. 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
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
no goals