url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply Forall_spec_id' (List.ofFn args_u)
|
case a.a.a.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
|
case a.a.a.h1.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.axiom_
|
case a.a.a.h1.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
|
case a.a.a.h1.h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsAxiom
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.h1.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
exact IsAxiom.eq_2_pred_const_ name n args_u args_v
|
case a.a.a.h1.h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsAxiom
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.h1.h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsAxiom
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
clear h2
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
clear h3
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [And_]
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
induction n
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
case a.a.zero
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
case a.a.succ
P_r P_s : Formula
r s : VarName
name : PredName
n✝ : ℕ
a✝ :
∀ (args_u args_v : Fin n✝ → VarName),
(∀ (i : Fin n✝), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n✝ + 1) → VarName
h1_1 : ∀ (i : Fin (n✝ + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
case _ =>
simp
SC
|
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp
|
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ true_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
SC
|
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ true_)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ true_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply
IsDeduct.mp_
((eq_ r s).imp_
(List.foldr and_ true_
(List.ofFn fun (i : Fin n) => eq_ (args_u i.succ) (args_v i.succ))))
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))
|
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))))
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))))
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))))
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [def_and_]
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))))
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).imp_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))).not_).not_)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
SC
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).imp_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))).not_).not_)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).imp_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))).not_).not_)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
specialize h1_1 0
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : args_u 0 = args_v 0 ∨ args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
cases h1_1
|
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : args_u 0 = args_v 0 ∨ args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
case a.a.inl
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h✝ : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
case a.a.inr
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h✝ : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : args_u 0 = args_v 0 ∨ args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
case _ c1 =>
apply IsDeduct.mp_ (eq_ (args_u 0) (args_v 0))
case _ =>
SC
case _ =>
simp only [c1]
apply specId (args_v 0)
apply IsDeduct.axiom_
apply IsAxiom.eq_1_
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
case _ c1 =>
cases c1
case _ c1_left c1_right =>
subst c1_left
subst c1_right
SC
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ (eq_ (args_u 0) (args_v 0))
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
case _ =>
SC
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
case _ =>
simp only [c1]
apply specId (args_v 0)
apply IsDeduct.axiom_
apply IsAxiom.eq_1_
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
SC
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [c1]
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_v 0) (args_v 0))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply specId (args_v 0)
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_v 0) (args_v 0))
|
case h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_v 0) (args_v 0))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.axiom_
|
case h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
|
case h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsAxiom (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsAxiom.eq_1_
|
case h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsAxiom (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsAxiom (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
cases c1
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
case intro
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
left✝ : args_u 0 = r
right✝ : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
case _ c1_left c1_right =>
subst c1_left
subst c1_right
SC
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1_left : args_u 0 = r
c1_right : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1_left : args_u 0 = r
c1_right : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
subst c1_left
|
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1_left : args_u 0 = r
c1_right : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
|
P_r P_s : Formula
s : VarName
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
c1_right : args_v 0 = s
ih :
∀ (args_u_1 args_v : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v i ∨ args_u_1 i = args_u 0 ∧ args_v i = s) →
IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (eq_ (args_u 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1_left : args_u 0 = r
c1_right : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
subst c1_right
|
P_r P_s : Formula
s : VarName
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
c1_right : args_v 0 = s
ih :
∀ (args_u_1 args_v : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v i ∨ args_u_1 i = args_u 0 ∧ args_v i = s) →
IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (eq_ (args_u 0) (args_v 0)))
|
P_r P_s : Formula
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
ih :
∀ (args_u_1 args_v_1 : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v_1 i ∨ args_u_1 i = args_u 0 ∧ args_v_1 i = args_v 0) →
IsDeduct ∅
((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v_1 i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (eq_ (args_u 0) (args_v 0)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
s : VarName
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
c1_right : args_v 0 = s
ih :
∀ (args_u_1 args_v : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v i ∨ args_u_1 i = args_u 0 ∧ args_v i = s) →
IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
SC
|
P_r P_s : Formula
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
ih :
∀ (args_u_1 args_v_1 : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v_1 i ∨ args_u_1 i = args_u 0 ∧ args_v_1 i = args_v 0) →
IsDeduct ∅
((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v_1 i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (eq_ (args_u 0) (args_v 0)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
ih :
∀ (args_u_1 args_v_1 : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v_1 i ∨ args_u_1 i = args_u 0 ∧ args_v_1 i = args_v 0) →
IsDeduct ∅
((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v_1 i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (eq_ (args_u 0) (args_v 0)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply ih
|
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))
|
case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ ∀ (i : Fin n), args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
intro i
|
case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ ∀ (i : Fin n), args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
|
case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
i : Fin n
⊢ args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ ∀ (i : Fin n), args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply h1_1
|
case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
i : Fin n
⊢ args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
i : Fin n
⊢ args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [isBoundIn] at h2
|
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u.not_
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
|
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u.not_
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [isBoundIn] at h3
|
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
|
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
specialize h1_ih h2 h3
|
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
|
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ ((eq_ r s).imp_ (P_u.iff_ P_v))
|
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
|
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (P_u.iff_ P_v)).imp_ ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)))
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [def_iff_]
|
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (P_u.iff_ P_v)).imp_ ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)))
|
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).and_ (P_v.not_.imp_ P_u.not_))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (P_u.iff_ P_v)).imp_ ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [def_and_]
|
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).and_ (P_v.not_.imp_ P_u.not_))))
|
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).imp_ (P_v.not_.imp_ P_u.not_).not_).not_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).and_ (P_v.not_.imp_ P_u.not_))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
SC
|
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).imp_ (P_v.not_.imp_ P_u.not_).not_).not_))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).imp_ (P_v.not_.imp_ P_u.not_).not_).not_))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
exact h1_ih
|
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [isBoundIn] at h2
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬isBoundIn r (P_u.imp_ Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬(isBoundIn r P_u ∨ isBoundIn r Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬isBoundIn r (P_u.imp_ Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
push_neg at h2
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬(isBoundIn r P_u ∨ isBoundIn r Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2 : ¬isBoundIn r P_u ∧ ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬(isBoundIn r P_u ∨ isBoundIn r Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
cases h2
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2 : ¬isBoundIn r P_u ∧ ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
case intro
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
left✝ : ¬isBoundIn r P_u
right✝ : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2 : ¬isBoundIn r P_u ∧ ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [isBoundIn] at h3
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬(isBoundIn s P_u ∨ isBoundIn s Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
push_neg at h3
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬(isBoundIn s P_u ∨ isBoundIn s Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3 : ¬isBoundIn s P_u ∧ ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬(isBoundIn s P_u ∨ isBoundIn s Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
cases h3
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3 : ¬isBoundIn s P_u ∧ ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
case intro
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
left✝ : ¬isBoundIn s P_u
right✝ : ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3 : ¬isBoundIn s P_u ∧ ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
specialize h1_ih_1 h2_left h3_left
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
specialize h1_ih_2 h2_right h3_right
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ ((eq_ r s).imp_ (Q_u.iff_ Q_v))
|
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
|
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))))
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (Q_u.iff_ Q_v))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ ((eq_ r s).imp_ (P_u.iff_ P_v))
|
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))))
|
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (P_u.iff_ P_v)).imp_
(((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))))
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [def_iff_]
|
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (P_u.iff_ P_v)).imp_
(((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))))
|
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).and_ (Q_v.imp_ Q_u))).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).and_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (P_u.iff_ P_v)).imp_
(((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [def_and_]
|
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).and_ (Q_v.imp_ Q_u))).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).and_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u))))))
|
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).imp_ (Q_v.imp_ Q_u).not_).not_).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).imp_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)).not_).not_)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).and_ (Q_v.imp_ Q_u))).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).and_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u))))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
SC
|
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).imp_ (Q_v.imp_ Q_u).not_).not_).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).imp_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)).not_).not_)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).imp_ (Q_v.imp_ Q_u).not_).not_).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).imp_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)).not_).not_)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
exact h1_ih_1
|
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
exact h1_ih_2
|
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (Q_u.iff_ Q_v))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (Q_u.iff_ Q_v))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [isBoundIn] at h2
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r (forall_ x P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬(r = x ∨ isBoundIn r P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r (forall_ x P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
push_neg at h2
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬(r = x ∨ isBoundIn r P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2 : r ≠ x ∧ ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬(r = x ∨ isBoundIn r P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
cases h2
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2 : r ≠ x ∧ ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
case intro
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
left✝ : r ≠ x
right✝ : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2 : r ≠ x ∧ ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [isBoundIn] at h3
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬(s = x ∨ isBoundIn s P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
push_neg at h3
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬(s = x ∨ isBoundIn s P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3 : s ≠ x ∧ ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬(s = x ∨ isBoundIn s P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
cases h3
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3 : s ≠ x ∧ ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
case intro
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
left✝ : s ≠ x
right✝ : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3 : s ≠ x ∧ ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply deduction_theorem
|
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct (∅ ∪ {eq_ r s}) ((forall_ x P_u).iff_ (forall_ x P_v))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp
|
case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct (∅ ∪ {eq_ r s}) ((forall_ x P_u).iff_ (forall_ x P_v))
|
case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x P_u).iff_ (forall_ x P_v))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct (∅ ∪ {eq_ r s}) ((forall_ x P_u).iff_ (forall_ x P_v))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ (forall_ x (P_u.iff_ P_v))
|
case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x P_u).iff_ (forall_ x P_v))
|
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (forall_ x (P_u.iff_ P_v))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x P_u).iff_ (forall_ x P_v))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply proof_imp_deduct
|
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
case h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply T_18_1
|
case h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ (eq_ r s)
|
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (forall_ x (P_u.iff_ P_v))
|
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (eq_ r s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (forall_ x (P_u.iff_ P_v))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply proof_imp_deduct
|
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
|
case h1.a.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.mp_ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v)))
|
case h1.a.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
|
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))).imp_ ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))))
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply T_19_TS_21_left
|
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))).imp_ ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))))
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬isFreeIn x (eq_ r s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))).imp_ ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [isFreeIn]
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬isFreeIn x (eq_ r s)
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬(x = r ∨ x = s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬isFreeIn x (eq_ r s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
push_neg
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬(x = r ∨ x = s)
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r ∧ x ≠ s
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬(x = r ∨ x = s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
constructor
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r ∧ x ≠ s
|
case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ s
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r ∧ x ≠ s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [ne_comm]
|
case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r
|
case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ r ≠ x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
exact h2_left
|
case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ r ≠ x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ r ≠ x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp only [ne_comm]
|
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ s
|
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ s ≠ x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
exact h3_left
|
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ s ≠ x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ s ≠ x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply generalization
|
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v)))
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ∀ H ∈ ∅, ¬isFreeIn x H
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
exact h1_ih h2_right h3_right
|
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
intro H a1
|
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ∀ H ∈ ∅, ¬isFreeIn x H
|
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ∀ H ∈ ∅, ¬isFreeIn x H
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp at a1
|
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
apply IsDeduct.assume_
|
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (eq_ r s)
|
case h1.a.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ eq_ r s ∈ {eq_ r s}
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (eq_ r s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
simp
|
case h1.a.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ eq_ r s ∈ {eq_ r s}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.a.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ eq_ r s ∈ {eq_ r s}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_21_8
|
[1404, 1]
|
[1543, 10]
|
sorry
|
case exists_
P_r P_s : Formula
r s x✝ : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r (exists_ x✝ P_u✝)
h3 : ¬isBoundIn s (exists_ x✝ P_u✝)
⊢ IsProof ((eq_ r s).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case exists_
P_r P_s : Formula
r s x✝ : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r (exists_ x✝ P_u✝)
h3 : ¬isBoundIn s (exists_ x✝ P_u✝)
⊢ IsProof ((eq_ r s).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝)))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.left_id_left_inverse
|
[74, 1]
|
[83, 22]
|
simp only [Function.LeftInverse]
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ LeftInverse g f
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ LeftInverse g f
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.left_id_left_inverse
|
[74, 1]
|
[83, 22]
|
intro x
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.left_id_left_inverse
|
[74, 1]
|
[83, 22]
|
exact congrFun h1 x
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.right_id_right_inverse
|
[86, 1]
|
[94, 45]
|
simp only [Function.RightInverse]
|
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ RightInverse g f
|
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ RightInverse g f
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.right_id_right_inverse
|
[86, 1]
|
[94, 45]
|
exact Function.left_id_left_inverse g f h1
|
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
funext x
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
⊢ updateITE f a b = Function.updateITE' f a b
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
⊢ updateITE f a b = Function.updateITE' f a b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
simp only [Function.updateITE']
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
split_ifs
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : x = a
h✝ : a = x
⊢ b = b
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : x = a
h✝ : ¬a = x
⊢ b = f x
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : ¬x = a
h✝ : a = x
⊢ f x = b
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : ¬x = a
h✝ : ¬a = x
⊢ f x = f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : a = x
⊢ b = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : a = x
⊢ b = b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
subst c1
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : ¬a = x
⊢ b = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : ¬a = x
⊢ b = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
subst c2
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : a = x
⊢ f x = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : a = x
⊢ f x = b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : ¬a = x
⊢ f x = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : ¬a = x
⊢ f x = f x
TACTIC:
|
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