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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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → 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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (hd :: tl) | hd : Definition
tl : List Definition
ih : Env.WellFormed tl → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ (∀ a' ∈ tl, hd.name = a'.name → ¬hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ (∀ a' ∈ tl, hd.name = a'.name → ¬hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) tl | case left
hd : Definition
tl : List Definition
ih : Env.WellFormed tl → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ ∀ a' ∈ tl, hd.name = a'.name → ¬hd.args.length = a'.args.length
case right
hd : Definition
tl : List Definition
ih : Env.WellFormed tl → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ ∀ a' ∈ tl, hd.name = a'.name → ¬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 → List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬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 : ∃ E1, E' = E1 ++ E
h2 : IsMetaVarOrAllDefInEnv E F
⊢ IsMetaVarOrAllDefInEnv E' F | case nil
E' : Env
F : Formula
h1 : ∃ E1, E' = E1 ++ []
h2 : IsMetaVarOrAllDefInEnv [] F
⊢ IsMetaVarOrAllDefInEnv E' F
case cons
E' : Env
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
F : Formula
h1 : ∃ 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 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
F : Formula
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
a✝ : Formula
a_ih✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝ → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
a✝¹ a✝ : Formula
a_ih✝¹ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝¹ → IsMetaVarOrAllDefInEnv E' a✝¹
a_ih✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝ → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
a✝¹ : VarName
a✝ : Formula
a_ih✝ : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) a✝ → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) true_
⊢ IsMetaVarOrAllDefInEnv E' true_ | E' : Env
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi.not_
⊢ IsMetaVarOrAllDefInEnv E' phi.not_ | E' : Env
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi
⊢ IsMetaVarOrAllDefInEnv E' phi.not_ | E' : Env
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → IsMetaVarOrAllDefInEnv E' psi
h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (phi.imp_ psi)
⊢ IsMetaVarOrAllDefInEnv E' (phi.imp_ psi) | E' : Env
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
phi psi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
psi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) psi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) (forall_ x phi)
⊢ IsMetaVarOrAllDefInEnv E' (forall_ x phi) | E' : Env
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → IsMetaVarOrAllDefInEnv E' phi
h2 : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi
⊢ IsMetaVarOrAllDefInEnv E' (forall_ x phi) | E' : Env
head✝ : Definition
tail✝ : List Definition
tail_ih✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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✝ : ∀ (F : Formula), (∃ E1, E' = E1 ++ tail✝) → IsMetaVarOrAllDefInEnv tail✝ F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ head✝ :: tail✝
x : VarName
phi : Formula
phi_ih : IsMetaVarOrAllDefInEnv (head✝ :: tail✝) phi → 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 : ∃ E1, E' = E1 ++ []
X : DefName
xs : List VarName
h2 : IsMetaVarOrAllDefInEnv [] (def_ X xs)
⊢ IsMetaVarOrAllDefInEnv E' (def_ X xs) | E' : Env
h1 : ∃ E1, E' = E1 ++ []
X : DefName
xs : List VarName
h2 : ∃ 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 : ∃ E1, E' = E1 ++ []
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length
⊢ ∀ (a : Env), E' = a ++ 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] | intro E1 h1_1 | E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length
⊢ ∀ (a : Env), E' = a ++ E_hd :: E_tl → IsMetaVarOrAllDefInEnv E' (def_ X xs) | E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E_hd :: E_tl
⊢ ∀ (a : Definition), a ∈ E_hd :: E_tl ∧ X = a.name ∧ xs.length = a.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] | intro d h2_1 | E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ d ∈ E_hd :: E_tl, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E_hd :: E_tl
⊢ ∀ (a : Definition), a ∈ E_hd :: E_tl ∧ X = a.name ∧ xs.length = a.args.length → IsMetaVarOrAllDefInEnv E' (def_ X xs) | E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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
⊢ ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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
⊢ ∃ d ∈ E', X = d.name ∧ xs.length = d.args.length | E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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
⊢ ∃ E1, E' = E1 ++ E_tl
case h2
E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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
⊢ ∃ E1, E' = E1 ++ E_tl | case h1
E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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] | apply Exists.intro d | case h2
E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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
E' : Env
E_hd : Definition
E_tl : List Definition
E_ih : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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 : ∀ (F : Formula), (∃ E1, E' = E1 ++ E_tl) → IsMetaVarOrAllDefInEnv E_tl F → IsMetaVarOrAllDefInEnv E' F
h1 : ∃ E1, E' = E1 ++ E_hd :: E_tl
X : DefName
xs : List VarName
h2 : ∃ 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✝ → d ∈ tail✝ → 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 → d ∈ tl → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d ∈ hd :: tl
h1 :
(∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d ∈ hd :: tl
h1 :
(∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False) ∧
IsMetaVarOrAllDefInEnv tl hd.F ∧ Env.WellFormed tl
⊢ IsMetaVarOrAllDefInEnv (hd :: tl) d.F | d hd : Definition
tl : List Definition
ih : Env.WellFormed tl → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1 :
(∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1 :
(∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
left✝ : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ ∃ E1, hd :: tl = E1 ++ tl
case h2
d hd : Definition
tl : List Definition
ih : Env.WellFormed tl → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
⊢ ∃ E1, hd :: tl = E1 ++ tl | case h1
d hd : Definition
tl : List Definition
ih : Env.WellFormed tl → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_right_right : Env.WellFormed tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → 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 → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_right_right : Env.WellFormed tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
⊢ 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] | exact ih h1_right_right c1 | d hd : Definition
tl : List Definition
ih : Env.WellFormed tl → d ∈ tl → IsMetaVarOrAllDefInEnv tl d.F
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → 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.holds_coincide_var | [1112, 1] | [1200, 17] | induction E generalizing F vs V V' | D : Type
I : Interpretation D
V V' : Valuation D
M : MetaValuation D
E : Env
F : Formula
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs F
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M E F ↔ Holds D I V' M E F | case nil
D : Type
I : Interpretation D
M : MetaValuation D
V V' : Valuation D
F : Formula
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs F
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M [] F ↔ Holds D I V' M [] F
case cons
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
V V' : Valuation D
F : Formula
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs F
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) F ↔ Holds D I V' M (head✝ :: tail✝) F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | case nil.def_ X xs =>
simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I V' M [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | induction F generalizing vs V V' | case cons
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
V V' : Valuation D
F : Formula
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs F
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) F ↔ Holds D I V' M (head✝ :: tail✝) F | case cons.meta_var_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
a✝ : MetaVarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ a✝) ↔ Holds D I V' M (head✝ :: tail✝) (meta_var_ a✝)
case cons.pred_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (pred_ a✝¹ a✝) ↔ Holds D I V' M (head✝ :: tail✝) (pred_ a✝¹ a✝)
case cons.eq_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
a✝¹ a✝ : VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' M (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) true_ ↔ Holds D I V' M (head✝ :: tail✝) true_
case cons.not_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
a✝ : Formula
a_ih✝ :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs a✝ →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) a✝ ↔ Holds D I V' M (head✝ :: tail✝) a✝)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs a✝.not_
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' M (head✝ :: tail✝) a✝.not_
case cons.imp_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs a✝¹ →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) a✝¹ ↔ Holds D I V' M (head✝ :: tail✝) a✝¹)
a_ih✝ :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs a✝ →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) a✝ ↔ Holds D I V' M (head✝ :: tail✝) a✝)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' M (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.forall_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs a✝ →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) a✝ ↔ Holds D I V' M (head✝ :: tail✝) a✝)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' M (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.def_
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' M (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | case meta_var_ X =>
unfold NoMetaVarAndAllFreeInList at h1
contradiction | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : MetaVarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V' M (head✝ :: tail✝) (meta_var_ X) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | case pred_ X xs =>
unfold NoMetaVarAndAllFreeInList at h1
simp only [Holds]
congr! 1
simp only [List.map_eq_map_iff]
tauto | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (pred_ X xs) ↔ Holds D I V' M (head✝ :: tail✝) (pred_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | case true_ =>
simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) true_ ↔ Holds D I V' M (head✝ :: tail✝) true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | case not_ phi phi_ih =>
unfold NoMetaVarAndAllFreeInList at h1
simp only [Holds]
congr! 1
exact phi_ih V V' vs h1 h2 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi.not_
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V' M (head✝ :: tail✝) phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList at h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : MetaVarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V' M (head✝ :: tail✝) (meta_var_ X) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : MetaVarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : False
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V' M (head✝ :: tail✝) (meta_var_ X) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | contradiction | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : MetaVarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : False
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V' M (head✝ :: tail✝) (meta_var_ X) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList at h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (pred_ X xs) ↔ Holds D I V' M (head✝ :: tail✝) (pred_ X xs) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ Holds D I V M (head✝ :: tail✝) (pred_ X xs) ↔ Holds D I V' M (head✝ :: tail✝) (pred_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ Holds D I V M (head✝ :: tail✝) (pred_ X xs) ↔ Holds D I V' M (head✝ :: tail✝) (pred_ X xs) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ I X (List.map V xs) ↔ I X (List.map V' xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | congr! 1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ I X (List.map V xs) ↔ I X (List.map V' xs) | case a.h.e'_2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ List.map V xs = List.map V' xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [List.map_eq_map_iff] | case a.h.e'_2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ List.map V xs = List.map V' xs | case a.h.e'_2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ ∀ x ∈ xs, V x = V' x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | tauto | case a.h.e'_2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
X : PredName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ ∀ x ∈ xs, V x = V' x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList at h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (eq_ x y)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V' M (head✝ :: tail✝) (eq_ x y) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : x ∈ vs ∧ y ∈ vs
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V' M (head✝ :: tail✝) (eq_ x y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | cases h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : x ∈ vs ∧ y ∈ vs
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V' M (head✝ :: tail✝) (eq_ x y) | case intro
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
left✝ : x ∈ vs
right✝ : y ∈ vs
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V' M (head✝ :: tail✝) (eq_ x y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V' M (head✝ :: tail✝) (eq_ x y) | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ V x = V y ↔ V' x = V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | congr! 1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ V x = V y ↔ V' x = V' y | case a.h.e'_2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ V x = V' x
case a.h.e'_3
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ V y = V' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact h2 x h1_left | case a.h.e'_2
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ V x = V' x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | exact h2 y h1_right | case a.h.e'_3
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x y : VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1_left : x ∈ vs
h1_right : y ∈ vs
⊢ V y = V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs true_
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) true_ ↔ Holds D I V' M (head✝ :: tail✝) true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | unfold NoMetaVarAndAllFreeInList at h1 | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs phi.not_
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V' M (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V' M (head✝ :: tail✝) phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/MM0/MM0.lean | MM0.holds_coincide_var | [1112, 1] | [1200, 17] | simp only [Holds] | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V' M (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
phi : Formula
phi_ih :
∀ (V V' : Valuation D) (vs : List VarName),
NoMetaVarAndAllFreeInList vs phi →
(∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList vs phi
⊢ ¬Holds D I V M (head✝ :: tail✝) phi ↔ ¬Holds D I V' M (head✝ :: tail✝) phi |
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