Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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stringclasses
147 values
commit
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101
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2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_2_a
[1766, 1]
[1837, 13]
exact h1
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h2 : βˆ€ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο„ X_1) h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ βˆƒ d ∈ E, X = d.name ∧ 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 Οƒ : Instantiation Ο„ : MetaInstantiation F : Formula h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ F) ⊒ IsMetaVarOrAllDefInEnv E F
case meta_var_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation a✝ : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (meta_var_ a✝)) ⊒ IsMetaVarOrAllDefInEnv E (meta_var_ a✝) case pred_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation a✝¹ : PredName a✝ : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (pred_ a✝¹ a✝)) ⊒ IsMetaVarOrAllDefInEnv E (pred_ a✝¹ a✝) case eq_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation a✝¹ a✝ : VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (eq_ a✝¹ a✝)) ⊒ IsMetaVarOrAllDefInEnv E (eq_ a✝¹ a✝) case true_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ true_) ⊒ IsMetaVarOrAllDefInEnv E true_ case not_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation a✝ : Formula a_ih✝ : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ a✝) β†’ IsMetaVarOrAllDefInEnv E a✝ h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ a✝.not_) ⊒ IsMetaVarOrAllDefInEnv E a✝.not_ case imp_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation a✝¹ a✝ : Formula a_ih✝¹ : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ a✝¹) β†’ IsMetaVarOrAllDefInEnv E a✝¹ a_ih✝ : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ a✝) β†’ IsMetaVarOrAllDefInEnv E a✝ h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (a✝¹.imp_ a✝)) ⊒ IsMetaVarOrAllDefInEnv E (a✝¹.imp_ a✝) case forall_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation a✝¹ : VarName a✝ : Formula a_ih✝ : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ a✝) β†’ IsMetaVarOrAllDefInEnv E a✝ h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (forall_ a✝¹ a✝)) ⊒ IsMetaVarOrAllDefInEnv E (forall_ a✝¹ a✝) case def_ E : Env Οƒ : Instantiation Ο„ : MetaInstantiation a✝¹ : DefName a✝ : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (def_ a✝¹ a✝)) ⊒ IsMetaVarOrAllDefInEnv E (def_ a✝¹ a✝)
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 Οƒ : Instantiation Ο„ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (meta_var_ X)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (pred_ X xs)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (eq_ x y)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ true_) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi.not_) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (forall_ x phi)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (def_ X xs)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (meta_var_ X)) ⊒ IsMetaVarOrAllDefInEnv E (meta_var_ X)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (meta_var_ X)) ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_2_b
[1840, 1]
[1889, 13]
simp only
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : MetaVarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (meta_var_ X)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (pred_ X xs)) ⊒ IsMetaVarOrAllDefInEnv E (pred_ X xs)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (pred_ X xs)) ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_2_b
[1840, 1]
[1889, 13]
simp only
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : PredName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (pred_ X xs)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (eq_ x y)) ⊒ IsMetaVarOrAllDefInEnv E (eq_ x y)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (eq_ x y)) ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_2_b
[1840, 1]
[1889, 13]
simp only
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation x y : VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (eq_ x y)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ true_) ⊒ IsMetaVarOrAllDefInEnv E true_
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ true_) ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_2_b
[1840, 1]
[1889, 13]
simp only
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ true_) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi.not_) ⊒ IsMetaVarOrAllDefInEnv E phi.not_
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi).not_ ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi).not_ ⊒ IsMetaVarOrAllDefInEnv E phi.not_
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ IsMetaVarOrAllDefInEnv E phi.not_
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (phi.imp_ psi)) ⊒ IsMetaVarOrAllDefInEnv E (phi.imp_ psi)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1 : IsMetaVarOrAllDefInEnv E ((sub Οƒ Ο„ phi).imp_ (sub Οƒ Ο„ psi)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1 : IsMetaVarOrAllDefInEnv E ((sub Οƒ Ο„ phi).imp_ (sub Οƒ Ο„ psi)) ⊒ IsMetaVarOrAllDefInEnv E (phi.imp_ psi)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ IsMetaVarOrAllDefInEnv E (phi.imp_ psi)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ IsMetaVarOrAllDefInEnv E phi ∧ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi
case intro E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi left✝ : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) right✝ : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ IsMetaVarOrAllDefInEnv E phi ∧ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1_left : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) h1_right : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi
case left E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1_left : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) h1_right : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ IsMetaVarOrAllDefInEnv E phi case right E : Env Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1_left : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) h1_right : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1_left : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) h1_right : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi psi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) β†’ IsMetaVarOrAllDefInEnv E psi h1_left : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) h1_right : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ psi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (forall_ x phi)) ⊒ IsMetaVarOrAllDefInEnv E (forall_ x phi)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (forall_ (↑σ x) (sub Οƒ Ο„ phi)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (forall_ (↑σ x) (sub Οƒ Ο„ phi)) ⊒ IsMetaVarOrAllDefInEnv E (forall_ x phi)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ IsMetaVarOrAllDefInEnv E (forall_ x phi)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) β†’ IsMetaVarOrAllDefInEnv E phi h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ phi) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (sub Οƒ Ο„ (def_ X xs)) ⊒ IsMetaVarOrAllDefInEnv E (def_ X xs)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (def_ X (List.map (↑σ) xs)) ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (def_ X (List.map (↑σ) xs)) ⊒ IsMetaVarOrAllDefInEnv E (def_ X xs)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ (List.map (↑σ) xs).length = d.args.length ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ (List.map (↑σ) xs).length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E (def_ X xs)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E (def_ X xs)
E : Env Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ βˆƒ d ∈ E, X = d.name ∧ 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 Οƒ : Instantiation Ο„ : MetaInstantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ βˆƒ d ∈ E, X = d.name ∧ 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 ⊒ IsMetaVarOrAllDefInEnv E F
case hyp E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ : Formula a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : phi✝ ∈ Ξ”βœ ⊒ IsMetaVarOrAllDefInEnv E phi✝ case mp E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula a✝¹ : IsProof E Ξ“βœ Ξ”βœ phi✝ a✝ : IsProof E Ξ“βœ Ξ”βœ (phi✝.imp_ psi✝) a_ih✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a_ih✝ : IsMetaVarOrAllDefInEnv E (phi✝.imp_ psi✝) ⊒ IsMetaVarOrAllDefInEnv E psi✝ case prop_1 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : IsMetaVarOrAllDefInEnv E psi✝ ⊒ IsMetaVarOrAllDefInEnv E (phi✝.imp_ (psi✝.imp_ phi✝)) case prop_2 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ chi✝ : Formula a✝² : IsMetaVarOrAllDefInEnv E phi✝ a✝¹ : IsMetaVarOrAllDefInEnv E psi✝ a✝ : IsMetaVarOrAllDefInEnv E chi✝ ⊒ IsMetaVarOrAllDefInEnv E ((phi✝.imp_ (psi✝.imp_ chi✝)).imp_ ((phi✝.imp_ psi✝).imp_ (phi✝.imp_ chi✝))) case prop_3 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : IsMetaVarOrAllDefInEnv E psi✝ ⊒ IsMetaVarOrAllDefInEnv E ((phi✝.not_.imp_ psi✝.not_).imp_ (psi✝.imp_ phi✝)) case gen E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ : Formula x✝ : VarName a✝ : IsProof E Ξ“βœ Ξ”βœ phi✝ a_ih✝ : IsMetaVarOrAllDefInEnv E phi✝ ⊒ IsMetaVarOrAllDefInEnv E (forall_ x✝ phi✝) case pred_1 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula x✝ : VarName a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : IsMetaVarOrAllDefInEnv E psi✝ ⊒ IsMetaVarOrAllDefInEnv E ((forall_ x✝ (phi✝.imp_ psi✝)).imp_ ((forall_ x✝ phi✝).imp_ (forall_ x✝ psi✝))) case pred_2 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ : Formula x✝ : VarName a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : NotFree Ξ“βœ x✝ phi✝ ⊒ IsMetaVarOrAllDefInEnv E (phi✝.imp_ (forall_ x✝ phi✝)) case eq_1 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula x✝ y✝ : VarName a✝ : Β¬y✝ = x✝ ⊒ IsMetaVarOrAllDefInEnv E (exists_ x✝ (eq_ x✝ y✝)) case eq_2 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula x✝ y✝ z✝ : VarName ⊒ IsMetaVarOrAllDefInEnv E ((eq_ x✝ y✝).imp_ ((eq_ x✝ z✝).imp_ (eq_ y✝ z✝))) case eq_3 E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula n✝ : β„• name✝ : PredName xs✝ ys✝ : Fin n✝ β†’ VarName ⊒ IsMetaVarOrAllDefInEnv E (eqSubPred name✝ n✝ xs✝ ys✝) case thm E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ Ξ“'✝ : List (VarName Γ— MetaVarName) Ξ”βœ Ξ”'✝ : List Formula phi✝ : Formula Οƒβœ : Instantiation Ο„βœ : MetaInstantiation a✝³ : βˆ€ X ∈ phi✝.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο„βœ X) a✝² : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ Ξ“βœ β†’ NotFree Ξ“'✝ (β†‘Οƒβœ x) (Ο„βœ X) a✝¹ : βˆ€ psi ∈ Ξ”βœ, IsProof E Ξ“'✝ Ξ”'✝ (sub Οƒβœ Ο„βœ psi) a✝ : IsProof E Ξ“βœ Ξ”βœ phi✝ a_ih✝¹ : βˆ€ psi ∈ Ξ”βœ, IsMetaVarOrAllDefInEnv E (sub Οƒβœ Ο„βœ psi) a_ih✝ : IsMetaVarOrAllDefInEnv E phi✝ ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒβœ Ο„βœ phi✝) case conv E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ phi'✝ : Formula a✝² : IsMetaVarOrAllDefInEnv E phi'✝ a✝¹ : IsProof E Ξ“βœ Ξ”βœ phi✝ a✝ : IsConv E phi✝ phi'✝ a_ih✝ : IsMetaVarOrAllDefInEnv E phi✝ ⊒ IsMetaVarOrAllDefInEnv E 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]
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 ∈ h1_Ξ” ⊒ 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) ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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_Οƒ h1_Ο„ h1_1 h1_2 h1_3 h1_4 h1_ih_1 h1_ih_2 => exact lem_2_a E h1_Οƒ 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_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, IsMetaVarOrAllDefInEnv E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : IsMetaVarOrAllDefInEnv E h1_phi ⊒ IsMetaVarOrAllDefInEnv E (sub h1_Οƒ 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 ⊒ 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 ∈ h1_Ξ” ⊒ 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) ⊒ 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 ∧ IsMetaVarOrAllDefInEnv E h1_psi ⊒ 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 ∧ IsMetaVarOrAllDefInEnv E h1_psi ⊒ 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 left✝ : IsMetaVarOrAllDefInEnv E h1_phi right✝ : IsMetaVarOrAllDefInEnv E h1_psi ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ 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 ⊒ IsMetaVarOrAllDefInEnv E (eq_ h1_x h1_y) ∧ 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 ⊒ IsMetaVarOrAllDefInEnv E (eq_ h1_x h1_y) ∧ 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 ⊒ 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 ⊒ 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 ⊒ 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_Οƒ 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_Οƒ : Instantiation h1_Ο„ : MetaInstantiation h1_1 : βˆ€ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_Ο„ X) h1_2 : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Ξ“ β†’ NotFree h1_Ξ“' (↑h1_Οƒ x) (h1_Ο„ X) h1_3 : βˆ€ psi ∈ h1_Ξ”, IsProof E h1_Ξ“' h1_Ξ”' (sub h1_Οƒ h1_Ο„ psi) h1_4 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih_1 : βˆ€ psi ∈ h1_Ξ”, IsMetaVarOrAllDefInEnv E (sub h1_Οƒ h1_Ο„ psi) h1_ih_2 : IsMetaVarOrAllDefInEnv E h1_phi ⊒ IsMetaVarOrAllDefInEnv E (sub h1_Οƒ 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 ⊒ 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 ∈ E h3 : name = d.name ∧ args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M E d.F ↔ 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 ∧ args.length = d.args.length h1 : Env.WellFormed [] h2 : d ∈ [] ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M [] d.F ↔ 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 ∧ args.length = d.args.length head✝ : Definition tail✝ : List Definition tail_ih✝ : Env.WellFormed tail✝ β†’ d ∈ tail✝ β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tail✝ d.F ↔ Holds D I V M tail✝ (def_ name args)) h1 : Env.WellFormed (head✝ :: tail✝) h2 : d ∈ head✝ :: tail✝ ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (head✝ :: tail✝) d.F ↔ Holds D I V M (head✝ :: tail✝) (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 ∧ args.length = d.args.length h1 : Env.WellFormed [] h2 : d ∈ [] ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M [] d.F ↔ 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 ∧ args.length = d.args.length h1 : Env.WellFormed [] h2 : d ∈ [] ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M [] d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 : βˆƒ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) ⊒ βˆƒ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) ⊒ βˆƒ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) ⊒ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) ⊒ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1 : (βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False) ∧ IsMetaVarOrAllDefInEnv tl hd.F ∧ Env.WellFormed tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h2 : d ∈ hd :: tl s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1 : (βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False) ∧ IsMetaVarOrAllDefInEnv tl hd.F ∧ Env.WellFormed tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1 : (βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False) ∧ IsMetaVarOrAllDefInEnv tl hd.F ∧ Env.WellFormed tl h2 : d = hd ∨ d ∈ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1 : (βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False) ∧ IsMetaVarOrAllDefInEnv tl hd.F ∧ Env.WellFormed tl h2 : d = hd ∨ d ∈ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl left✝ : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False right✝ : IsMetaVarOrAllDefInEnv tl hd.F ∧ Env.WellFormed tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right : IsMetaVarOrAllDefInEnv tl hd.F ∧ Env.WellFormed tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False left✝ : IsMetaVarOrAllDefInEnv tl hd.F right✝ : Env.WellFormed tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ 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 ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ 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 ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ if name = hd.name ∧ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ 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 ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ if name = hd.name ∧ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ 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 h✝ : name = hd.name ∧ args.length = hd.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ 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 ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ 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 h✝ : Β¬(name = hd.name ∧ args.length = hd.args.length) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)