβ nβ xsβ ysβ)
case thm
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ Ξ'β : List (VarName Γ MetaVarName)
Ξβ Ξ'β : List Formula
phiβ : Formula
Οβ : Instantiation
Οβ : MetaInstantiation
aβΒ³ : β X β phiβ.metaVarSet, IsMetaVarOrAllDefInEnv E (Οβ X)
aβΒ² : β (x : VarName) (X : MetaVarName), (x, X) β Ξβ β NotFree Ξ'β (βΟβ x) (Οβ X)
aβΒΉ : β psi β Ξβ, IsProof E Ξ'β Ξ'β (sub Οβ Οβ psi)
aβ : IsProof E Ξβ Ξβ phiβ
a_ihβΒΉ : β psi β Ξβ, IsMetaVarOrAllDefInEnv E (sub Οβ Οβ psi)
a_ihβ : IsMetaVarOrAllDefInEnv E phiβ
β’ IsMetaVarOrAllDefInEnv E (sub Οβ Οβ phiβ)
case conv
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
Ξβ : List (VarName Γ MetaVarName)
Ξβ : List Formula
phiβ phi'β : Formula
aβΒ² : IsMetaVarOrAllDefInEnv E phi'β
aβΒΉ : IsProof E Ξβ Ξβ phiβ
aβ : IsConv E phiβ phi'β
a_ihβ : IsMetaVarOrAllDefInEnv E phiβ
β’ IsMetaVarOrAllDefInEnv E phi'β |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case hyp h1_Ξ h1_Ξ h1_phi h1_1 h1_2 =>
exact h1_1 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : h1_phi β h1_Ξ
β’ IsMetaVarOrAllDefInEnv E h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case mp h1_Ξ h1_Ξ h1_phi h1_psi h1_1 h1_2 h1_ih_1 h1_ih_2 =>
unfold IsMetaVarOrAllDefInEnv at h1_ih_2
cases h1_ih_2
case intro h1_ih_2_left h1_ih_2_right =>
exact h1_ih_2_right | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_2 : IsProof E h1_Ξ h1_Ξ (h1_phi.imp_ h1_psi)
h1_ih_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_ih_2 : IsMetaVarOrAllDefInEnv E (h1_phi.imp_ h1_psi)
β’ IsMetaVarOrAllDefInEnv E h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case prop_1 h1_Ξ h1_Ξ h1_phi h1_psi h1_1 h1_2 =>
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E (h1_phi.imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case prop_2 h1_Ξ h1_Ξ h1_phi h1_psi h1_chi h1_1 h1_2 h1_3 =>
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
β’ IsMetaVarOrAllDefInEnv E ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case prop_3 h1_Ξ h1_Ξ h1_phi h1_psi h1_1 h1_2 =>
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case gen h1_Ξ h1_Ξ h1_phi h1_x h1_1 h1_ih =>
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_ih : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case pred_1 h1_Ξ h1_Ξ h1_phi h1_psi h1_x h1_1 h1_2 =>
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E ((forall_ h1_x (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_x h1_phi).imp_ (forall_ h1_x h1_psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case pred_2 h1_Ξ h1_Ξ h1_phi h1_x h1_1 h1_2 =>
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Ξ h1_x h1_phi
β’ IsMetaVarOrAllDefInEnv E (h1_phi.imp_ (forall_ h1_x h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case eq_1 h1_Ξ h1_Ξ h1_x h1_y h1_1 =>
unfold exists_
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y : VarName
h1_1 : Β¬h1_y = h1_x
β’ IsMetaVarOrAllDefInEnv E (exists_ h1_x (eq_ h1_x h1_y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case eq_2 h1_Ξ h1_Ξ h1_x h1_y h1_z =>
unfold IsMetaVarOrAllDefInEnv
repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y h1_z : VarName
β’ IsMetaVarOrAllDefInEnv E ((eq_ h1_x h1_y).imp_ ((eq_ h1_x h1_z).imp_ (eq_ h1_y h1_z))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case eq_3 h1_Ξ h1_Ξ h1_n h1_name h1_xs h1_ys =>
sorry | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_n : β
h1_name : PredName
h1_xs h1_ys : Fin h1_n β VarName
β’ IsMetaVarOrAllDefInEnv E (eqSubPred h1_name h1_n h1_xs h1_ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case thm h1_Ξ h1_Ξ' h1_Ξ h1_Ξ' h1_phi h1_Ο h1_Ο h1_1 h1_2 h1_3 h1_4 h1_ih_1 h1_ih_2 =>
exact lem_2_a E h1_Ο h1_Ο h1_phi h1_ih_2 h1_1 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ h1_Ξ' : List (VarName Γ MetaVarName)
h1_Ξ h1_Ξ' : List Formula
h1_phi : Formula
h1_Ο : Instantiation
h1_Ο : MetaInstantiation
h1_1 : β X β h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο X)
h1_2 : β (x : VarName) (X : MetaVarName), (x, X) β h1_Ξ β NotFree h1_Ξ' (βh1_Ο x) (h1_Ο X)
h1_3 : β psi β h1_Ξ, IsProof E h1_Ξ' h1_Ξ' (sub h1_Ο h1_Ο psi)
h1_4 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_ih_1 : β psi β h1_Ξ, IsMetaVarOrAllDefInEnv E (sub h1_Ο h1_Ο psi)
h1_ih_2 : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E (sub h1_Ο h1_Ο h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case conv h1_Ξ h1_Ξ h1_phi h1_phi' h1_1 h1_2 h1_3 h1_ih =>
exact h1_1 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | exact h1_1 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : h1_phi β h1_Ξ
β’ IsMetaVarOrAllDefInEnv E h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | unfold IsMetaVarOrAllDefInEnv at h1_ih_2 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_2 : IsProof E h1_Ξ h1_Ξ (h1_phi.imp_ h1_psi)
h1_ih_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_ih_2 : IsMetaVarOrAllDefInEnv E (h1_phi.imp_ h1_psi)
β’ IsMetaVarOrAllDefInEnv E h1_psi | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_2 : IsProof E h1_Ξ h1_Ξ (h1_phi.imp_ h1_psi)
h1_ih_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_ih_2 : IsMetaVarOrAllDefInEnv E h1_phi β§ IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | cases h1_ih_2 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_2 : IsProof E h1_Ξ h1_Ξ (h1_phi.imp_ h1_psi)
h1_ih_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_ih_2 : IsMetaVarOrAllDefInEnv E h1_phi β§ IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_psi | case intro
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_2 : IsProof E h1_Ξ h1_Ξ (h1_phi.imp_ h1_psi)
h1_ih_1 leftβ : IsMetaVarOrAllDefInEnv E h1_phi
rightβ : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | case intro h1_ih_2_left h1_ih_2_right =>
exact h1_ih_2_right | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_2 : IsProof E h1_Ξ h1_Ξ (h1_phi.imp_ h1_psi)
h1_ih_1 h1_ih_2_left : IsMetaVarOrAllDefInEnv E h1_phi
h1_ih_2_right : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | exact h1_ih_2_right | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_2 : IsProof E h1_Ξ h1_Ξ (h1_phi.imp_ h1_psi)
h1_ih_1 h1_ih_2_left : IsMetaVarOrAllDefInEnv E h1_phi
h1_ih_2_right : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E (h1_phi.imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | first | constructor | assumption | case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | constructor | case right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E (h1_psi.imp_ h1_phi) | case right.left
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_psi
case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | assumption | case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
β’ IsMetaVarOrAllDefInEnv E ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | first | constructor | assumption | case right.right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
β’ IsMetaVarOrAllDefInEnv E h1_chi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | constructor | case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
β’ IsMetaVarOrAllDefInEnv E (h1_phi.imp_ h1_chi) | case right.right.left
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
β’ IsMetaVarOrAllDefInEnv E h1_phi
case right.right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
β’ IsMetaVarOrAllDefInEnv E h1_chi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | assumption | case right.right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
β’ IsMetaVarOrAllDefInEnv E h1_chi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_ih : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_ih : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_ih : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E ((forall_ h1_x (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_x h1_phi).imp_ (forall_ h1_x h1_psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | first | constructor | assumption | case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | constructor | case right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E ((forall_ h1_x h1_phi).imp_ (forall_ h1_x h1_psi)) | case right.left
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi)
case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | assumption | case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Ξ h1_x h1_phi
β’ IsMetaVarOrAllDefInEnv E (h1_phi.imp_ (forall_ h1_x h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | first | constructor | assumption | case right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Ξ h1_x h1_phi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | constructor | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Ξ h1_x h1_phi
β’ IsMetaVarOrAllDefInEnv E (h1_phi.imp_ (forall_ h1_x h1_phi)) | case left
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Ξ h1_x h1_phi
β’ IsMetaVarOrAllDefInEnv E h1_phi
case right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Ξ h1_x h1_phi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | assumption | case right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Ξ h1_x h1_phi
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | unfold exists_ | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y : VarName
h1_1 : Β¬h1_y = h1_x
β’ IsMetaVarOrAllDefInEnv E (exists_ h1_x (eq_ h1_x h1_y)) | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y : VarName
h1_1 : Β¬h1_y = h1_x
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x (eq_ h1_x h1_y).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y : VarName
h1_1 : Β¬h1_y = h1_x
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x (eq_ h1_x h1_y).not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y : VarName
h1_1 : Β¬h1_y = h1_x
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x (eq_ h1_x h1_y).not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | constructor | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y : VarName
h1_1 : Β¬h1_y = h1_x
β’ IsMetaVarOrAllDefInEnv E (forall_ h1_x (eq_ h1_x h1_y).not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | unfold IsMetaVarOrAllDefInEnv | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y h1_z : VarName
β’ IsMetaVarOrAllDefInEnv E ((eq_ h1_x h1_y).imp_ ((eq_ h1_x h1_z).imp_ (eq_ h1_y h1_z))) | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y h1_z : VarName
β’ IsMetaVarOrAllDefInEnv E (eq_ h1_x h1_y) β§ IsMetaVarOrAllDefInEnv E ((eq_ h1_x h1_z).imp_ (eq_ h1_y h1_z)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | repeat first | constructor | assumption | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y h1_z : VarName
β’ IsMetaVarOrAllDefInEnv E (eq_ h1_x h1_y) β§ IsMetaVarOrAllDefInEnv E ((eq_ h1_x h1_z).imp_ (eq_ h1_y h1_z)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | first | constructor | assumption | case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y h1_z : VarName
β’ IsMetaVarOrAllDefInEnv E (eq_ h1_y h1_z) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | constructor | case right.right
E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_x h1_y h1_z : VarName
β’ IsMetaVarOrAllDefInEnv E (eq_ h1_y h1_z) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | sorry | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_n : β
h1_name : PredName
h1_xs h1_ys : Fin h1_n β VarName
β’ IsMetaVarOrAllDefInEnv E (eqSubPred h1_name h1_n h1_xs h1_ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | exact lem_2_a E h1_Ο h1_Ο h1_phi h1_ih_2 h1_1 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ h1_Ξ' : List (VarName Γ MetaVarName)
h1_Ξ h1_Ξ' : List Formula
h1_phi : Formula
h1_Ο : Instantiation
h1_Ο : MetaInstantiation
h1_1 : β X β h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο X)
h1_2 : β (x : VarName) (X : MetaVarName), (x, X) β h1_Ξ β NotFree h1_Ξ' (βh1_Ο x) (h1_Ο X)
h1_3 : β psi β h1_Ξ, IsProof E h1_Ξ' h1_Ξ' (sub h1_Ο h1_Ο psi)
h1_4 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_ih_1 : β psi β h1_Ξ, IsMetaVarOrAllDefInEnv E (sub h1_Ο h1_Ο psi)
h1_ih_2 : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E (sub h1_Ο h1_Ο h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.is_proof_imp_is_meta_var_or_all_def_in_env | [1892, 1] | [1932, 15] | exact h1_1 | E : Env
Ξ : List (VarName Γ MetaVarName)
Ξ : List Formula
F : Formula
h1_Ξ : List (VarName Γ MetaVarName)
h1_Ξ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Ξ h1_Ξ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih : IsMetaVarOrAllDefInEnv E h1_phi
β’ IsMetaVarOrAllDefInEnv E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | induction E | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
E : Env
d : Definition
name : DefName
args : List VarName
h1 : E.WellFormed
h2 : d β E
h3 : name = d.name β§ args.length = d.args.length
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M E d.F β Holds D I V M E (def_ name args) | case nil
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
h1 : Env.WellFormed []
h2 : d β []
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M [] d.F β Holds D I V M [] (def_ name args)
case cons
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
headβ : Definition
tailβ : List Definition
tail_ihβ :
Env.WellFormed tailβ β
d β tailβ β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tailβ d.F β Holds D I V M tailβ (def_ name args))
h1 : Env.WellFormed (headβ :: tailβ)
h2 : d β headβ :: tailβ
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (headβ :: tailβ) d.F β
Holds D I V M (headβ :: tailβ) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | case nil =>
simp at h2 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
h1 : Env.WellFormed []
h2 : d β []
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M [] d.F β Holds D I V M [] (def_ name args) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | simp at h2 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
h1 : Env.WellFormed []
h2 : d β []
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M [] d.F β Holds D I V M [] (def_ name args) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | have s1 : Env.Nodup (hd :: tl) := well_formed_env_has_no_dup (hd :: tl) h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | have s2 : β E1 : Env, (hd :: tl) = E1 ++ tl | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | case s2
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
β’ β E1, hd :: tl = E1 ++ tl
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | apply Exists.intro [hd] | case s2
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
β’ β E1, hd :: tl = E1 ++ tl
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | case s2
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
β’ hd :: tl = [hd] ++ tl
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | simp | case s2
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
β’ hd :: tl = [hd] ++ tl
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | unfold Env.WellFormed at h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h1 : Env.WellFormed (hd :: tl)
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h1 :
(β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False) β§
IsMetaVarOrAllDefInEnv tl hd.F β§ Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | simp at h2 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
h2 : d β hd :: tl
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h1 :
(β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False) β§
IsMetaVarOrAllDefInEnv tl hd.F β§ Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h1 :
(β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False) β§
IsMetaVarOrAllDefInEnv tl hd.F β§ Env.WellFormed tl
h2 : d = hd β¨ d β tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | cases h1 | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h1 :
(β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False) β§
IsMetaVarOrAllDefInEnv tl hd.F β§ Env.WellFormed tl
h2 : d = hd β¨ d β tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | case intro
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
leftβ : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
rightβ : IsMetaVarOrAllDefInEnv tl hd.F β§ Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | cases h1_right | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
h1_left : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
h1_right : IsMetaVarOrAllDefInEnv tl hd.F β§ Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | case intro
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
h1_left : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
leftβ : IsMetaVarOrAllDefInEnv tl hd.F
rightβ : Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | simp only [Holds] | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
h1_left : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I V M (hd :: tl) (def_ name args) | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
h1_left : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
if name = hd.name β§ args.length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
else Holds D I V M tl (def_ name args) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.lem_4 | [1935, 1] | [1999, 21] | split_ifs | D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
h1_left : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
if name = hd.name β§ args.length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
else Holds D I V M tl (def_ name args) | case pos
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
h1_left : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
hβ : name = hd.name β§ args.length = hd.args.length
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β
Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
case neg
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name β§ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
Env.WellFormed tl β
d β tl β
(Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F β Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : β E1, hd :: tl = E1 ++ tl
h2 : d = hd β¨ d β tl
h1_left : β d' β tl, hd.name = d'.name β hd.args.length = d'.args.length β False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
hβ : Β¬(name = hd.name β§ args.length = hd.args.length)
β’ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F β Holds D I V M tl (def_ name args) |
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