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/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | . trivial | case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ refl_trans_clos (fun x y => S x y) z w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ refl_trans_clos (fun x y => S x y) z w
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | apply refl_trans_clos_monotone | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos S x y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ ∀ (x y : A), ?R x y → S x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos ?R x y
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos S x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | intros _ _ | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ ∀ (x y : A), ?R x y → S x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos ?R x y
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ A → A → Prop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
x✝ y✝ : A
⊢ ?R x✝ y✝ → S x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos ?R x y
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ ∀ (x y : A), ?R x y → S x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos ?R x y
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | apply (equiv _ _).2 | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
x✝ y✝ : A
⊢ ?R x✝ y✝ → S x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos ?R x y
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ A → A → Prop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos (fun x y => R x y) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
x✝ y✝ : A
⊢ ?R x✝ y✝ → S x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos ?R x y
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos (fun x y => R x y) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
⊢ refl_trans_clos (fun x y => R x y) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | apply refl_trans_clos_monotone | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos S x z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ ∀ (x y : A), ?R x y → S x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos ?R x z
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos S x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | intros _ _ | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ ∀ (x y : A), ?R x y → S x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos ?R x z
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ A → A → Prop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
x✝ y✝ : A
⊢ ?R x✝ y✝ → S x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos ?R x z
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ ∀ (x y : A), ?R x y → S x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos ?R x z
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | apply (equiv _ _).2 | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
x✝ y✝ : A
⊢ ?R x✝ y✝ → S x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos ?R x z
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ A → A → Prop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos (fun x y => R x y) x z | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
x✝ y✝ : A
⊢ ?R x✝ y✝ → S x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos ?R x z
case R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos (fun x y => R x y) x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1 : refl_trans_clos S x y
⊢ refl_trans_clos (fun x y => R x y) x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | intros x y | case intro.intro.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ ∀ (x y : A), ?intro.intro.left.R x y → R x y | case intro.intro.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y✝ z x✝ : A
red_x_y : refl_trans_clos R x✝ y✝
red_y_z : refl_trans_clos R x✝ z
h1✝ : refl_trans_clos S x✝ y✝
h2✝ : refl_trans_clos S x✝ z
w : A
h1 : refl_trans_clos S y✝ w
h2 : refl_trans_clos S z w
x y : A
⊢ ?intro.intro.left.R x y → R x y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ ∀ (x y : A), ?intro.intro.left.R x y → R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | apply (equiv _ _).1 | case intro.intro.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y✝ z x✝ : A
red_x_y : refl_trans_clos R x✝ y✝
red_y_z : refl_trans_clos R x✝ z
h1✝ : refl_trans_clos S x✝ y✝
h2✝ : refl_trans_clos S x✝ z
w : A
h1 : refl_trans_clos S y✝ w
h2 : refl_trans_clos S z w
x y : A
⊢ ?intro.intro.left.R x y → R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y✝ z x✝ : A
red_x_y : refl_trans_clos R x✝ y✝
red_y_z : refl_trans_clos R x✝ z
h1✝ : refl_trans_clos S x✝ y✝
h2✝ : refl_trans_clos S x✝ z
w : A
h1 : refl_trans_clos S y✝ w
h2 : refl_trans_clos S z w
x y : A
⊢ ?intro.intro.left.R x y → R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | trivial | case intro.intro.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ refl_trans_clos (fun x y => S x y) y w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ refl_trans_clos (fun x y => S x y) y w
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | intros x y | case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ ∀ (x y : A), ?intro.intro.right.R x y → R x y | case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y✝ z x✝ : A
red_x_y : refl_trans_clos R x✝ y✝
red_y_z : refl_trans_clos R x✝ z
h1✝ : refl_trans_clos S x✝ y✝
h2✝ : refl_trans_clos S x✝ z
w : A
h1 : refl_trans_clos S y✝ w
h2 : refl_trans_clos S z w
x y : A
⊢ ?intro.intro.right.R x y → R x y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ ∀ (x y : A), ?intro.intro.right.R x y → R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | apply (equiv _ _).1 | case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y✝ z x✝ : A
red_x_y : refl_trans_clos R x✝ y✝
red_y_z : refl_trans_clos R x✝ z
h1✝ : refl_trans_clos S x✝ y✝
h2✝ : refl_trans_clos S x✝ z
w : A
h1 : refl_trans_clos S y✝ w
h2 : refl_trans_clos S z w
x y : A
⊢ ?intro.intro.right.R x y → R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y✝ z x✝ : A
red_x_y : refl_trans_clos R x✝ y✝
red_y_z : refl_trans_clos R x✝ z
h1✝ : refl_trans_clos S x✝ y✝
h2✝ : refl_trans_clos S x✝ z
w : A
h1 : refl_trans_clos S y✝ w
h2 : refl_trans_clos S z w
x y : A
⊢ ?intro.intro.right.R x y → R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_conf | [717, 1] | [738, 12] | trivial | case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ refl_trans_clos (fun x y => S x y) z w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
equiv : ∀ (x y : A), S x y ↔ R x y
h :
∀ (y z x : A),
refl_trans_clos S x y → refl_trans_clos S x z → ∃ z_1, refl_trans_clos S y z_1 ∧ refl_trans_clos S z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_y_z : refl_trans_clos R x z
h1✝ : refl_trans_clos S x y
h2✝ : refl_trans_clos S x z
w : A
h1 : refl_trans_clos S y w
h2 : refl_trans_clos S z w
⊢ refl_trans_clos (fun x y => S x y) z w
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | simp | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (R : A → A → Prop),
(fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_trans_clos (refl_trans_clos R)) (refl_trans_clos R) | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (R : A → A → Prop) (x y : A), refl_trans_clos (refl_trans_clos R) x y ↔ refl_trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (R : A → A → Prop),
(fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_trans_clos (refl_trans_clos R)) (refl_trans_clos R)
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | intros R x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (R : A → A → Prop) (x y : A), refl_trans_clos (refl_trans_clos R) x y ↔ refl_trans_clos R x y | A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y : A
⊢ refl_trans_clos (refl_trans_clos R) x y ↔ refl_trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (R : A → A → Prop) (x y : A), refl_trans_clos (refl_trans_clos R) x y ↔ refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | constructor <;> intros steps <;> induction' steps with z a b c red_a_c _red_b_c ih | A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y : A
⊢ refl_trans_clos (refl_trans_clos R) x y ↔ refl_trans_clos R x y | case mp.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos R z z
case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y : A
⊢ refl_trans_clos (refl_trans_clos R) x y ↔ refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | . constructor | case mp.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos R z z
case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos R z z
case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | . apply refl_trans_clos_transitive <;> trivial | case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | . constructor | case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | . apply refl_trans_clos.step
. apply refl_trans_clos.step; trivial
constructor
. trivial | case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | constructor | case mp.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos R z z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos R z z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | apply refl_trans_clos_transitive <;> trivial | case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : refl_trans_clos R a b
_red_b_c : refl_trans_clos (refl_trans_clos R) b c
ih : refl_trans_clos R b c
⊢ refl_trans_clos R a c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | constructor | case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y z : A
⊢ refl_trans_clos (refl_trans_clos R) z z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | apply refl_trans_clos.step | case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c | case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R a ?mpr.step.b
case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) ?mpr.step.b c
case mpr.step.b
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ A | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) a c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | . apply refl_trans_clos.step; trivial
constructor | case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R a ?mpr.step.b
case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) ?mpr.step.b c
case mpr.step.b
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ A | case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) b c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R a ?mpr.step.b
case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) ?mpr.step.b c
case mpr.step.b
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ A
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | . trivial | case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) b c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | apply refl_trans_clos.step | case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R a ?mpr.step.b | case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ R a ?mpr.step.a.b
case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R ?mpr.step.a.b ?mpr.step.b
case mpr.step.a.b
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ A | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R a ?mpr.step.b
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | trivial | case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ R a ?mpr.step.a.b
case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R ?mpr.step.a.b ?mpr.step.b
case mpr.step.a.b
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ A | case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R b ?mpr.step.b | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ R a ?mpr.step.a.b
case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R ?mpr.step.a.b ?mpr.step.b
case mpr.step.a.b
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ A
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | constructor | case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R b ?mpr.step.b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step.a.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos R b ?mpr.step.b
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_idempotent | [741, 1] | [751, 14] | trivial | case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
S R : A → A → Prop
x y a b c : A
red_a_c : R a b
_red_b_c : refl_trans_clos R b c
ih : refl_trans_clos (refl_trans_clos R) b c
⊢ refl_trans_clos (refl_trans_clos R) b c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | simp [confluent, joins, wedge] | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ confluent (refl_trans_clos R) ↔ confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) ↔
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ confluent (refl_trans_clos R) ↔ confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | constructor | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) ↔
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) →
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) ↔
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | . intros h y z x red_x_y red_x_z
have h1 : refl_trans_clos (refl_trans_clos R) x y :=
by
apply (trans_clos_idempotent _ _ _).2; trivial
have h2 : refl_trans_clos (refl_trans_clos R) x z :=
by
apply (trans_clos_idempotent _ _ _).2; trivial
cases' (h _ _ _ h1 h2) with w h
cases' h with h1 h2
exists w; constructor <;> apply (trans_clos_idempotent _ _ _).1 <;> trivial | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) →
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) →
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | . intros h y z x red_x_y red_x_z
have h1 : refl_trans_clos R x y :=
by
apply (trans_clos_idempotent _ _ _).1; trivial
have h2 : refl_trans_clos R x z :=
by
apply (trans_clos_idempotent _ _ _).1; trivial
cases' (h _ _ _ h1 h2) with w h
cases' h with h1 h2
exists w; constructor <;> apply (trans_clos_idempotent _ _ _).2 <;> trivial | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | intros h y z x red_x_y red_x_z | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) →
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1) →
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | have h1 : refl_trans_clos (refl_trans_clos R) x y :=
by
apply (trans_clos_idempotent _ _ _).2; trivial | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | have h2 : refl_trans_clos (refl_trans_clos R) x z :=
by
apply (trans_clos_idempotent _ _ _).2; trivial | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
h2 : refl_trans_clos (refl_trans_clos R) x z
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | cases' (h _ _ _ h1 h2) with w h | case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
h2 : refl_trans_clos (refl_trans_clos R) x z
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | case mp.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h✝ :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
h2 : refl_trans_clos (refl_trans_clos R) x z
w : A
h : refl_trans_clos (refl_trans_clos R) y w ∧ refl_trans_clos (refl_trans_clos R) z w
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
h2 : refl_trans_clos (refl_trans_clos R) x z
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | cases' h with h1 h2 | case mp.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h✝ :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
h2 : refl_trans_clos (refl_trans_clos R) x z
w : A
h : refl_trans_clos (refl_trans_clos R) y w ∧ refl_trans_clos (refl_trans_clos R) z w
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | case mp.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1✝ : refl_trans_clos (refl_trans_clos R) x y
h2✝ : refl_trans_clos (refl_trans_clos R) x z
w : A
h1 : refl_trans_clos (refl_trans_clos R) y w
h2 : refl_trans_clos (refl_trans_clos R) z w
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h✝ :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
h2 : refl_trans_clos (refl_trans_clos R) x z
w : A
h : refl_trans_clos (refl_trans_clos R) y w ∧ refl_trans_clos (refl_trans_clos R) z w
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | exists w | case mp.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1✝ : refl_trans_clos (refl_trans_clos R) x y
h2✝ : refl_trans_clos (refl_trans_clos R) x z
w : A
h1 : refl_trans_clos (refl_trans_clos R) y w
h2 : refl_trans_clos (refl_trans_clos R) z w
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1 | case mp.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1✝ : refl_trans_clos (refl_trans_clos R) x y
h2✝ : refl_trans_clos (refl_trans_clos R) x z
w : A
h1 : refl_trans_clos (refl_trans_clos R) y w
h2 : refl_trans_clos (refl_trans_clos R) z w
⊢ refl_trans_clos R y w ∧ refl_trans_clos R z w | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1✝ : refl_trans_clos (refl_trans_clos R) x y
h2✝ : refl_trans_clos (refl_trans_clos R) x z
w : A
h1 : refl_trans_clos (refl_trans_clos R) y w
h2 : refl_trans_clos (refl_trans_clos R) z w
⊢ ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | constructor <;> apply (trans_clos_idempotent _ _ _).1 <;> trivial | case mp.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1✝ : refl_trans_clos (refl_trans_clos R) x y
h2✝ : refl_trans_clos (refl_trans_clos R) x z
w : A
h1 : refl_trans_clos (refl_trans_clos R) y w
h2 : refl_trans_clos (refl_trans_clos R) z w
⊢ refl_trans_clos R y w ∧ refl_trans_clos R z w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1✝ : refl_trans_clos (refl_trans_clos R) x y
h2✝ : refl_trans_clos (refl_trans_clos R) x z
w : A
h1 : refl_trans_clos (refl_trans_clos R) y w
h2 : refl_trans_clos (refl_trans_clos R) z w
⊢ refl_trans_clos R y w ∧ refl_trans_clos R z w
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | apply (trans_clos_idempotent _ _ _).2 | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ refl_trans_clos (refl_trans_clos R) x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ refl_trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ refl_trans_clos (refl_trans_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | trivial | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ refl_trans_clos R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
⊢ refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | apply (trans_clos_idempotent _ _ _).2 | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ refl_trans_clos (refl_trans_clos R) x z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ refl_trans_clos R x z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ refl_trans_clos (refl_trans_clos R) x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | trivial | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ refl_trans_clos R x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
y z x : A
red_x_y : refl_trans_clos R x y
red_x_z : refl_trans_clos R x z
h1 : refl_trans_clos (refl_trans_clos R) x y
⊢ refl_trans_clos R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | intros h y z x red_x_y red_x_z | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
∀ (y z x : A),
refl_trans_clos (refl_trans_clos R) x y →
refl_trans_clos (refl_trans_clos R) x z →
∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | have h1 : refl_trans_clos R x y :=
by
apply (trans_clos_idempotent _ _ _).1; trivial | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | have h2 : refl_trans_clos R x z :=
by
apply (trans_clos_idempotent _ _ _).1; trivial | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
h2 : refl_trans_clos R x z
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | cases' (h _ _ _ h1 h2) with w h | case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
h2 : refl_trans_clos R x z
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | case mpr.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h✝ :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
h2 : refl_trans_clos R x z
w : A
h : refl_trans_clos R y w ∧ refl_trans_clos R z w
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
h2 : refl_trans_clos R x z
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | cases' h with h1 h2 | case mpr.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h✝ :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
h2 : refl_trans_clos R x z
w : A
h : refl_trans_clos R y w ∧ refl_trans_clos R z w
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | case mpr.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1✝ : refl_trans_clos R x y
h2✝ : refl_trans_clos R x z
w : A
h1 : refl_trans_clos R y w
h2 : refl_trans_clos R z w
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h✝ :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
h2 : refl_trans_clos R x z
w : A
h : refl_trans_clos R y w ∧ refl_trans_clos R z w
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | exists w | case mpr.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1✝ : refl_trans_clos R x y
h2✝ : refl_trans_clos R x z
w : A
h1 : refl_trans_clos R y w
h2 : refl_trans_clos R z w
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1 | case mpr.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1✝ : refl_trans_clos R x y
h2✝ : refl_trans_clos R x z
w : A
h1 : refl_trans_clos R y w
h2 : refl_trans_clos R z w
⊢ refl_trans_clos (refl_trans_clos R) y w ∧ refl_trans_clos (refl_trans_clos R) z w | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1✝ : refl_trans_clos R x y
h2✝ : refl_trans_clos R x z
w : A
h1 : refl_trans_clos R y w
h2 : refl_trans_clos R z w
⊢ ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | constructor <;> apply (trans_clos_idempotent _ _ _).2 <;> trivial | case mpr.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1✝ : refl_trans_clos R x y
h2✝ : refl_trans_clos R x z
w : A
h1 : refl_trans_clos R y w
h2 : refl_trans_clos R z w
⊢ refl_trans_clos (refl_trans_clos R) y w ∧ refl_trans_clos (refl_trans_clos R) z w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1✝ : refl_trans_clos R x y
h2✝ : refl_trans_clos R x z
w : A
h1 : refl_trans_clos R y w
h2 : refl_trans_clos R z w
⊢ refl_trans_clos (refl_trans_clos R) y w ∧ refl_trans_clos (refl_trans_clos R) z w
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | apply (trans_clos_idempotent _ _ _).1 | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ refl_trans_clos R x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ refl_trans_clos (refl_trans_clos R) x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | trivial | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ refl_trans_clos (refl_trans_clos R) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
⊢ refl_trans_clos (refl_trans_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | apply (trans_clos_idempotent _ _ _).1 | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ refl_trans_clos R x z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ refl_trans_clos (refl_trans_clos R) x z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ refl_trans_clos R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | conf_trans_clos | [753, 1] | [776, 80] | trivial | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ refl_trans_clos (refl_trans_clos R) x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
h :
∀ (y z x : A),
refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1
y z x : A
red_x_y : refl_trans_clos (refl_trans_clos R) x y
red_x_z : refl_trans_clos (refl_trans_clos R) x z
h1 : refl_trans_clos R x y
⊢ refl_trans_clos (refl_trans_clos R) x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | simp | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ((fun R R' => ∀ (x y : A), R x y → R' x y) (fun x x_1 => R x x_1) fun x x_1 => S x x_1) →
((fun R R' => ∀ (x y : A), R x y → R' x y) (fun x x_1 => S x x_1) fun x x_1 => refl_trans_clos R x x_1) →
confluent S → confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (x y : A), R x y → S x y) → (∀ (x y : A), S x y → refl_trans_clos R x y) → confluent S → confluent R | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ((fun R R' => ∀ (x y : A), R x y → R' x y) (fun x x_1 => R x x_1) fun x x_1 => S x x_1) →
((fun R R' => ∀ (x y : A), R x y → R' x y) (fun x x_1 => S x x_1) fun x x_1 => refl_trans_clos R x x_1) →
confluent S → confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | intros | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (x y : A), R x y → S x y) → (∀ (x y : A), S x y → refl_trans_clos R x y) → confluent S → confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent R | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (x y : A), R x y → S x y) → (∀ (x y : A), S x y → refl_trans_clos R x y) → confluent S → confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | apply (conf_trans_clos _).1 | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent (refl_trans_clos R) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | apply equiv_conf | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent (refl_trans_clos R) | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), ?S x y ↔ refl_trans_clos R x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent ?S
case S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent (refl_trans_clos R)
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | . apply inc_refl_trans_eq; simp
. trivial
. simp; trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), ?S x y ↔ refl_trans_clos R x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent ?S
case S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent fun x y => (fun x x_1 => refl_trans_clos (fun x y => S x y) x x_1) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), ?S x y ↔ refl_trans_clos R x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent ?S
case S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | . aesop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent fun x y => (fun x x_1 => refl_trans_clos (fun x y => S x y) x x_1) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent fun x y => (fun x x_1 => refl_trans_clos (fun x y => S x y) x x_1) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | apply inc_refl_trans_eq | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), ?S x y ↔ refl_trans_clos R x y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => R x x_1) x y → (fun x x_1 => ?a.S✝ x x_1) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => ?a.S✝ x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
case a.S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), ?S x y ↔ refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | simp | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => R x x_1) x y → (fun x x_1 => ?a.S✝ x x_1) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => ?a.S✝ x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
case a.S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), R x y → ?a.S✝ x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => ?a.S✝ x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
case a.S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => R x x_1) x y → (fun x x_1 => ?a.S✝ x x_1) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => ?a.S✝ x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
case a.S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | . trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), R x y → ?a.S✝ x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => ?a.S✝ x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
case a.S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => S x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), R x y → ?a.S✝ x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => ?a.S✝ x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
case a.S
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | . simp; trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => S x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => S x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), R x y → ?a.S✝ x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), R x y → ?a.S✝ x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | simp | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => S x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), S x y → refl_trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), (fun x x_1 => S x x_1) x y → (fun x x_1 => refl_trans_clos R x x_1) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), S x y → refl_trans_clos R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ ∀ (x y : A), S x y → refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | inc_refl_trans_confl | [778, 1] | [787, 10] | aesop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent fun x y => (fun x x_1 => refl_trans_clos (fun x y => S x y) x x_1) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
a✝² : ∀ (x y : A), R x y → S x y
a✝¹ : ∀ (x y : A), S x y → refl_trans_clos R x y
a✝ : confluent S
⊢ confluent fun x y => (fun x x_1 => refl_trans_clos (fun x y => S x y) x x_1) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_is_refl_trans | [789, 1] | [789, 57] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
x y : A
⊢ refl_clos R x y → refl_trans_clos R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
x y : A
⊢ refl_clos R x y → refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | intros str_conf | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ strongly_confluent R → confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ confluent R | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ strongly_confluent R → confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | apply semi_confluent_implies_confluent | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ confluent R | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ semi_confluent R | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | unfold semi_confluent | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ semi_confluent R | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ ∀ (x y z : A), R x y → refl_trans_clos R x z → joins R y z | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ semi_confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | intros x y z red_x_y red_x_z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ ∀ (x y z : A), R x y → refl_trans_clos R x z → joins R y z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x y z : A
red_x_y : R x y
red_x_z : refl_trans_clos R x z
⊢ joins R y z | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
⊢ ∀ (x y z : A), R x y → refl_trans_clos R x z → joins R y z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | revert red_x_y y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x y z : A
red_x_y : R x y
red_x_z : refl_trans_clos R x z
⊢ joins R y z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z : A
red_x_z : refl_trans_clos R x z
⊢ ∀ (y : A), R x y → joins R y z | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x y z : A
red_x_y : R x y
red_x_z : refl_trans_clos R x z
⊢ joins R y z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | induction' red_x_z with a x x₂ xₙ red_x_x2 red_x2_xn ih | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z : A
red_x_z : refl_trans_clos R x z
⊢ ∀ (y : A), R x y → joins R y z | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a : A
⊢ ∀ (y : A), R a y → joins R y a
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z : A
red_x_z : refl_trans_clos R x z
⊢ ∀ (y : A), R x y → joins R y z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . intros y _; exists y; aesop | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a : A
⊢ ∀ (y : A), R a y → joins R y a
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a : A
⊢ ∀ (y : A), R a y → joins R y a
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . intros y red_x_y
cases' (str_conf _ _ _ red_x_y red_x_x2) with y₂ h
cases h.2
. exists xₙ; constructor
. apply refl_trans_clos_transitive
. apply h.1
. trivial
. constructor
. have ih : y₂ ~>*.*<~ xₙ :=
by apply ih; trivial
cases' ih with w _
exists w; constructor
. apply refl_trans_clos_transitive
. apply h.1
. aesop
. aesop | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | intros y _ | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a : A
⊢ ∀ (y : A), R a y → joins R y a | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a y : A
red_x_y✝ : R a y
⊢ joins R y a | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a : A
⊢ ∀ (y : A), R a y → joins R y a
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | exists y | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a y : A
red_x_y✝ : R a y
⊢ joins R y a | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a y : A
red_x_y✝ : R a y
⊢ refl_trans_clos R y y ∧ refl_trans_clos R a y | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a y : A
red_x_y✝ : R a y
⊢ joins R y a
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | aesop | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a y : A
red_x_y✝ : R a y
⊢ refl_trans_clos R y y ∧ refl_trans_clos R a y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x z a y : A
red_x_y✝ : R a y
⊢ refl_trans_clos R y y ∧ refl_trans_clos R a y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | intros y red_x_y | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
⊢ joins R y xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
⊢ ∀ (y : A), R x y → joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | cases' (str_conf _ _ _ red_x_y red_x_x2) with y₂ h | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
⊢ joins R y xₙ | case a.step.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
⊢ joins R y xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
⊢ joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | cases h.2 | case a.step.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
⊢ joins R y xₙ | case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ joins R y xₙ
case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
⊢ joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . exists xₙ; constructor
. apply refl_trans_clos_transitive
. apply h.1
. trivial
. constructor | case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ joins R y xₙ
case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ | case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ joins R y xₙ
case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . have ih : y₂ ~>*.*<~ xₙ :=
by apply ih; trivial
cases' ih with w _
exists w; constructor
. apply refl_trans_clos_transitive
. apply h.1
. aesop
. aesop | case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | exists xₙ | case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ joins R y xₙ | case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ ∧ refl_trans_clos R xₙ xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ joins R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | constructor | case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ ∧ refl_trans_clos R xₙ xₙ | case a.step.intro.refl.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ
case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ ∧ refl_trans_clos R xₙ xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . apply refl_trans_clos_transitive
. apply h.1
. trivial | case a.step.intro.refl.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ
case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ | case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ
case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . constructor | case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | apply refl_trans_clos_transitive | case a.step.intro.refl.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ | case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y ?a.step.intro.refl.left.y✝
case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R ?a.step.intro.refl.left.y✝ xₙ
case a.step.intro.refl.left.y
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ A | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . apply h.1 | case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y ?a.step.intro.refl.left.y✝
case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R ?a.step.intro.refl.left.y✝ xₙ
case a.step.intro.refl.left.y
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ A | case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R x₂ xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y ?a.step.intro.refl.left.y✝
case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R ?a.step.intro.refl.left.y✝ xₙ
case a.step.intro.refl.left.y
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ A
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . trivial | case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R x₂ xₙ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R x₂ xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | apply h.1 | case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y ?a.step.intro.refl.left.y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R y ?a.step.intro.refl.left.y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | trivial | case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R x₂ xₙ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R x₂ xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | constructor | case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.refl.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
h : refl_trans_clos R y x₂ ∧ refl_clos R x₂ x₂
⊢ refl_trans_clos R xₙ xₙ
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | have ih : y₂ ~>*.*<~ xₙ :=
by apply ih; trivial | case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ | case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih✝ : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
ih : joins R y₂ xₙ
⊢ joins R y xₙ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.intro.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
str_conf : strongly_confluent R
x✝ z x x₂ xₙ : A
red_x_x2 : R x x₂
red_x2_xn : refl_trans_clos R x₂ xₙ
ih : ∀ (y : A), R x₂ y → joins R y xₙ
y : A
red_x_y : R x y
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ joins R y xₙ
TACTIC:
|
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