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 |
|---|---|---|---|---|---|---|---|---|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | cases' ih with w _ | 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 y... | case a.step.intro.base.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
a✝ : R... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | exists w | case a.step.intro.base.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
a✝ : R... | case a.step.intro.base.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
a✝ : R... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | constructor | case a.step.intro.base.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
a✝ : R... | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a... |
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
. aesop | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a... | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . aesop | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | apply ih | 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 y₂ xₙ | case red_x_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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ R x₂ y₂ |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | trivial | case red_x_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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a✝ : R x₂ y₂
⊢ R x₂ y₂ | no goals |
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.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
a... | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂... |
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.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂... | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | . aesop | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂... | no goals |
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.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | aesop | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | strong_confluent_implies_confluent | [793, 1] | [817, 14] | aesop | case a.step.intro.base.intro.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
y₂ : A
h : refl_trans_clos R y y₂ ∧ refl_clos R x₂ y₂
... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_union_left | [836, 1] | [839, 30] | 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 => refl_trans_clos R x x_1)
(refl_trans_clos ((fun R S x y => R x y ∨ S x y) R S)) | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (x y : A), refl_trans_clos R x y → refl_trans_clos (fun x y => R x y ∨ S x y) x y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_union_left | [836, 1] | [839, 30] | intros x y red_x_y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (x y : A), refl_trans_clos R x y → refl_trans_clos (fun x y => R x y ∨ S x y) x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
x y : A
red_x_y : refl_trans_clos R x y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_union_left | [836, 1] | [839, 30] | induction red_x_y <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
x y : A
red_x_y : refl_trans_clos R x y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_union_right | [841, 1] | [844, 30] | 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 => refl_trans_clos S x x_1)
(refl_trans_clos ((fun R S x y => R x y ∨ S x y) R S)) | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (x y : A), refl_trans_clos S x y → refl_trans_clos (fun x y => R x y ∨ S x y) x y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_union_right | [841, 1] | [844, 30] | intros x y red_x_y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ ∀ (x y : A), refl_trans_clos S x y → refl_trans_clos (fun x y => R x y ∨ S x y) x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
x y : A
red_x_y : refl_trans_clos S x y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_union_right | [841, 1] | [844, 30] | induction red_x_y <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
x y : A
red_x_y : refl_trans_clos S x y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | intros commut_r_s conf_r conf_s | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ commute R S → confluent R → confluent S → confluent ((fun R S x y => R x y ∨ S x y) R S) | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ confluent ((fun R S x y => R x y ∨ S x y) R S) |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | apply inc_refl_trans_confl | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ confluent ((fun R S x y => R x y ∨ S x y) R S) | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A), (fun x x_1 => (fun R S x y => R x y ∨ S x y) R S x x_1) x y → (fun x x_1 => ?S x x_1) x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Pr... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . have h : R ∪ S ⊆ (. ~>₁* .) ∘ (. ~>₂* .) :=
by
simp; intros x y red_or
cases red_or
. exists y; aesop
. exists x; aesop
apply h | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A), (fun x x_1 => (fun R S x y => R x y ∨ S x y) R S x x_1) x y → (fun x x_1 => ?S x x_1) x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Pr... | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A),
(fun x x_1 => ∃ z, (fun x x_2 => refl_trans_clos R x x_2) x z ∧ (fun x x_2 => refl_trans_clos S x x_2) z x_1) x y →
(fun x x_1 => refl_trans_clos ((fun R... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . simp
intros x y z red_x_z red_z_y
apply refl_trans_clos_transitive
. apply refl_trans_union_left
trivial
. apply refl_trans_union_right; trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A),
(fun x x_1 => ∃ z, (fun x x_2 => refl_trans_clos R x x_2) x z ∧ (fun x x_2 => refl_trans_clos S x x_2) z x_1) x y →
(fun x x_1 => refl_trans_clos ((fun R... | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ confluent fun x y => ∃ z, (fun x x_1 => refl_trans_clos R x x_1) x z ∧ (fun x x_1 => refl_trans_clos S x x_1) z y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . apply diamond_implies_confluent
simp [diamond]
intros x y z y1 red_x_y1 red_y1_y
intros z1 red_x_z1 red_z1_z
have wedge_y1_z1 : wedge R y1 z1 :=
by exists x
have join_y1_z1 := conf_r _ _ wedge_y1_z1
cases' join_y1_z1 with w h
clear wedge_y1_z1
have h1 := commut_r_s _ _ _ h.1 red_y1_y
have h2 := ... | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ confluent fun x y => ∃ z, (fun x x_1 => refl_trans_clos R x x_1) x z ∧ (fun x x_1 => refl_trans_clos S x x_1) z y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | have h : R ∪ S ⊆ (. ~>₁* .) ∘ (. ~>₂* .) :=
by
simp; intros x y red_or
cases red_or
. exists y; aesop
. exists x; aesop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A), (fun x x_1 => (fun R S x y => R x y ∨ S x y) R S x x_1) x y → (fun x x_1 => ?S x x_1) x y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
h :
(fun R R' => ∀ (x y : A), R x y → R' x y) ((fun R S x y => R x y ∨ S x y) R S)
((fun R S x y => ∃ z, R x z ∧ S z y) (fun x x_1 => refl_trans_clos R x x_1) fun x x_1 => ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | apply h | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
h :
(fun R R' => ∀ (x y : A), R x y → R' x y) ((fun R S x y => R x y ∨ S x y) R S)
((fun R S x y => ∃ z, R x z ∧ S z y) (fun x x_1 => refl_trans_clos R x x_1) fun x x_1 => ... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | simp | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) ((fun R S x y => R x y ∨ S x y) R S)
((fun R S x y => ∃ z, R x z ∧ S z y) (fun x x_1 => refl_trans_clos R x x_1) fun x x_1 => refl_trans_... | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A), R x y ∨ S x y → ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | intros x y red_or | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A), R x y ∨ S x y → ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
red_or : R x y ∨ S x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | cases red_or | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
red_or : R x y ∨ S x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y | case inl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : R x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y
case inr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
c... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . exists y; aesop | case inl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : R x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y
case inr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
c... | case inr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : S x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . exists x; aesop | case inr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : S x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | exists y | case inl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : R x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y | case inl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : R x y
⊢ refl_trans_clos R x y ∧ refl_trans_clos S y y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | aesop | case inl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : R x y
⊢ refl_trans_clos R x y ∧ refl_trans_clos S y y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | exists x | case inr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : S x y
⊢ ∃ z, refl_trans_clos R x z ∧ refl_trans_clos S z y | case inr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : S x y
⊢ refl_trans_clos R x x ∧ refl_trans_clos S x y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | aesop | case inr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y : A
h✝ : S x y
⊢ refl_trans_clos R x x ∧ refl_trans_clos S x y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | simp | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y : A),
(fun x x_1 => ∃ z, (fun x x_2 => refl_trans_clos R x x_2) x z ∧ (fun x x_2 => refl_trans_clos S x x_2) z x_1) x y →
(fun x x_1 => refl_trans_clos ((fun R... | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y x_1 : A), refl_trans_clos R x x_1 → refl_trans_clos S x_1 y → refl_trans_clos (fun x y => R x y ∨ S x y) x y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | intros x y z red_x_z red_z_y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y x_1 : A), refl_trans_clos R x x_1 → refl_trans_clos S x_1 y → refl_trans_clos (fun x y => R x y ∨ S x y) x y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | apply refl_trans_clos_transitive | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x ?a.y✝
case a.a
A : Type
R : A → A → Prop
inhabited_A :... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . apply refl_trans_union_left
trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x ?a.y✝
case a.a
A : Type
R : A → A → Prop
inhabited_A :... | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) z y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . apply refl_trans_union_right; trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) z y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | apply refl_trans_union_left | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) x ?a.y✝ | case a.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y) x ?a.y✝ |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | trivial | case a.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y) x ?a.y✝ | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | apply refl_trans_union_right | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => R x y ∨ S x y) z y | case a.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => S x y) z y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | trivial | case a.a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z : A
red_x_z : refl_trans_clos R x z
red_z_y : refl_trans_clos S z y
⊢ refl_trans_clos (fun x y => S x y) z y | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | apply diamond_implies_confluent | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ confluent fun x y => ∃ z, (fun x x_1 => refl_trans_clos R x x_1) x z ∧ (fun x x_1 => refl_trans_clos S x x_1) z y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ diamond fun x y => ∃ z, (fun x x_1 => refl_trans_clos R x x_1) x z ∧ (fun x x_1 => refl_trans_clos S x x_1) z y |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | simp [diamond] | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ diamond fun x y => ∃ z, (fun x x_1 => refl_trans_clos R x x_1) x z ∧ (fun x x_1 => refl_trans_clos S x x_1) z y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y z x_1 : A),
refl_trans_clos R x x_1 →
refl_trans_clos S x_1 y →
∀ (x_2 : A),
refl_trans_clos R x x_2 →
refl_trans_clos S x_2 z ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | intros x y z y1 red_x_y1 red_y1_y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
⊢ ∀ (x y z x_1 : A),
refl_trans_clos R x x_1 →
refl_trans_clos S x_1 y →
∀ (x_2 : A),
refl_trans_clos R x x_2 →
refl_trans_clos S x_2 z ... | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
⊢ ∀ (x_1 : A),
refl_trans_clos R x x_1 →
refl_trans_clos S x_1 z →
∃ w,
... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | intros z1 red_x_z1 red_z1_z | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
⊢ ∀ (x_1 : A),
refl_trans_clos R x x_1 →
refl_trans_clos S x_1 z →
∃ w,
... | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
⊢ ∃ w, (∃ z, refl_tr... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | have wedge_y1_z1 : wedge R y1 z1 :=
by exists x | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
⊢ ∃ w, (∃ z, refl_tr... | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
wedge_y1_z1 : wedge ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | have join_y1_z1 := conf_r _ _ wedge_y1_z1 | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
wedge_y1_z1 : wedge ... | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
wedge_y1_z1 : wedge ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | cases' join_y1_z1 with w h | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
wedge_y1_z1 : wedge ... | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
wedge_y1_z1 : ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | clear wedge_y1_z1 | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
wedge_y1_z1 : ... | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | have h1 := commut_r_s _ _ _ h.1 red_y1_y | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl... | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | have h2 := commut_r_s _ _ _ h.2 red_z1_z | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl... | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | cases' h1 with w1 h1 | case a.a.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl... | case a.a.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | cases' h2 with w2 h2 | case a.a.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h ... | case a.a.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | have wedge_w1_w2 : wedge S w1 w2 :=
by exists w; aesop | case a.a.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w ... | case a.a.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | have join_w1_w2 : joins S w1 w2 := by apply conf_s; trivial | case a.a.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w ... | case a.a.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w ... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | cases' join_w1_w2 with omega h3 | case a.a.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w ... | case a.a.intro.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | clear wedge_w1_w2 | case a.a.intro.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z... | case a.a.intro.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | exists omega | case a.a.intro.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z... | case a.a.intro.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | constructor | case a.a.intro.intro.intro.intro
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z... | case a.a.intro.intro.intro.intro.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clo... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . exists w1; aesop | case a.a.intro.intro.intro.intro.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clo... | case a.a.intro.intro.intro.intro.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_cl... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | . exists w2; aesop | case a.a.intro.intro.intro.intro.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_cl... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | exists x | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
⊢ wedge R y1 z1 | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | exists w | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl_trans_clos R y... | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl_trans_clos R y... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl_trans_clos R y... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | apply conf_s | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl_trans_clos R y... | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl_trans_c... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clos S z1 z
w : A
h : refl_trans_c... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | exists w1 | case a.a.intro.intro.intro.intro.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clo... | case a.a.intro.intro.intro.intro.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clo... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | aesop | case a.a.intro.intro.intro.intro.left
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_clo... | no goals |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | exists w2 | case a.a.intro.intro.intro.intro.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_cl... | case a.a.intro.intro.intro.intro.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_cl... |
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | commuting_confluence_implies_confluence | [847, 1] | [886, 25] | aesop | case a.a.intro.intro.intro.intro.right
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
commut_r_s : commute R S
conf_r : confluent R
conf_s : confluent S
x y z y1 : A
red_x_y1 : refl_trans_clos R x y1
red_y1_y : refl_trans_clos S y1 y
z1 : A
red_x_z1 : refl_trans_clos R x z1
red_z1_z : refl_trans_cl... | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.left_mem_pair | [7, 1] | [8, 10] | simp | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∈ pair x y | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.right_mem_pair | [10, 1] | [11, 10] | simp | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈ pair x y | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.mem_sInter_iff | [34, 1] | [37, 8] | unfold sInter NonemptySet | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈ ⋂ x ↔ NonemptySet x ∧ ∀ (t : V), t ∈ x → y ∈ t | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈
sep
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr 1) (Sum.inl ()))
(BoundedFormula.mem (Sum.inr 0) (Sum.inr 1))))
(fun x_1 => x) (⋃ x) ↔
(∃ y, y ∈ x) ∧ ∀ (t : V), t ∈ x → y ∈ t |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.mem_sInter_iff | [34, 1] | [37, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈
sep
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr 1) (Sum.inl ()))
(BoundedFormula.mem (Sum.inr 0) (Sum.inr 1))))
(fun x_1 => x) (⋃ x) ↔
(∃ y, y ∈ x) ∧ ∀ (t : V), t ∈ x → y ∈ t | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.subset_sInter | [39, 1] | [41, 8] | intro hx h z hz | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ NonemptySet x → y ∈ x → ⋂ x ⊆ y | V : Type u_1
inst✝ : Zermelo V
x y z✝ : V
hx : NonemptySet x
h : y ∈ x
z : V
hz : z ∈ ⋂ x
⊢ z ∈ y |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.subset_sInter | [39, 1] | [41, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x y z✝ : V
hx : NonemptySet x
h : y ∈ x
z : V
hz : z ∈ ⋂ x
⊢ z ∈ y | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.mem_inter_iff | [48, 1] | [51, 7] | show z ∈ ⋂ pair x y ↔ z ∈ x ∧ z ∈ y | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈ x ∩ y ↔ z ∈ x ∧ z ∈ y | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈ ⋂ pair x y ↔ z ∈ x ∧ z ∈ y |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.mem_inter_iff | [48, 1] | [51, 7] | simp | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈ ⋂ pair x y ↔ z ∈ x ∧ z ∈ y | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_self | [60, 1] | [61, 15] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ x = x | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ x ↔ z✝ ∈ x |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_self | [60, 1] | [61, 15] | simp | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ x ↔ z✝ ∈ x | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_self | [64, 1] | [65, 15] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ x = x | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ x ↔ z✝ ∈ x |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_self | [64, 1] | [65, 15] | simp | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ x ↔ z✝ ∈ x | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | constructor | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ {x} ∪ pair x y = {x} ∩ pair x y ↔ x = y | case mp
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ {x} ∪ pair x y = {x} ∩ pair x y → x = y
case mpr
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x = y → {x} ∪ pair x y = {x} ∩ pair x y |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | intro h | case mp
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ {x} ∪ pair x y = {x} ∩ pair x y → x = y | case mp
V : Type u_1
inst✝ : Zermelo V
x y z : V
h : {x} ∪ pair x y = {x} ∩ pair x y
⊢ x = y |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | rw [ext_iff] at h | case mp
V : Type u_1
inst✝ : Zermelo V
x y z : V
h : {x} ∪ pair x y = {x} ∩ pair x y
⊢ x = y | case mp
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ {x} ∩ pair x y
⊢ x = y |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | have := (h y).mp ?_ | case mp
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ {x} ∩ pair x y
⊢ x = y | case mp.refine_2
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ {x} ∩ pair x y
this : y ∈ {x} ∩ pair x y
⊢ x = y
case mp.refine_1
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ ... |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | rw [mem_inter_iff, mem_singleton_iff] at this | case mp.refine_2
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ {x} ∩ pair x y
this : y ∈ {x} ∩ pair x y
⊢ x = y | case mp.refine_2
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ {x} ∩ pair x y
this : y = x ∧ y ∈ pair x y
⊢ x = y |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | exact this.1.symm | case mp.refine_2
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ {x} ∩ pair x y
this : y = x ∧ y ∈ pair x y
⊢ x = y | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | simp | case mp.refine_1
V : Type u_1
inst✝ : Zermelo V
x y z : V
h✝ : {x} ∪ pair x y = {x} ∩ pair x y
h : ∀ (z : V), z ∈ {x} ∪ pair x y ↔ z ∈ {x} ∩ pair x y
⊢ y ∈ {x} ∪ pair x y | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | rintro rfl | case mpr
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x = y → {x} ∪ pair x y = {x} ∩ pair x y | case mpr
V : Type u_1
inst✝ : Zermelo V
x z : V
⊢ {x} ∪ pair x x = {x} ∩ pair x x |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_pair_eq_inter_pair | [67, 1] | [77, 9] | simp | case mpr
V : Type u_1
inst✝ : Zermelo V
x z : V
⊢ {x} ∪ pair x x = {x} ∩ pair x x | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.forall_not_mem | [79, 1] | [80, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
h : ∀ (y : V), ¬y ∈ x
⊢ x = ∅ | case h
V : Type u_1
inst✝ : Zermelo V
x y z : V
h : ∀ (y : V), ¬y ∈ x
z✝ : V
⊢ z✝ ∈ x ↔ z✝ ∈ ∅ |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.forall_not_mem | [79, 1] | [80, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z : V
h : ∀ (y : V), ¬y ∈ x
z✝ : V
⊢ z✝ ∈ x ↔ z✝ ∈ ∅ | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.eq_empty_iff_forall_not_mem | [82, 1] | [83, 29] | aesop | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x = ∅ → ∀ (y : V), ¬y ∈ x | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.subset_of_eq | [89, 1] | [90, 31] | subst h | V : Type u_1
inst✝ : Zermelo V
x y z : V
h : x = y
⊢ x ⊆ y | V : Type u_1
inst✝ : Zermelo V
x z : V
⊢ x ⊆ x |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.subset_of_eq | [89, 1] | [90, 31] | exact subset_rfl | V : Type u_1
inst✝ : Zermelo V
x z : V
⊢ x ⊆ x | no goals |
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.subset_antisymm | [92, 1] | [93, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
h₁ : x ⊆ y
h₂ : y ⊆ x
⊢ x = y | case h
V : Type u_1
inst✝ : Zermelo V
x y z : V
h₁ : x ⊆ y
h₂ : y ⊆ x
z✝ : V
⊢ z✝ ∈ x ↔ z✝ ∈ y |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.