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 | normalizing_normal | [320, 1] | [333, 14] | . exists x; constructor
. constructor
. apply h | case neg
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | cases' h with y h' | case pos
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ∃ y, R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | case pos.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ∃ y, R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | cases' h _ h' with z h'' | case pos.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | case pos.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
h'' : refl_trans_clos R y z ∧ normal R z
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | cases h'' | case pos.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
h'' : refl_trans_clos R y z ∧ normal R z
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | case pos.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
left✝ : refl_trans_clos R y z
right✝ : normal R z
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
h'' : refl_trans_clos R y z ∧ normal R z
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | exists z | case pos.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
left✝ : refl_trans_clos R y z
right✝ : normal R z
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | case pos.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
left✝ : refl_trans_clos R y z
right✝ : normal R z
⊢ refl_trans_clos R x z ∧ normal R z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
left✝ : refl_trans_clos R y z
right✝ : normal R z
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | constructor <;> aesop | case pos.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
left✝ : refl_trans_clos R y z
right✝ : normal R z
⊢ refl_trans_clos R x z ∧ normal R z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
y : A
h' : R x y
z : A
left✝ : refl_trans_clos R y z
right✝ : normal R z
⊢ refl_trans_clos R x z ∧ normal R z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | exists x | case neg
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y | case neg
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x ∧ normal R x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ ∃ y, refl_trans_clos R x y ∧ normal R y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | constructor | case neg
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x ∧ normal R x | case neg.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x
case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x ∧ normal R x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | . constructor | case neg.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x
case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x | case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x
case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | . apply h | case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | constructor | case neg.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ refl_trans_clos R x x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normalizing_normal | [320, 1] | [333, 14] | apply h | case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ x : A
h✝ : ∀ (y : A), R x y → ∃ y_1, refl_trans_clos R y y_1 ∧ normal R y_1
h : ¬∃ y, R x y
⊢ normal R x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_refl_trans | [357, 1] | [363, 8] | intros R x y red | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ ∀ (R : A → A → Prop), (fun R R' => ∀ (x y : A), R x y → R' x y) (refl_trans_clos R) (refl_trans_sym_clos R) | A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y : A
red : refl_trans_clos R x y
⊢ refl_trans_sym_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
⊢ ∀ (R : A → A → Prop), (fun R R' => ∀ (x y : A), R x y → R' x y) (refl_trans_clos R) (refl_trans_sym_clos R)
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_refl_trans | [357, 1] | [363, 8] | induction red | A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y : A
red : refl_trans_clos R x y
⊢ refl_trans_sym_clos R x y | case refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝ : A
⊢ refl_trans_sym_clos R a✝ a✝
case step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y : A
red : refl_trans_clos R x y
⊢ refl_trans_sym_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_refl_trans | [357, 1] | [363, 8] | constructor | case refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝ : A
⊢ refl_trans_sym_clos R a✝ a✝
case step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² c✝ | case step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝ : A
⊢ refl_trans_sym_clos R a✝ a✝
case step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_refl_trans | [357, 1] | [363, 8] | apply refl_trans_sym_clos.trans <;> try trivial | case step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² c✝ | case step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² b✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case step
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_refl_trans | [357, 1] | [363, 8] | aesop | case step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² b✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R a✝² b✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_refl_trans | [357, 1] | [363, 8] | trivial | case step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R ?step.b c✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step.a
A : Type
R✝ : A → A → Prop
inhabited_A : Nonempty A
R : A → A → Prop
x y a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : refl_trans_sym_clos R b✝ c✝
⊢ refl_trans_sym_clos R ?step.b c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | intros x y wedge | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) (wedge R) fun x x_1 => refl_trans_sym_clos R x x_1 | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
wedge : _root_.wedge R x y
⊢ refl_trans_sym_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
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) (wedge R) fun x x_1 => refl_trans_sym_clos R x x_1
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | cases' wedge with w h | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
wedge : _root_.wedge R x y
⊢ refl_trans_sym_clos R x y | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h : refl_trans_clos R w x ∧ refl_trans_clos R w y
⊢ refl_trans_sym_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
x y : A
wedge : _root_.wedge R x y
⊢ refl_trans_sym_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | cases' h with h1 h2 | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h : refl_trans_clos R w x ∧ refl_trans_clos R w y
⊢ refl_trans_sym_clos R x y | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h : refl_trans_clos R w x ∧ refl_trans_clos R w y
⊢ refl_trans_sym_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | apply (refl_trans_sym_clos.trans _ w) | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x y | case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x w
case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | . apply refl_trans_sym_clos.inv
apply refl_trans_sym_refl_trans; trivial | case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x w
case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y | case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x w
case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | . apply refl_trans_sym_refl_trans; trivial | case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | apply refl_trans_sym_clos.inv | case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x w | case intro.intro.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R x w
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | apply refl_trans_sym_refl_trans | case intro.intro.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w x | case intro.intro.a.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_clos R w x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | trivial | case intro.intro.a.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_clos R w x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_clos R w x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | apply refl_trans_sym_refl_trans | case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y | case intro.intro.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_clos R w y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_sym_clos R w y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | wedge_inc_refl_sym_trans | [365, 1] | [374, 47] | trivial | case intro.intro.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_clos R w y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ refl_trans_clos R w y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | church_rosser_implies_confluent | [377, 1] | [380, 10] | intros cr y z wedge | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ church_rosser R → confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
cr : church_rosser R
y z : A
wedge : _root_.wedge R y z
⊢ joins R y z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ church_rosser R → confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | church_rosser_implies_confluent | [377, 1] | [380, 10] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
cr : church_rosser R
y z : A
wedge : _root_.wedge R y z
⊢ joins R y z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
cr : church_rosser R
y z : A
wedge : _root_.wedge R y z
⊢ joins R y z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | swap_joins | [383, 1] | [388, 11] | intros joins | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ joins R x y → joins R y x | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
joins : _root_.joins R x y
⊢ _root_.joins R y x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ joins R x y → joins R y x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | swap_joins | [383, 1] | [388, 11] | cases' joins with z h | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
joins : _root_.joins R x y
⊢ _root_.joins R y x | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
h : refl_trans_clos R x z ∧ refl_trans_clos R y z
⊢ joins R y x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
joins : _root_.joins R x y
⊢ _root_.joins R y x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | swap_joins | [383, 1] | [388, 11] | cases h | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
h : refl_trans_clos R x z ∧ refl_trans_clos R y z
⊢ joins R y x | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
left✝ : refl_trans_clos R x z
right✝ : refl_trans_clos R y z
⊢ joins R y x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
h : refl_trans_clos R x z ∧ refl_trans_clos R y z
⊢ joins R y x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | swap_joins | [383, 1] | [388, 11] | exists z | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
left✝ : refl_trans_clos R x z
right✝ : refl_trans_clos R y z
⊢ joins R y x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
left✝ : refl_trans_clos R x z
right✝ : refl_trans_clos R y z
⊢ joins R y x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | red_joins | [390, 1] | [395, 18] | intros red_x_y joins_y_z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ R x y → joins R y z → joins R x z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
joins_y_z : joins R y z
⊢ joins R x z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ R x y → joins R y z → joins R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | red_joins | [390, 1] | [395, 18] | cases' joins_y_z with w h | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
joins_y_z : joins R y z
⊢ joins R x z | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
h : refl_trans_clos R y w ∧ refl_trans_clos R z w
⊢ joins R x z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
joins_y_z : joins R y z
⊢ joins R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | red_joins | [390, 1] | [395, 18] | cases h | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
h : refl_trans_clos R y w ∧ refl_trans_clos R z w
⊢ joins R x z | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
left✝ : refl_trans_clos R y w
right✝ : refl_trans_clos R z w
⊢ joins R x z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
h : refl_trans_clos R y w ∧ refl_trans_clos R z w
⊢ joins R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | red_joins | [390, 1] | [395, 18] | exists w | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
left✝ : refl_trans_clos R y w
right✝ : refl_trans_clos R z w
⊢ joins R x z | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
left✝ : refl_trans_clos R y w
right✝ : refl_trans_clos R z w
⊢ refl_trans_clos R x w ∧ refl_trans_clos R z w | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
left✝ : refl_trans_clos R y w
right✝ : refl_trans_clos R z w
⊢ joins R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | red_joins | [390, 1] | [395, 18] | aesop | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
left✝ : refl_trans_clos R y w
right✝ : refl_trans_clos R z w
⊢ refl_trans_clos R x w ∧ refl_trans_clos R z w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : R x y
w : A
left✝ : refl_trans_clos R y w
right✝ : refl_trans_clos R z w
⊢ refl_trans_clos R x w ∧ refl_trans_clos R z w
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | intros red_x_y joins_y_z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ refl_trans_clos R x y → joins R y z → joins R x z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : refl_trans_clos R x y
joins_y_z : joins R y z
⊢ joins R x z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ refl_trans_clos R x y → joins R y z → joins R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | induction' red_x_y with _ _ _ _ _ _ ih | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : refl_trans_clos R x y
joins_y_z : joins R y z
⊢ joins R x z | case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝ : A
joins_y_z : joins R a✝ z
⊢ joins R a✝ z
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
red_x_y : refl_trans_clos R x y
joins_y_z : joins R y z
⊢ joins R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | . trivial | case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝ : A
joins_y_z : joins R a✝ z
⊢ joins R a✝ z
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z | case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝ : A
joins_y_z : joins R a✝ z
⊢ joins R a✝ z
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | . apply red_joins
. trivial
. have h' := ih joins_y_z
trivial | case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | trivial | case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝ : A
joins_y_z : joins R a✝ z
⊢ joins R a✝ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝ : A
joins_y_z : joins R a✝ z
⊢ joins R a✝ z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | apply red_joins | case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ R a✝² ?step.y
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R ?step.y z
case step.y
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ A | Please generate a tactic in lean4 to solve the state.
STATE:
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R a✝² z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | . trivial | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ R a✝² ?step.y
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R ?step.y z
case step.y
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ A | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R b✝ z | Please generate a tactic in lean4 to solve the state.
STATE:
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ R a✝² ?step.y
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R ?step.y z
case step.y
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ A
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | . have h' := ih joins_y_z
trivial | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R b✝ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R b✝ z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | trivial | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ R a✝² ?step.y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ R a✝² ?step.y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | have h' := ih joins_y_z | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R b✝ z | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
h' : joins R b✝ z
⊢ joins R b✝ z | Please generate a tactic in lean4 to solve the state.
STATE:
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
⊢ joins R b✝ z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_left | [397, 1] | [406, 14] | trivial | case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
h' : joins R b✝ z
⊢ joins R b✝ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
joins_y_z✝ : joins R y z
a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
ih : joins R c✝ z → joins R b✝ z
joins_y_z : joins R c✝ z
h' : joins R b✝ z
⊢ joins R b✝ z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_right | [408, 1] | [414, 40] | intros red_x_y joins_y_z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
⊢ refl_trans_clos R x z → joins R y z → joins R y x | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
⊢ joins R y x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
⊢ refl_trans_clos R x z → joins R y z → joins R y x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_right | [408, 1] | [414, 40] | apply swap_joins | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
⊢ joins R y x | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
⊢ joins R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
⊢ joins R y x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_right | [408, 1] | [414, 40] | have joins_z_y := swap_joins _ joins_y_z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
⊢ joins R x y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
joins_z_y : joins R z y
⊢ joins R 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
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
⊢ joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | reds_joins_right | [408, 1] | [414, 40] | apply (reds_joins_left _) <;> trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
joins_z_y : joins R z y
⊢ joins R 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
x z y : A
red_x_y : refl_trans_clos R x z
joins_y_z : joins R y z
joins_z_y : joins R z y
⊢ joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | intros semi x y wedge | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ semi_confluent R → confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y : A
wedge : _root_.wedge R x y
⊢ joins R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ semi_confluent R → confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | cases' wedge with w h | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y : A
wedge : _root_.wedge R x y
⊢ joins R x y | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y w : A
h : refl_trans_clos R w x ∧ refl_trans_clos R w y
⊢ joins R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y : A
wedge : _root_.wedge R x y
⊢ joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | cases' h with h1 h2 | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y w : A
h : refl_trans_clos R w x ∧ refl_trans_clos R w y
⊢ joins R x y | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ joins R x y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y w : A
h : refl_trans_clos R w x ∧ refl_trans_clos R w y
⊢ joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | revert h2 y | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ joins R x y | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w : A
h1 : refl_trans_clos R w x
⊢ ∀ (y : A), refl_trans_clos R w y → joins R x y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x y w : A
h1 : refl_trans_clos R w x
h2 : refl_trans_clos R w y
⊢ joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | induction' h1 with z x y y' red_x_y _red_y_y' ih <;> intros z red_x_z | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w : A
h1 : refl_trans_clos R w x
⊢ ∀ (y : A), refl_trans_clos R w y → joins R x y | case intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ joins R z✝ z
case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
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 intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w : A
h1 : refl_trans_clos R w x
⊢ ∀ (y : A), refl_trans_clos R w y → joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | . exists z; aesop | case intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ joins R z✝ z
case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
⊢ joins R y' z | case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
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 intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ joins R z✝ z
case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
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 | semi_confluent_implies_confluent | [417, 1] | [429, 39] | . have h := semi x y z red_x_y red_x_z
cases' h with z' h
cases' h with h1 h2
have ih := ih z' h1
apply reds_joins_right <;> trivial | case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
⊢ joins R y' z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
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 | semi_confluent_implies_confluent | [417, 1] | [429, 39] | exists z | case intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ joins R z✝ z | case intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ refl_trans_clos R z✝ z ∧ refl_trans_clos R z z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ joins R z✝ z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | aesop | case intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ refl_trans_clos R z✝ z ∧ refl_trans_clos R z z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x w z✝ z : A
red_x_z : refl_trans_clos R z✝ z
⊢ refl_trans_clos R z✝ z ∧ refl_trans_clos R z z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | have h := semi x y z red_x_y red_x_z | case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
⊢ joins R y' z | case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
h : joins R y z
⊢ joins R y' z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
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 | semi_confluent_implies_confluent | [417, 1] | [429, 39] | cases' h with z' h | case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
h : joins R y z
⊢ joins R y' z | case intro.intro.step.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h : refl_trans_clos R y z' ∧ refl_trans_clos R z z'
⊢ joins R y' z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
h : joins R y z
⊢ joins R y' z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | cases' h with h1 h2 | case intro.intro.step.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h : refl_trans_clos R y z' ∧ refl_trans_clos R z z'
⊢ joins R y' z | case intro.intro.step.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h1 : refl_trans_clos R y z'
h2 : refl_trans_clos R z z'
⊢ joins R y' z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.step.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h : refl_trans_clos R y z' ∧ refl_trans_clos R z z'
⊢ joins R y' z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | have ih := ih z' h1 | case intro.intro.step.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h1 : refl_trans_clos R y z'
h2 : refl_trans_clos R z z'
⊢ joins R y' z | case intro.intro.step.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih✝ : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h1 : refl_trans_clos R y z'
h2 : refl_trans_clos R z z'
ih : joins R y' z'
⊢ joins R y' z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.step.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h1 : refl_trans_clos R y z'
h2 : refl_trans_clos R z z'
⊢ joins R y' z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | semi_confluent_implies_confluent | [417, 1] | [429, 39] | apply reds_joins_right <;> trivial | case intro.intro.step.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih✝ : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h1 : refl_trans_clos R y z'
h2 : refl_trans_clos R z z'
ih : joins R y' z'
⊢ joins R y' z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.step.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
semi : semi_confluent R
x✝ w x y y' : A
red_x_y : R x y
_red_y_y' : refl_trans_clos R y y'
ih✝ : ∀ (y_1 : A), refl_trans_clos R y y_1 → joins R y' y_1
z : A
red_x_z : refl_trans_clos R x z
z' : A
h1 : refl_trans_clos R y z'
h2 : refl_trans_clos R z z'
ih : joins R y' z'
⊢ joins R y' z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | intros sc | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ diamond R → confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
⊢ confluent R | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ diamond R → confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | apply semi_confluent_implies_confluent | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
⊢ confluent R | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond 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
sc : diamond R
⊢ confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | unfold semi_confluent | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
⊢ semi_confluent R | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond 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
sc : diamond R
⊢ semi_confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | intros x y z red h | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond 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
sc : diamond R
x y z : A
red : R x y
h : 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
sc : diamond 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 | diamond_implies_confluent | [442, 1] | [459, 30] | revert y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x y z : A
red : R x y
h : refl_trans_clos R x z
⊢ joins R y z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z : A
h : 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
sc : diamond R
x y z : A
red : R x y
h : refl_trans_clos R x z
⊢ joins R y z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | induction h | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z : A
h : 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
sc : diamond 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
sc : diamond R
x z a✝² b✝ c✝ : A
a✝¹ : R a✝² b✝
a✝ : refl_trans_clos R b✝ c✝
a_ih✝ : ∀ (y : A), R b✝ y → joins R y c✝
⊢ ∀ (y : A), R a✝² y → joins R y c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z : A
h : 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 | diamond_implies_confluent | [442, 1] | [459, 30] | case a.refl _ =>
intros y _
exists y; aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ : A
⊢ ∀ (y : A), R a✝ y → joins R y a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond 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 | diamond_implies_confluent | [442, 1] | [459, 30] | case a.step a b c red_a_b _red_b_c ih =>
intros d red_a_d
have h := (sc _ _ _ red_a_b red_a_d)
cases' h with e h
cases' h with red_b_e red_d_e
apply red_joins <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
⊢ ∀ (y : A), R a y → joins R y c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
⊢ ∀ (y : A), R a y → joins R y c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | intros y _ | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ : A
⊢ ∀ (y : A), R a✝ y → joins R y a✝ | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ y : A
red✝ : R a✝ y
⊢ joins R y a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond 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 | diamond_implies_confluent | [442, 1] | [459, 30] | exists y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ y : A
red✝ : R a✝ y
⊢ joins R y a✝ | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ y : A
red✝ : 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:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ y : A
red✝ : R a✝ y
⊢ joins R y a✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ y : A
red✝ : 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:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a✝ y : A
red✝ : 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 | diamond_implies_confluent | [442, 1] | [459, 30] | intros d red_a_d | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
⊢ ∀ (y : A), R a y → joins R y c | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
⊢ joins R d c | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
⊢ ∀ (y : A), R a y → joins R y c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | have h := (sc _ _ _ red_a_b red_a_d) | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
⊢ joins R d c | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
h : ∃ w, R b w ∧ R d w
⊢ joins R d c | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
⊢ joins R d c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | cases' h with e h | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
h : ∃ w, R b w ∧ R d w
⊢ joins R d c | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
e : A
h : R b e ∧ R d e
⊢ joins R d c | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
h : ∃ w, R b w ∧ R d w
⊢ joins R d c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | cases' h with red_b_e red_d_e | case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
e : A
h : R b e ∧ R d e
⊢ joins R d c | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
e : A
red_b_e : R b e
red_d_e : R d e
⊢ joins R d c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
e : A
h : R b e ∧ R d e
⊢ joins R d c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | diamond_implies_confluent | [442, 1] | [459, 30] | apply red_joins <;> aesop | case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
e : A
red_b_e : R b e
red_d_e : R d e
⊢ joins R d c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
sc : diamond R
x z a b c : A
red_a_b : R a b
_red_b_c : refl_trans_clos R b c
ih : ∀ (y : A), R b y → joins R y c
d : A
red_a_d : R a d
e : A
red_b_e : R b e
red_d_e : R d e
⊢ joins R d c
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | intros confl x y z red_x_y red_x_z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ confluent R → weakly_confluent R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ joins R y z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ confluent R → weakly_confluent R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | apply confl | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ joins R y z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ wedge R y z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ joins R y z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | exists x | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ wedge R y z | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ refl_trans_clos R x y ∧ refl_trans_clos R 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
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ wedge R y z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | constructor <;> apply refl_trans_clos.step <;> try constructor | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ refl_trans_clos R x y ∧ refl_trans_clos R x z | case a.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x y
case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R 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
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ refl_trans_clos R x y ∧ refl_trans_clos R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | . trivial | case a.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x y
case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x z | case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x z | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x y
case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | . trivial | case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | constructor | case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ refl_trans_clos R ?a.right.b✝ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ refl_trans_clos R ?a.right.b✝ z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | trivial | case a.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_weakly_confluent | [462, 1] | [467, 12] | trivial | case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y z : A
red_x_y : R x y
red_x_z : R x z
⊢ R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_church_rosser | [471, 1] | [495, 13] | intros confl | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ confluent R → church_rosser R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
⊢ church_rosser R | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ confluent R → church_rosser R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_church_rosser | [471, 1] | [495, 13] | unfold church_rosser | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
⊢ church_rosser R | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
⊢ ∀ (x y : A), refl_trans_sym_clos R x y → joins R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
⊢ church_rosser R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_church_rosser | [471, 1] | [495, 13] | intros x y red_x_y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
⊢ ∀ (x y : A), refl_trans_sym_clos R x y → joins R x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y : A
red_x_y : refl_trans_sym_clos R x y
⊢ joins R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
⊢ ∀ (x y : A), refl_trans_sym_clos R x y → joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_church_rosser | [471, 1] | [495, 13] | induction red_x_y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y : A
red_x_y : refl_trans_sym_clos R x y
⊢ joins R x y | case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y a✝ : A
⊢ joins R a✝ a✝
case base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y a✝¹ b✝ : A
a✝ : R a✝¹ b✝
⊢ joins R a✝¹ b✝
case trans
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y a✝² b✝ c✝ : A
a✝¹ : refl_trans_sym_clos R a✝² b✝
a✝ : refl_trans_sym_clos R b✝ c✝
a_ih✝¹ : joins R a✝² b✝
a_ih✝ : joins R b✝ c✝
⊢ joins R a✝² c✝
case inv
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y a✝¹ b✝ : A
a✝ : refl_trans_sym_clos R b✝ a✝¹
a_ih✝ : joins R b✝ a✝¹
⊢ joins R a✝¹ b✝ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x y : A
red_x_y : refl_trans_sym_clos R x y
⊢ joins R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_church_rosser | [471, 1] | [495, 13] | case refl x =>
exists x ; repeat constructor | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x✝ y x : A
⊢ joins R x x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x✝ y x : A
⊢ joins R x x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | confluent_implies_church_rosser | [471, 1] | [495, 13] | case base x y red_x_y =>
exists y; constructor <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
confl : confluent R
x✝ y✝ x y : A
red_x_y : R x y
⊢ joins 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
confl : confluent R
x✝ y✝ x y : A
red_x_y : R x y
⊢ joins R x y
TACTIC:
|
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