Theorem: Batteries.RBNode.Path.Ordered.ins
File: .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean

Proof steps:
state_before: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left red parent x b)) a
⊢ RBNode.Ordered cmp ((left red parent x b).ins a)
state_after: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left red parent x b)) a
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
tactic: unfold ins
state_before: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left red parent x b)) a
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
state_after: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left red parent x b)) a
ax : All (fun a => cmpLT cmp a x) a
ap : All (RootOrdered cmp parent) a
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
tactic: have ⟨ax, ap⟩ := All_and.1 H
state_before: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left red parent x b)) a
ax : All (fun a => cmpLT cmp a x) a
ap : All (RootOrdered cmp parent) a
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
state_after: no goals
tactic: exact hp.ins ⟨ax, xb, ha, hb⟩ ⟨xp, ap, bp⟩
state_before: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right red a x parent)) b
⊢ RBNode.Ordered cmp ((right red a x parent).ins b)
state_after: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right red a x parent)) b
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
tactic: unfold ins
state_before: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right red a x parent)) b
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
state_after: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right red a x parent)) b
xb : All (cmpLT cmp x) b
bp : All (RootOrdered cmp parent) b
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
tactic: have ⟨xb, bp⟩ := All_and.1 H
state_before: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right red a x parent)) b
xb : All (cmpLT cmp x) b
bp : All (RootOrdered cmp parent) b
⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
state_after: no goals
tactic: exact hp.ins ⟨ax, xb, ha, hb⟩ ⟨xp, ap, bp⟩
state_before: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left black parent x b)) a
⊢ RBNode.Ordered cmp ((left black parent x b).ins a)
state_after: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left black parent x b)) a
⊢ RBNode.Ordered cmp (parent.ins (a.balance1 x b))
tactic: unfold ins
state_before: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left black parent x b)) a
⊢ RBNode.Ordered cmp (parent.ins (a.balance1 x b))
state_after: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left black parent x b)) a
ax : All (fun a => cmpLT cmp a x) a
ap : All (RootOrdered cmp parent) a
⊢ RBNode.Ordered cmp (parent.ins (a.balance1 x b))
tactic: have ⟨ax, ap⟩ := All_and.1 H
state_before: α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left black parent x b)) a
ax : All (fun a => cmpLT cmp a x) a
ap : All (RootOrdered cmp parent) a
⊢ RBNode.Ordered cmp (parent.ins (a.balance1 x b))
state_after: no goals
tactic: exact hp.ins (ha.balance1 ax xb hb) (balance1_All.2 ⟨xp, ap, bp⟩)
state_before: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right black a x parent)) b
⊢ RBNode.Ordered cmp ((right black a x parent).ins b)
state_after: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right black a x parent)) b
⊢ RBNode.Ordered cmp (parent.ins (a.balance2 x b))
tactic: unfold ins
state_before: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right black a x parent)) b
⊢ RBNode.Ordered cmp (parent.ins (a.balance2 x b))
state_after: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right black a x parent)) b
xb : All (cmpLT cmp x) b
bp : All (RootOrdered cmp parent) b
⊢ RBNode.Ordered cmp (parent.ins (a.balance2 x b))
tactic: have ⟨xb, bp⟩ := All_and.1 H
state_before: α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right black a x parent)) b
xb : All (cmpLT cmp x) b
bp : All (RootOrdered cmp parent) b
⊢ RBNode.Ordered cmp (parent.ins (a.balance2 x b))
state_after: no goals
tactic: exact hp.ins (ha.balance2 ax xb hb) (balance2_All.2 ⟨xp, ap, bp⟩)