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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_preserves_empty_meta_var_set | [775, 1] | [817, 8] | exact phi_ih h1_left | case left
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
psi_ih : psi.metaVarSet = ∅ → (sub σ τ psi).metaVarSet = ∅
h1_left : phi.metaVarSet = ∅
h1_right : psi.metaVarSet = ∅
⊢ (sub σ τ phi).metaVarSet = ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_preserves_empty_meta_var_set | [775, 1] | [817, 8] | exact psi_ih h1_right | case right
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
psi_ih : psi.metaVarSet = ∅ → (sub σ τ psi).metaVarSet = ∅
h1_left : phi.metaVarSet = ∅
h1_right : psi.metaVarSet = ∅
⊢ (sub σ τ psi).metaVarSet = ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_preserves_empty_meta_var_set | [775, 1] | [817, 8] | unfold Formula.metaVarSet at h1 | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
h1 : (forall_ x phi).metaVarSet = ∅
⊢ (sub σ τ (forall_ x phi)).metaVarSet = ∅ | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
h1 : phi.metaVarSet = ∅
⊢ (sub σ τ (forall_ x phi)).metaVarSet = ∅ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_preserves_empty_meta_var_set | [775, 1] | [817, 8] | unfold sub | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
h1 : phi.metaVarSet = ∅
⊢ (sub σ τ (forall_ x phi)).metaVarSet = ∅ | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
h1 : phi.metaVarSet = ∅
⊢ (forall_ (↑σ x) (sub σ τ phi)).metaVarSet = ∅ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_preserves_empty_meta_var_set | [775, 1] | [817, 8] | unfold Formula.metaVarSet | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
h1 : phi.metaVarSet = ∅
⊢ (forall_ (↑σ x) (sub σ τ phi)).metaVarSet = ∅ | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
h1 : phi.metaVarSet = ∅
⊢ (sub σ τ phi).metaVarSet = ∅ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_preserves_empty_meta_var_set | [775, 1] | [817, 8] | exact phi_ih h1 | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → (sub σ τ phi).metaVarSet = ∅
h1 : phi.metaVarSet = ∅
⊢ (sub σ τ phi).metaVarSet = ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_preserves_empty_meta_var_set | [775, 1] | [817, 8] | rfl | σ : Instantiation
τ : MetaInstantiation
X : DefName
xs : List VarName
h1 : (def_ X xs).metaVarSet = ∅
⊢ (sub σ τ (def_ X xs)).metaVarSet = ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | induction F | F : Formula
σ : Instantiation
τ τ' : MetaInstantiation
h1 : F.metaVarSet = ∅
⊢ sub σ τ F = sub σ τ' F | case meta_var_
σ : Instantiation
τ τ' : MetaInstantiation
a✝ : MetaVarName
h1 : (meta_var_ a✝).metaVarSet = ∅
⊢ sub σ τ (meta_var_ a✝) = sub σ τ' (meta_var_ a✝)
case pred_
σ : Instantiation
τ τ' : MetaInstantiation
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_ a✝¹ a✝).metaVarSet = ∅
⊢ sub σ τ (pred_ a✝¹ a✝) = sub σ τ' (pred_ a✝¹ a✝)
case eq_
σ : Instantiation
τ τ' : MetaInstantiation
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).metaVarSet = ∅
⊢ sub σ τ (eq_ a✝¹ a✝) = sub σ τ' (eq_ a✝¹ a✝)
case true_
σ : Instantiation
τ τ' : MetaInstantiation
h1 : true_.metaVarSet = ∅
⊢ sub σ τ true_ = sub σ τ' true_
case not_
σ : Instantiation
τ τ' : MetaInstantiation
a✝ : Formula
a_ih✝ : a✝.metaVarSet = ∅ → sub σ τ a✝ = sub σ τ' a✝
h1 : a✝.not_.metaVarSet = ∅
⊢ sub σ τ a✝.not_ = sub σ τ' a✝.not_
case imp_
σ : Instantiation
τ τ' : MetaInstantiation
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.metaVarSet = ∅ → sub σ τ a✝¹ = sub σ τ' a✝¹
a_ih✝ : a✝.metaVarSet = ∅ → sub σ τ a✝ = sub σ τ' a✝
h1 : (a✝¹.imp_ a✝).metaVarSet = ∅
⊢ sub σ τ (a✝¹.imp_ a✝) = sub σ τ' (a✝¹.imp_ a✝)
case forall_
σ : Instantiation
τ τ' : MetaInstantiation
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.metaVarSet = ∅ → sub σ τ a✝ = sub σ τ' a✝
h1 : (forall_ a✝¹ a✝).metaVarSet = ∅
⊢ sub σ τ (forall_ a✝¹ a✝) = sub σ τ' (forall_ a✝¹ a✝)
case def_
σ : Instantiation
τ τ' : MetaInstantiation
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).metaVarSet = ∅
⊢ sub σ τ (def_ a✝¹ a✝) = sub σ τ' (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | case meta_var_ X =>
unfold Formula.metaVarSet at h1
simp at h1 | σ : Instantiation
τ τ' : MetaInstantiation
X : MetaVarName
h1 : (meta_var_ X).metaVarSet = ∅
⊢ sub σ τ (meta_var_ X) = sub σ τ' (meta_var_ X) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | case pred_ X xs =>
rfl | σ : Instantiation
τ τ' : MetaInstantiation
X : PredName
xs : List VarName
h1 : (pred_ X xs).metaVarSet = ∅
⊢ sub σ τ (pred_ X xs) = sub σ τ' (pred_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | case eq_ x y =>
rfl | σ : Instantiation
τ τ' : MetaInstantiation
x y : VarName
h1 : (eq_ x y).metaVarSet = ∅
⊢ sub σ τ (eq_ x y) = sub σ τ' (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | case true_ =>
rfl | σ : Instantiation
τ τ' : MetaInstantiation
h1 : true_.metaVarSet = ∅
⊢ sub σ τ true_ = sub σ τ' true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | case not_ phi phi_ih =>
unfold Formula.metaVarSet at h1
unfold sub
congr! 1
exact phi_ih h1 | σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.not_.metaVarSet = ∅
⊢ sub σ τ phi.not_ = sub σ τ' phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | case forall_ x phi phi_ih =>
unfold Formula.metaVarSet at h1
unfold sub
congr! 1
exact phi_ih h1 | σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : (forall_ x phi).metaVarSet = ∅
⊢ sub σ τ (forall_ x phi) = sub σ τ' (forall_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | case def_ X xs =>
rfl | σ : Instantiation
τ τ' : MetaInstantiation
X : DefName
xs : List VarName
h1 : (def_ X xs).metaVarSet = ∅
⊢ sub σ τ (def_ X xs) = sub σ τ' (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | unfold Formula.metaVarSet at h1 | σ : Instantiation
τ τ' : MetaInstantiation
X : MetaVarName
h1 : (meta_var_ X).metaVarSet = ∅
⊢ sub σ τ (meta_var_ X) = sub σ τ' (meta_var_ X) | σ : Instantiation
τ τ' : MetaInstantiation
X : MetaVarName
h1 : {X} = ∅
⊢ sub σ τ (meta_var_ X) = sub σ τ' (meta_var_ X) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | simp at h1 | σ : Instantiation
τ τ' : MetaInstantiation
X : MetaVarName
h1 : {X} = ∅
⊢ sub σ τ (meta_var_ X) = sub σ τ' (meta_var_ X) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | rfl | σ : Instantiation
τ τ' : MetaInstantiation
X : PredName
xs : List VarName
h1 : (pred_ X xs).metaVarSet = ∅
⊢ sub σ τ (pred_ X xs) = sub σ τ' (pred_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | rfl | σ : Instantiation
τ τ' : MetaInstantiation
x y : VarName
h1 : (eq_ x y).metaVarSet = ∅
⊢ sub σ τ (eq_ x y) = sub σ τ' (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | rfl | σ : Instantiation
τ τ' : MetaInstantiation
h1 : true_.metaVarSet = ∅
⊢ sub σ τ true_ = sub σ τ' true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | unfold Formula.metaVarSet at h1 | σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.not_.metaVarSet = ∅
⊢ sub σ τ phi.not_ = sub σ τ' phi.not_ | σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ phi.not_ = sub σ τ' phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | unfold sub | σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ phi.not_ = sub σ τ' phi.not_ | σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ (sub σ τ phi).not_ = (sub σ τ' phi).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | congr! 1 | σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ (sub σ τ phi).not_ = (sub σ τ' phi).not_ | case h.e'_1
σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ phi = sub σ τ' phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | exact phi_ih h1 | case h.e'_1
σ : Instantiation
τ τ' : MetaInstantiation
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ phi = sub σ τ' phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | unfold Formula.metaVarSet at h1 | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1 : (phi.imp_ psi).metaVarSet = ∅
⊢ sub σ τ (phi.imp_ psi) = sub σ τ' (phi.imp_ psi) | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1 : phi.metaVarSet ∪ psi.metaVarSet = ∅
⊢ sub σ τ (phi.imp_ psi) = sub σ τ' (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | simp only [Finset.union_eq_empty] at h1 | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1 : phi.metaVarSet ∪ psi.metaVarSet = ∅
⊢ sub σ τ (phi.imp_ psi) = sub σ τ' (phi.imp_ psi) | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1 : phi.metaVarSet = ∅ ∧ psi.metaVarSet = ∅
⊢ sub σ τ (phi.imp_ psi) = sub σ τ' (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | unfold sub | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1 : phi.metaVarSet = ∅ ∧ psi.metaVarSet = ∅
⊢ sub σ τ (phi.imp_ psi) = sub σ τ' (phi.imp_ psi) | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1 : phi.metaVarSet = ∅ ∧ psi.metaVarSet = ∅
⊢ (sub σ τ phi).imp_ (sub σ τ psi) = (sub σ τ' phi).imp_ (sub σ τ' psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | cases h1 | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1 : phi.metaVarSet = ∅ ∧ psi.metaVarSet = ∅
⊢ (sub σ τ phi).imp_ (sub σ τ psi) = (sub σ τ' phi).imp_ (sub σ τ' psi) | case intro
σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
left✝ : phi.metaVarSet = ∅
right✝ : psi.metaVarSet = ∅
⊢ (sub σ τ phi).imp_ (sub σ τ psi) = (sub σ τ' phi).imp_ (sub σ τ' psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | congr! 1 | σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1_left : phi.metaVarSet = ∅
h1_right : psi.metaVarSet = ∅
⊢ (sub σ τ phi).imp_ (sub σ τ psi) = (sub σ τ' phi).imp_ (sub σ τ' psi) | case h.e'_1
σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1_left : phi.metaVarSet = ∅
h1_right : psi.metaVarSet = ∅
⊢ sub σ τ phi = sub σ τ' phi
case h.e'_2
σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1_left : phi.metaVarSet = ∅
h1_right : psi.metaVarSet = ∅
⊢ sub σ τ psi = sub σ τ' psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | exact phi_ih h1_left | case h.e'_1
σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1_left : phi.metaVarSet = ∅
h1_right : psi.metaVarSet = ∅
⊢ sub σ τ phi = sub σ τ' phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | exact psi_ih h1_right | case h.e'_2
σ : Instantiation
τ τ' : MetaInstantiation
phi psi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
psi_ih : psi.metaVarSet = ∅ → sub σ τ psi = sub σ τ' psi
h1_left : phi.metaVarSet = ∅
h1_right : psi.metaVarSet = ∅
⊢ sub σ τ psi = sub σ τ' psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | unfold Formula.metaVarSet at h1 | σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : (forall_ x phi).metaVarSet = ∅
⊢ sub σ τ (forall_ x phi) = sub σ τ' (forall_ x phi) | σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ (forall_ x phi) = sub σ τ' (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | unfold sub | σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ (forall_ x phi) = sub σ τ' (forall_ x phi) | σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ forall_ (↑σ x) (sub σ τ phi) = forall_ (↑σ x) (sub σ τ' phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | congr! 1 | σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ forall_ (↑σ x) (sub σ τ phi) = forall_ (↑σ x) (sub σ τ' phi) | case h.e'_2
σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ phi = sub σ τ' phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | exact phi_ih h1 | case h.e'_2
σ : Instantiation
τ τ' : MetaInstantiation
x : VarName
phi : Formula
phi_ih : phi.metaVarSet = ∅ → sub σ τ phi = sub σ τ' phi
h1 : phi.metaVarSet = ∅
⊢ sub σ τ phi = sub σ τ' phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_imp_meta_instantiation_irrelevance_in_sub | [820, 1] | [860, 8] | rfl | σ : Instantiation
τ τ' : MetaInstantiation
X : DefName
xs : List VarName
h1 : (def_ X xs).metaVarSet = ∅
⊢ sub σ τ (def_ X xs) = sub σ τ' (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | induction F generalizing vs | F : Formula
vs : List VarName
σ : Instantiation
τ : MetaInstantiation
h1 : NoMetaVarAndAllFreeInList vs F
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ F) | case meta_var_
σ : Instantiation
τ : MetaInstantiation
a✝ : MetaVarName
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (meta_var_ a✝))
case pred_
σ : Instantiation
τ : MetaInstantiation
a✝¹ : PredName
a✝ vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (pred_ a✝¹ a✝))
case eq_
σ : Instantiation
τ : MetaInstantiation
a✝¹ a✝ : VarName
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (eq_ a✝¹ a✝))
case true_
σ : Instantiation
τ : MetaInstantiation
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ true_)
case not_
σ : Instantiation
τ : MetaInstantiation
a✝ : Formula
a_ih✝ :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ a✝)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs a✝.not_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ a✝.not_)
case imp_
σ : Instantiation
τ : MetaInstantiation
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝¹ → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ a✝¹)
a_ih✝ :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ a✝)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (a✝¹.imp_ a✝))
case forall_
σ : Instantiation
τ : MetaInstantiation
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ a✝)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (forall_ a✝¹ a✝))
case def_
σ : Instantiation
τ : MetaInstantiation
a✝¹ : DefName
a✝ vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | case meta_var_ X =>
unfold NoMetaVarAndAllFreeInList at h1
contradiction | σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (meta_var_ X)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | case pred_ X xs =>
unfold sub
unfold NoMetaVarAndAllFreeInList
exact List.map_subset σ.1 h1 | σ : Instantiation
τ : MetaInstantiation
X : PredName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (pred_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | case true_ =>
unfold sub
unfold NoMetaVarAndAllFreeInList
simp | σ : Instantiation
τ : MetaInstantiation
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ true_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | case not_ phi phi_ih =>
unfold NoMetaVarAndAllFreeInList at h1
unfold sub
unfold NoMetaVarAndAllFreeInList
exact phi_ih vs h1 | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi.not_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | case forall_ x phi phi_ih =>
unfold NoMetaVarAndAllFreeInList at h1
unfold sub
unfold NoMetaVarAndAllFreeInList
simp only [← List.map_cons]
exact phi_ih (x :: vs) h1 | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (forall_ x phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | case def_ X xs =>
unfold sub
unfold NoMetaVarAndAllFreeInList
exact List.map_subset σ.val h1 | σ : Instantiation
τ : MetaInstantiation
X : DefName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (def_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList at h1 | σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (meta_var_ X)) | σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
vs : List VarName
h1 : False
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (meta_var_ X)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | contradiction | σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
vs : List VarName
h1 : False
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (meta_var_ X)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold sub | σ : Instantiation
τ : MetaInstantiation
X : PredName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (pred_ X xs)) | σ : Instantiation
τ : MetaInstantiation
X : PredName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (pred_ X (List.map (↑σ) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList | σ : Instantiation
τ : MetaInstantiation
X : PredName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (pred_ X (List.map (↑σ) xs)) | σ : Instantiation
τ : MetaInstantiation
X : PredName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
⊢ List.map (↑σ) xs ⊆ List.map (↑σ) vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact List.map_subset σ.1 h1 | σ : Instantiation
τ : MetaInstantiation
X : PredName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
⊢ List.map (↑σ) xs ⊆ List.map (↑σ) vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold sub | σ : Instantiation
τ : MetaInstantiation
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ true_) | σ : Instantiation
τ : MetaInstantiation
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) true_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList | σ : Instantiation
τ : MetaInstantiation
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) true_ | σ : Instantiation
τ : MetaInstantiation
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
⊢ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | simp | σ : Instantiation
τ : MetaInstantiation
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
⊢ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList at h1 | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi.not_
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi.not_) | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold sub | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi.not_) | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi).not_ | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact phi_ih vs h1 | σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList at h1 | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (phi.imp_ psi)) | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (phi.imp_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold sub | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (phi.imp_ psi)) | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) ((sub σ τ phi).imp_ (sub σ τ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) ((sub σ τ phi).imp_ (sub σ τ psi)) | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi) ∧
NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | cases h1 | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi) ∧
NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi) | case intro
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
left✝ : NoMetaVarAndAllFreeInList vs phi
right✝ : NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi) ∧
NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | constructor | σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi) ∧
NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi) | case left
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
case right
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact phi_ih vs h1_left | case left
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact psi_ih vs h1_right | case right
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
psi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi)
vs : List VarName
h1_left : NoMetaVarAndAllFreeInList vs phi
h1_right : NoMetaVarAndAllFreeInList vs psi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList at h1 | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (eq_ x y)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (eq_ x y)) | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1 : x ∈ vs ∧ y ∈ vs
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (eq_ x y)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold sub | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1 : x ∈ vs ∧ y ∈ vs
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (eq_ x y)) | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1 : x ∈ vs ∧ y ∈ vs
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (eq_ (↑σ x) (↑σ y)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1 : x ∈ vs ∧ y ∈ vs
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (eq_ (↑σ x) (↑σ y)) | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1 : x ∈ vs ∧ y ∈ vs
⊢ ↑σ x ∈ List.map (↑σ) vs ∧ ↑σ y ∈ List.map (↑σ) vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | cases h1 | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1 : x ∈ vs ∧ y ∈ vs
⊢ ↑σ x ∈ List.map (↑σ) vs ∧ ↑σ y ∈ List.map (↑σ) vs | case intro
σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
left✝ : x ∈ vs
right✝ : y ∈ vs
⊢ ↑σ x ∈ List.map (↑σ) vs ∧ ↑σ y ∈ List.map (↑σ) vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | constructor | σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ ↑σ x ∈ List.map (↑σ) vs ∧ ↑σ y ∈ List.map (↑σ) vs | case left
σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ ↑σ x ∈ List.map (↑σ) vs
case right
σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ ↑σ y ∈ List.map (↑σ) vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact List.mem_map_of_mem σ.val h1_left | case left
σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ ↑σ x ∈ List.map (↑σ) vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact List.mem_map_of_mem σ.val h1_right | case right
σ : Instantiation
τ : MetaInstantiation
x y : VarName
vs : List VarName
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ ↑σ y ∈ List.map (↑σ) vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList at h1 | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (forall_ x phi)) | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (forall_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold sub | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (forall_ x phi)) | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (forall_ (↑σ x) (sub σ τ phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (forall_ (↑σ x) (sub σ τ phi)) | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (↑σ x :: List.map (↑σ) vs) (sub σ τ phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | simp only [← List.map_cons] | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (↑σ x :: List.map (↑σ) vs) (sub σ τ phi) | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) (x :: vs)) (sub σ τ phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact phi_ih (x :: vs) h1 | σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ phi)
vs : List VarName
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) (x :: vs)) (sub σ τ phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold sub | σ : Instantiation
τ : MetaInstantiation
X : DefName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (sub σ τ (def_ X xs)) | σ : Instantiation
τ : MetaInstantiation
X : DefName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (def_ X (List.map (↑σ) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | unfold NoMetaVarAndAllFreeInList | σ : Instantiation
τ : MetaInstantiation
X : DefName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
⊢ NoMetaVarAndAllFreeInList (List.map (↑σ) vs) (def_ X (List.map (↑σ) xs)) | σ : Instantiation
τ : MetaInstantiation
X : DefName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
⊢ List.map (↑σ) xs ⊆ List.map (↑σ) vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.no_meta_var_and_all_free_in_list_map_sub | [863, 1] | [920, 35] | exact List.map_subset σ.val h1 | σ : Instantiation
τ : MetaInstantiation
X : DefName
xs vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
⊢ List.map (↑σ) xs ⊆ List.map (↑σ) vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | induction F | F : Formula
E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E F
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ F) | case meta_var_
E : Env
σ : Instantiation
a✝ : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ a✝)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (meta_var_ a✝))
case pred_
E : Env
σ : Instantiation
a✝¹ : PredName
a✝ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ a✝¹ a✝)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (pred_ a✝¹ a✝))
case eq_
E : Env
σ : Instantiation
a✝¹ a✝ : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ a✝¹ a✝)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (eq_ a✝¹ a✝))
case true_
E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E true_
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ true_)
case not_
E : Env
σ : Instantiation
a✝ : Formula
a_ih✝ : IsMetaVarOrAllDefInEnv E a✝ → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ a✝)
h1 : IsMetaVarOrAllDefInEnv E a✝.not_
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ a✝.not_)
case imp_
E : Env
σ : Instantiation
a✝¹ a✝ : Formula
a_ih✝¹ : IsMetaVarOrAllDefInEnv E a✝¹ → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ a✝¹)
a_ih✝ : IsMetaVarOrAllDefInEnv E a✝ → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ a✝)
h1 : IsMetaVarOrAllDefInEnv E (a✝¹.imp_ a✝)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (a✝¹.imp_ a✝))
case forall_
E : Env
σ : Instantiation
a✝¹ : VarName
a✝ : Formula
a_ih✝ : IsMetaVarOrAllDefInEnv E a✝ → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ a✝)
h1 : IsMetaVarOrAllDefInEnv E (forall_ a✝¹ a✝)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (forall_ a✝¹ a✝))
case def_
E : Env
σ : Instantiation
a✝¹ : DefName
a✝ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ a✝¹ a✝)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | case meta_var_ X =>
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
σ : Instantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (meta_var_ X)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | case pred_ X xs =>
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
σ : Instantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (pred_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | case eq_ x y =>
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
σ : Instantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (eq_ x y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | case true_ =>
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp only | E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E true_
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ true_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | case not_ phi phi_ih =>
unfold IsMetaVarOrAllDefInEnv at h1
unfold sub
unfold IsMetaVarOrAllDefInEnv
exact phi_ih h1 | E : Env
σ : Instantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E phi.not_
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | case forall_ x phi phi_ih =>
unfold IsMetaVarOrAllDefInEnv at h1
unfold sub
unfold IsMetaVarOrAllDefInEnv
exact phi_ih h1 | E : Env
σ : Instantiation
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (forall_ x phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | case def_ X xs =>
unfold IsMetaVarOrAllDefInEnv at h1
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp
exact h1 | E : Env
σ : Instantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ X xs)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (def_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold sub | E : Env
σ : Instantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (meta_var_ X)) | E : Env
σ : Instantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
⊢ IsMetaVarOrAllDefInEnv E (meta_var_ X) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
σ : Instantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
⊢ IsMetaVarOrAllDefInEnv E (meta_var_ X) | E : Env
σ : Instantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
⊢ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | simp only | E : Env
σ : Instantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
⊢ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold sub | E : Env
σ : Instantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (pred_ X xs)) | E : Env
σ : Instantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
⊢ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (↑σ) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
σ : Instantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
⊢ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (↑σ) xs)) | E : Env
σ : Instantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
⊢ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | simp only | E : Env
σ : Instantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
⊢ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold sub | E : Env
σ : Instantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ (eq_ x y)) | E : Env
σ : Instantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
⊢ IsMetaVarOrAllDefInEnv E (eq_ (↑σ x) (↑σ y)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
σ : Instantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
⊢ IsMetaVarOrAllDefInEnv E (eq_ (↑σ x) (↑σ y)) | E : Env
σ : Instantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
⊢ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | simp only | E : Env
σ : Instantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
⊢ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold sub | E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E true_
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ true_) | E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E true_
⊢ IsMetaVarOrAllDefInEnv E true_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E true_
⊢ IsMetaVarOrAllDefInEnv E true_ | E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E true_
⊢ True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | simp only | E : Env
σ : Instantiation
h1 : IsMetaVarOrAllDefInEnv E true_
⊢ True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold IsMetaVarOrAllDefInEnv at h1 | E : Env
σ : Instantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E phi.not_
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi.not_) | E : Env
σ : Instantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold sub | E : Env
σ : Instantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi.not_) | E : Env
σ : Instantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.sub_meta_id_preserves_is_meta_var_or_all_def_in_env | [923, 1] | [975, 13] | unfold IsMetaVarOrAllDefInEnv | E : Env
σ : Instantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi).not_ | E : Env
σ : Instantiation
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv E phi → IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
⊢ IsMetaVarOrAllDefInEnv E (sub σ meta_var_ phi) |
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