Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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stringclasses
147 values
commit
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7
101
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2.09M
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]
exact phi_ih h1
E : Env Οƒ : Instantiation phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E phi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ 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]
unfold IsMetaVarOrAllDefInEnv at h1
E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1 : IsMetaVarOrAllDefInEnv E (phi.imp_ psi) ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (phi.imp_ psi))
E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (phi.imp_ psi))
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 Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (phi.imp_ psi))
E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E ((sub Οƒ meta_var_ phi).imp_ (sub Οƒ meta_var_ psi))
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 Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E ((sub Οƒ meta_var_ phi).imp_ (sub Οƒ meta_var_ psi))
E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi)
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]
cases h1
E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi)
case intro E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) left✝ : IsMetaVarOrAllDefInEnv E phi right✝ : IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi)
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]
constructor
E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) ∧ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi)
case left E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) case right E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi)
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]
exact phi_ih h1_left
case left E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ 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]
exact psi_ih h1_right
case right E : Env Οƒ : Instantiation phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) psi_ih : IsMetaVarOrAllDefInEnv E psi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi) h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ psi)
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 Οƒ : Instantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi) ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (forall_ x phi))
E : Env Οƒ : Instantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E phi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (forall_ x phi))
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 Οƒ : Instantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E phi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (forall_ x phi))
E : Env Οƒ : Instantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E phi ⊒ IsMetaVarOrAllDefInEnv E (forall_ (↑σ x) (sub Οƒ meta_var_ phi))
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 Οƒ : Instantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E phi ⊒ IsMetaVarOrAllDefInEnv E (forall_ (↑σ x) (sub Οƒ meta_var_ phi))
E : Env Οƒ : Instantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E phi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi)
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]
exact phi_ih h1
E : Env Οƒ : Instantiation x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv E phi β†’ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ phi) h1 : IsMetaVarOrAllDefInEnv E phi ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ 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]
unfold IsMetaVarOrAllDefInEnv at h1
E : Env Οƒ : Instantiation X : DefName xs : List VarName h1 : IsMetaVarOrAllDefInEnv E (def_ X xs) ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (def_ X xs))
E : Env Οƒ : Instantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (def_ X 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 sub
E : Env Οƒ : Instantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E (sub Οƒ meta_var_ (def_ X xs))
E : Env Οƒ : Instantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E (def_ 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 Οƒ : Instantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E (def_ X (List.map (↑σ) xs))
E : Env Οƒ : Instantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ βˆƒ d ∈ E, X = d.name ∧ (List.map (↑σ) xs).length = d.args.length
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
E : Env Οƒ : Instantiation X : DefName xs : List VarName h1 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ βˆƒ d ∈ E, X = d.name ∧ (List.map (↑σ) xs).length = d.args.length
E : Env Οƒ : Instantiation 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.sub_meta_id_preserves_is_meta_var_or_all_def_in_env
[923, 1]
[975, 13]
exact h1
E : Env Οƒ : Instantiation 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.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
induction E
E : Env h1 : E.WellFormed ⊒ E.Nodup
case nil h1 : Env.WellFormed [] ⊒ Env.Nodup [] case cons head✝ : Definition tail✝ : List Definition tail_ih✝ : Env.WellFormed tail✝ β†’ Env.Nodup tail✝ h1 : Env.WellFormed (head✝ :: tail✝) ⊒ Env.Nodup (head✝ :: tail✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
case nil => unfold Env.Nodup simp
h1 : Env.WellFormed [] ⊒ Env.Nodup []
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
unfold Env.Nodup
h1 : Env.WellFormed [] ⊒ Env.Nodup []
h1 : Env.WellFormed [] ⊒ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) []
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
simp
h1 : Env.WellFormed [] ⊒ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) []
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
unfold Env.Nodup at ih
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ Env.Nodup tl h1 : Env.WellFormed (hd :: tl) ⊒ Env.Nodup (hd :: tl)
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) tl h1 : Env.WellFormed (hd :: tl) ⊒ Env.Nodup (hd :: tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
unfold Env.WellFormed at h1
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) tl h1 : Env.WellFormed (hd :: tl) ⊒ Env.Nodup (hd :: tl)
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ Env.Nodup (hd :: tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
cases h1
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ Env.Nodup (hd :: tl)
case intro hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ Env.Nodup (hd :: tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
cases h1_right
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ Env.Nodup (hd :: tl)
case intro hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ Env.Nodup (hd :: tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
unfold Env.Nodup
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ Env.Nodup (hd :: tl)
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (hd :: tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
simp
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (hd :: tl)
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ (βˆ€ a' ∈ tl, hd.name = a'.name β†’ Β¬hd.args.length = a'.args.length) ∧ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
constructor
hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ (βˆ€ a' ∈ tl, hd.name = a'.name β†’ Β¬hd.args.length = a'.args.length) ∧ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) tl
case left hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ βˆ€ a' ∈ tl, hd.name = a'.name β†’ Β¬hd.args.length = a'.args.length case right hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
exact h1_left
case left hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ βˆ€ a' ∈ tl, hd.name = a'.name β†’ Β¬hd.args.length = a'.args.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.well_formed_env_has_no_dup
[978, 1]
[1000, 34]
exact ih h1_right_right
case right hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) 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 ⊒ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) tl
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
induction E generalizing F
E E' : Env F : Formula h1 : βˆƒ E1, E' = E1 ++ E h2 : IsMetaVarOrAllDefInEnv E F ⊒ IsMetaVarOrAllDefInEnv E' F
case nil E' : Env F : Formula h1 : βˆƒ E1, E' = E1 ++ [] h2 : IsMetaVarOrAllDefInEnv [] F ⊒ IsMetaVarOrAllDefInEnv E' F case cons E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F F : Formula h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) F ⊒ IsMetaVarOrAllDefInEnv E' F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
case nil.def_ X xs => unfold IsMetaVarOrAllDefInEnv at h2 simp at h2
E' : Env h1 : βˆƒ E1, E' = E1 ++ [] X : DefName xs : List VarName h2 : IsMetaVarOrAllDefInEnv [] (def_ X xs) ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
induction F
case cons E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F F : Formula h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) F ⊒ IsMetaVarOrAllDefInEnv E' F
case cons.meta_var_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ a✝ : MetaVarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (meta_var_ a✝) ⊒ IsMetaVarOrAllDefInEnv E' (meta_var_ a✝) case cons.pred_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ a✝¹ : PredName a✝ : List VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (pred_ a✝¹ a✝) ⊒ IsMetaVarOrAllDefInEnv E' (pred_ a✝¹ a✝) case cons.eq_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ a✝¹ a✝ : VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (eq_ a✝¹ a✝) ⊒ IsMetaVarOrAllDefInEnv E' (eq_ a✝¹ a✝) case cons.true_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) true_ ⊒ IsMetaVarOrAllDefInEnv E' true_ case cons.not_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ a✝ : Formula a_ih✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝ β†’ IsMetaVarOrAllDefInEnv E' a✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝.not_ ⊒ IsMetaVarOrAllDefInEnv E' a✝.not_ case cons.imp_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ a✝¹ a✝ : Formula a_ih✝¹ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝¹ β†’ IsMetaVarOrAllDefInEnv E' a✝¹ a_ih✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝ β†’ IsMetaVarOrAllDefInEnv E' a✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (a✝¹.imp_ a✝) ⊒ IsMetaVarOrAllDefInEnv E' (a✝¹.imp_ a✝) case cons.forall_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ a✝¹ : VarName a✝ : Formula a_ih✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝ β†’ IsMetaVarOrAllDefInEnv E' a✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (forall_ a✝¹ a✝) ⊒ IsMetaVarOrAllDefInEnv E' (forall_ a✝¹ a✝) case cons.def_ E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ a✝¹ : DefName a✝ : List VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (def_ a✝¹ a✝) ⊒ IsMetaVarOrAllDefInEnv E' (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
case meta_var_ X => unfold IsMetaVarOrAllDefInEnv simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : MetaVarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (meta_var_ X) ⊒ IsMetaVarOrAllDefInEnv E' (meta_var_ X)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
case pred_ X xs => unfold IsMetaVarOrAllDefInEnv simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : PredName xs : List VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (pred_ X xs) ⊒ IsMetaVarOrAllDefInEnv E' (pred_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
case eq_ x y => unfold IsMetaVarOrAllDefInEnv simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x y : VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (eq_ x y) ⊒ IsMetaVarOrAllDefInEnv E' (eq_ x y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
case true_ => unfold IsMetaVarOrAllDefInEnv simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) true_ ⊒ IsMetaVarOrAllDefInEnv E' true_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
case not_ phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h2 unfold IsMetaVarOrAllDefInEnv exact phi_ih h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi.not_ ⊒ IsMetaVarOrAllDefInEnv E' phi.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
case forall_ x phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h2 unfold IsMetaVarOrAllDefInEnv exact phi_ih h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (forall_ x phi) ⊒ IsMetaVarOrAllDefInEnv E' (forall_ x phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : MetaVarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (meta_var_ X) ⊒ IsMetaVarOrAllDefInEnv E' (meta_var_ X)
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : MetaVarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (meta_var_ X) ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : MetaVarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (meta_var_ X) ⊒ True
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : PredName xs : List VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (pred_ X xs) ⊒ IsMetaVarOrAllDefInEnv E' (pred_ X xs)
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : PredName xs : List VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (pred_ X xs) ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ X : PredName xs : List VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (pred_ X xs) ⊒ True
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x y : VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (eq_ x y) ⊒ IsMetaVarOrAllDefInEnv E' (eq_ x y)
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x y : VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (eq_ x y) ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x y : VarName h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (eq_ x y) ⊒ True
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) true_ ⊒ IsMetaVarOrAllDefInEnv E' true_
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) true_ ⊒ True
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp only
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) true_ ⊒ True
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv at h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi.not_ ⊒ IsMetaVarOrAllDefInEnv E' phi.not_
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' phi.not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' phi.not_
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact phi_ih h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv at h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (phi.imp_ psi) ⊒ IsMetaVarOrAllDefInEnv E' (phi.imp_ psi)
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ∧ IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' (phi.imp_ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ∧ IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' (phi.imp_ psi)
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ∧ IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' phi ∧ IsMetaVarOrAllDefInEnv E' psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
cases h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ∧ IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' phi ∧ IsMetaVarOrAllDefInEnv E' psi
case intro E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi left✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi right✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' phi ∧ IsMetaVarOrAllDefInEnv E' psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
constructor
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2_left : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi h2_right : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' phi ∧ IsMetaVarOrAllDefInEnv E' psi
case left E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2_left : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi h2_right : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' phi case right E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2_left : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi h2_right : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact phi_ih h2_left
case left E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2_left : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi h2_right : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact psi_ih h2_right
case right E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ phi psi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi β†’ IsMetaVarOrAllDefInEnv E' psi h2_left : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi h2_right : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi ⊒ IsMetaVarOrAllDefInEnv E' psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv at h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (forall_ x phi) ⊒ IsMetaVarOrAllDefInEnv E' (forall_ x phi)
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' (forall_ x phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' (forall_ x phi)
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact phi_ih h2
E' : Env head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ tail✝) β†’ IsMetaVarOrAllDefInEnv tail✝ F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ head✝ :: tail✝ x : VarName phi : Formula phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi β†’ IsMetaVarOrAllDefInEnv E' phi h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi ⊒ IsMetaVarOrAllDefInEnv E' phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv at h2
E' : Env h1 : βˆƒ E1, E' = E1 ++ [] X : DefName xs : List VarName h2 : IsMetaVarOrAllDefInEnv [] (def_ X xs) ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env h1 : βˆƒ E1, E' = E1 ++ [] X : DefName xs : List VarName h2 : βˆƒ d ∈ [], 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.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp at h2
E' : Env h1 : βˆƒ E1, E' = E1 ++ [] X : DefName xs : List VarName h2 : βˆƒ d ∈ [], X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv at h2
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs) ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, 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.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
apply Exists.elim h1
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length ⊒ βˆ€ (a : Env), E' = a ++ E_hd :: E_tl β†’ IsMetaVarOrAllDefInEnv E' (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
intro E1 h1_1
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length ⊒ βˆ€ (a : Env), E' = a ++ E_hd :: E_tl β†’ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
apply Exists.elim h2
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl ⊒ βˆ€ (a : Definition), a ∈ E_hd :: E_tl ∧ X = a.name ∧ xs.length = a.args.length β†’ IsMetaVarOrAllDefInEnv E' (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
intro d h2_1
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl ⊒ βˆ€ (a : Definition), a ∈ E_hd :: E_tl ∧ X = a.name ∧ xs.length = a.args.length β†’ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1 : d ∈ E_hd :: E_tl ∧ 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.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
cases h2_1
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1 : d ∈ E_hd :: E_tl ∧ X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
case intro E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition left✝ : d ∈ E_hd :: E_tl right✝ : 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.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp at h2_1_left
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_left : d ∈ E_hd :: E_tl h2_1_right : X = d.name ∧ xs.length = d.args.length ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length h2_1_left : d = E_hd ∨ d ∈ E_tl ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
cases h2_1_left
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length h2_1_left : d = E_hd ∨ d ∈ E_tl ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
case inl E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length h✝ : d = E_hd ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs) case inr E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length h✝ : d ∈ E_tl ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ βˆƒ d ∈ E', X = d.name ∧ xs.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
apply Exists.intro E_hd
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ βˆƒ d ∈ E', X = d.name ∧ xs.length = d.args.length
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ E_hd ∈ E' ∧ X = E_hd.name ∧ xs.length = E_hd.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp only [h1_1]
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ E_hd ∈ E' ∧ X = E_hd.name ∧ xs.length = E_hd.args.length
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ E_hd ∈ E1 ++ E_hd :: E_tl ∧ X = E_hd.name ∧ xs.length = E_hd.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
constructor
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ E_hd ∈ E1 ++ E_hd :: E_tl ∧ X = E_hd.name ∧ xs.length = E_hd.args.length
case left E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ E_hd ∈ E1 ++ E_hd :: E_tl case right E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ X = E_hd.name ∧ xs.length = E_hd.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp
case left E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ E_hd ∈ E1 ++ E_hd :: E_tl
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp only [← c1]
case right E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ X = E_hd.name ∧ xs.length = E_hd.args.length
case right E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ X = d.name ∧ xs.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact h2_1_right
case right E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d = E_hd ⊒ X = d.name ∧ xs.length = d.args.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
apply E_ih
E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ IsMetaVarOrAllDefInEnv E' (def_ X xs)
case h1 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ βˆƒ E1, E' = E1 ++ E_tl case h2 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ IsMetaVarOrAllDefInEnv E_tl (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
apply Exists.intro (E1 ++ [E_hd])
case h1 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ βˆƒ E1, E' = E1 ++ E_tl
case h1 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ E' = E1 ++ [E_hd] ++ E_tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
simp
case h1 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ E' = E1 ++ [E_hd] ++ E_tl
case h1 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ E' = E1 ++ E_hd :: E_tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact h1_1
case h1 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ E' = E1 ++ E_hd :: E_tl
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
unfold IsMetaVarOrAllDefInEnv
case h2 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ IsMetaVarOrAllDefInEnv E_tl (def_ X xs)
case h2 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ βˆƒ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
apply Exists.intro d
case h2 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ βˆƒ d ∈ E_tl, X = d.name ∧ xs.length = d.args.length
case h2 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ d ∈ E_tl ∧ X = d.name ∧ xs.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
constructor
case h2 E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ d ∈ E_tl ∧ X = d.name ∧ xs.length = d.args.length
case h2.left E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ d ∈ E_tl case h2.right E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ X = d.name ∧ xs.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact c1
case h2.left E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ d ∈ E_tl
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.concat_env_preserves_is_meta_var_or_all_def_in_env
[1003, 1]
[1079, 29]
exact h2_1_right
case h2.right E' : Env E_hd : Definition E_tl : List Definition E_ih : βˆ€ (F : Formula), (βˆƒ E1, E' = E1 ++ E_tl) β†’ IsMetaVarOrAllDefInEnv E_tl F β†’ IsMetaVarOrAllDefInEnv E' F h1 : βˆƒ E1, E' = E1 ++ E_hd :: E_tl X : DefName xs : List VarName h2 : βˆƒ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length E1 : Env h1_1 : E' = E1 ++ E_hd :: E_tl d : Definition h2_1_right : X = d.name ∧ xs.length = d.args.length c1 : d ∈ E_tl ⊒ X = d.name ∧ xs.length = d.args.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
induction E
E : Env d : Definition h1 : E.WellFormed h2 : d ∈ E ⊒ IsMetaVarOrAllDefInEnv E d.F
case nil d : Definition h1 : Env.WellFormed [] h2 : d ∈ [] ⊒ IsMetaVarOrAllDefInEnv [] d.F case cons d head✝ : Definition tail✝ : List Definition tail_ih✝ : Env.WellFormed tail✝ β†’ d ∈ tail✝ β†’ IsMetaVarOrAllDefInEnv tail✝ d.F h1 : Env.WellFormed (head✝ :: tail✝) h2 : d ∈ head✝ :: tail✝ ⊒ IsMetaVarOrAllDefInEnv (head✝ :: tail✝) d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
case nil => simp at h2
d : Definition h1 : Env.WellFormed [] h2 : d ∈ [] ⊒ IsMetaVarOrAllDefInEnv [] d.F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
simp at h2
d : Definition h1 : Env.WellFormed [] h2 : d ∈ [] ⊒ IsMetaVarOrAllDefInEnv [] d.F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
unfold Env.WellFormed at h1
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F h1 : Env.WellFormed (hd :: tl) h2 : d ∈ hd :: tl ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F h2 : d ∈ hd :: 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
simp at h2
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F h2 : d ∈ hd :: 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
cases h1
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
case intro d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
cases h1_right
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
case intro d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
apply concat_env_preserves_is_meta_var_or_all_def_in_env tl (hd :: tl)
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv (hd :: tl) d.F
case h1 d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ βˆƒ E1, hd :: tl = E1 ++ tl case h2 d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv tl d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
apply Exists.intro [hd]
case h1 d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ βˆƒ E1, hd :: tl = E1 ++ tl
case h1 d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ hd :: tl = [hd] ++ tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
simp
case h1 d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ hd :: tl = [hd] ++ tl
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
cases h2
case h2 d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 ⊒ IsMetaVarOrAllDefInEnv tl d.F
case h2.inl d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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✝ : d = hd ⊒ IsMetaVarOrAllDefInEnv tl d.F case h2.inr d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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✝ : d ∈ tl ⊒ IsMetaVarOrAllDefInEnv tl d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
case _ c1 => subst c1 exact h1_right_left
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 c1 : d = hd ⊒ IsMetaVarOrAllDefInEnv tl d.F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
case _ c1 => exact ih h1_right_right c1
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 c1 : d ∈ tl ⊒ IsMetaVarOrAllDefInEnv tl d.F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
subst c1
d hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F 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 c1 : d = hd ⊒ IsMetaVarOrAllDefInEnv tl d.F
d : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F h1_right_right : Env.WellFormed tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F ⊒ IsMetaVarOrAllDefInEnv tl d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_in_well_formed_env_is_meta_var_or_all_def_in_env
[1082, 1]
[1109, 39]
exact h1_right_left
d : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ IsMetaVarOrAllDefInEnv tl d.F h1_right_right : Env.WellFormed tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F ⊒ IsMetaVarOrAllDefInEnv tl d.F
no goals