internlm/Lean-Github huggingface dataset:
Theorem: newmans_lemma
Proof state before:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
Tactic:
intros y z wedge
Proof state after:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ joins R y z
Theorem: newmans_lemma
Proof state before:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ joins R y z
Tactic:
(apply h <;> try trivial) <;> try apply norm_trans
Proof state after:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ ∀ (x : A),
    (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
      ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
Theorem: newmans_lemma
Proof state before:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ ∀ (x : A),
    (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
      ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
Tactic:
clear h
Proof state after:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ ∀ (x : A),
    (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
      ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
Theorem: newmans_lemma
Proof state before:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ ∀ (x : A),
    (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
      ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
Tactic:
intros x ih y z wedge
Proof state after:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ joins R y z
Theorem: newmans_lemma
Proof state before:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ joins R y z
Tactic:
cases wedge.1
Proof state after:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y z✝ : A
wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
z : A
wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z
⊢ joins R x z

case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ joins R y z
Theorem: newmans_lemma
Proof state before:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y z✝ : A
wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
z : A
wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z
⊢ joins R x z

case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ joins R y z
Tactic:
. exists z; aesop
Proof state after:
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ joins R y z
Theorem: newmans_lemma
Proof state before:
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ joins R y z
Tactic:
. case a.step y' red_x_y' red_y'_y =>
  cases wedge.2
  . exists y; aesop
  . case step z' red_x_z' red_z'_z =>
    have join1 := weak _ _ _ red_x_y' red_x_z'
    cases' join1 with w h'
    have join2 : y ~>*.*<~ w :=
    by
      apply ih
      . apply trans_clos.base
        apply red_x_y'
      . aesop
    have join3 : z ~>*.*<~ w :=
    by
      apply ih
      . apply trans_clos.base
        apply red_x_z'
      . aesop
    cases' join2 with w1 h1
    cases' join3 with w2 h2
    have red_x_w : x ~>+ w :=
      by
        apply refl_trans_step_is_trans
        . apply red_x_z'
        . apply h'.2
    have join4 : w1 ~>*.*<~ w2 :=
    by
      apply ih
      . apply red_x_w
      . aesop
     
    cases' join4 with omega h3
    exists omega; constructor
    . apply refl_trans_clos_transitive
      . apply h1.1
      . apply h3.1
    . apply refl_trans_clos_transitive
      . apply h2.1
      . apply h3.2
Proof state after:
no goals
Theorem: newmans_lemma
Proof state before:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ joins R y z
Tactic:
apply h <;> try trivial
Proof state after:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ ∀ (x : A),
    (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
      ∀ (y z : A), refl_trans_clos 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
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ normalizes (fun x x_1 => trans_clos R x x_1) x
Theorem: newmans_lemma
Proof state before:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ refl_trans_clos R ?x y ∧ refl_trans_clos R ?x z
Tactic:
trivial
Proof state after:
no goals
Theorem: newmans_lemma
Proof state before:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
h :
  (∀ (x : A),
      (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) →
    ∀ (x : A),
      normalizes (fun x x_1 => trans_clos R x x_1) x →
        ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
⊢ normalizes (fun x x_1 => trans_clos R x x_1) x
Tactic:
apply norm_trans
Proof state after:
no goals
Theorem: newmans_lemma
Proof state before:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y z✝ : A
wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
z : A
wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z
⊢ joins R x z
Tactic:
exists z
Proof state after:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y z✝ : A
wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
z : A
wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z
⊢ refl_trans_clos R x z ∧ refl_trans_clos R z z
Theorem: newmans_lemma
Proof state before:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y z✝ : A
wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
z : A
wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z
⊢ refl_trans_clos R x z ∧ refl_trans_clos R z z
Tactic:
aesop
Proof state after:
no goals
Theorem: newmans_lemma
Proof state before:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
⊢ joins R y z
Tactic:
case a.step y' red_x_y' red_y'_y =>
cases wedge.2
. exists y; aesop
. case step z' red_x_z' red_z'_z =>
  have join1 := weak _ _ _ red_x_y' red_x_z'
  cases' join1 with w h'
  have join2 : y ~>*.*<~ w :=
  by
    apply ih
    . apply trans_clos.base
      apply red_x_y'
    . aesop
  have join3 : z ~>*.*<~ w :=
  by
    apply ih
    . apply trans_clos.base
      apply red_x_z'
    . aesop
  cases' join2 with w1 h1
  cases' join3 with w2 h2
  have red_x_w : x ~>+ w :=
    by
      apply refl_trans_step_is_trans
      . apply red_x_z'
      . apply h'.2
  have join4 : w1 ~>*.*<~ w2 :=
  by
    apply ih
    . apply red_x_w
    . aesop
   
  cases' join4 with omega h3
  exists omega; constructor
  . apply refl_trans_clos_transitive
    . apply h1.1
    . apply h3.1
  . apply refl_trans_clos_transitive
    . apply h2.1
    . apply h3.2
Proof state after:
no goals
Theorem: newmans_lemma
Proof state before:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
⊢ joins R y z
Tactic:
cases wedge.2
Proof state after:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x x
⊢ joins R y x

case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ z
⊢ joins R y z
Theorem: newmans_lemma
Proof state before:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x x
⊢ joins R y x

case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ z
⊢ joins R y z
Tactic:
. exists y; aesop
Proof state after:
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ z
⊢ joins R y z
Theorem: newmans_lemma
Proof state before:
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ z
⊢ joins R y z
Tactic:
. case step z' red_x_z' red_z'_z =>
  have join1 := weak _ _ _ red_x_y' red_x_z'
  cases' join1 with w h'
  have join2 : y ~>*.*<~ w :=
  by
    apply ih
    . apply trans_clos.base
      apply red_x_y'
    . aesop
  have join3 : z ~>*.*<~ w :=
  by
    apply ih
    . apply trans_clos.base
      apply red_x_z'
    . aesop
  cases' join2 with w1 h1
  cases' join3 with w2 h2
  have red_x_w : x ~>+ w :=
    by
      apply refl_trans_step_is_trans
      . apply red_x_z'
      . apply h'.2
  have join4 : w1 ~>*.*<~ w2 :=
  by
    apply ih
    . apply red_x_w
    . aesop
   
  cases' join4 with omega h3
  exists omega; constructor
  . apply refl_trans_clos_transitive
    . apply h1.1
    . apply h3.1
  . apply refl_trans_clos_transitive
    . apply h2.1
    . apply h3.2
Proof state after:
no goals
Theorem: newmans_lemma
Proof state before:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x x
⊢ joins R y x
Tactic:
exists y
Proof state after:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x x
⊢ refl_trans_clos R y y ∧ refl_trans_clos R x y
Theorem: newmans_lemma
Proof state before:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x x
⊢ refl_trans_clos R y y ∧ refl_trans_clos R x y
Tactic:
aesop
Proof state after:
no goals
Theorem: newmans_lemma
Proof state before:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
norm_R : normalizing R
weak : weakly_confluent R
x✝ : A
norm_trans : normalizing fun x x_1 => trans_clos R x x_1
y✝ z✝ : A
wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝
x : A
ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z
y z : A
wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z
y' : A
red_x_y' : R x y'
red_y'_y : refl_trans_clos R y' y
z' : A
red_x_z' : R x z'
red_z'_z : refl_trans_clos R z' z
⊢ joins R y z
Tactic:
case step z' red_x_z' red_z'_z =>
have join1 := weak _ _ _ red_x_y' red_x_z'
cases' join1 with w h'
have join2 : y ~>*.*<~ w :=
by
  apply ih
  . apply trans_clos.base
    apply red_x_y'
  . aesop
have join3 : z ~>*.*<~ w :=
by
  apply ih
  . apply trans_clos.base
    apply red_x_z'
  . aesop
cases' join2 with w1 h1
cases' join3 with w2 h2
have red_x_w : x ~>+ w :=
  by
    apply refl_trans_step_is_trans
    . apply red_x_z'
    . apply h'.2
have join4 : w1 ~>*.*<~ w2 :=
by
  apply ih
  . apply red_x_w
  . aesop
 
cases' join4 with omega h3
exists omega; constructor
. apply refl_trans_clos_transitive
  . apply h1.1
  . apply h3.1
. apply refl_trans_clos_transitive
  . apply h2.1
  . apply h3.2
Proof state after:
no goals