mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case node.nil.node.node
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : ((node lls lll llx llr).node' lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
⊢ (if lrs < ratio * lls then (node lls lll llx llr).node3R lx (node lrs lrl lrx lrr) x nil
    else (node lls lll llx llr).node4R lx (node lrs lrl lrx lrr) x nil) =
    if lrs < ratio * lls then
      node (lls + lrs + 1 + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
    else
      node (lls + lrs + 1 + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
        (node (lrr.size + 1) lrr x nil)

Tactic:
;

Goal state after:
case node.nil.node.node
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : ((node lls lll llx llr).node' lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
⊢ (if lrs < ratio * lls then (node lls lll llx llr).node3R lx (node lrs lrl lrx lrr) x nil
    else (node lls lll llx llr).node4R lx (node lrs lrl lrx lrr) x nil) =
    if lrs < ratio * lls then
      node (lls + lrs + 1 + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
    else
      node (lls + lrs + 1 + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
        (node (lrr.size + 1) lrr x nil)
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case pos
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : ((node lls lll llx llr).node' lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
h✝ : lrs < ratio * lls
⊢ lls + (lrs + 1) = lls + lrs + 1

Tactic:
;

Goal state after:
case pos
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : ((node lls lll llx llr).node' lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
h✝ : lrs < ratio * lls
⊢ lls + (lrs + 1) = lls + lrs + 1
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case neg
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : ((node lls lll llx llr).node' lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
h✝ : ¬lrs < ratio * lls
⊢ lls + lrl.size + 1 + (lrr.size + 1) = lls + (lrl.size + lrr.size + 1) + 1

Tactic:
;

Goal state after:
case neg
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : ((node lls lll llx llr).node' lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
h✝ : ¬lrs < ratio * lls
⊢ lls + lrl.size + 1 + (lrr.size + 1) = lls + (lrl.size + lrr.size + 1) + 1
Module: Mathlib.Data.Ordmap.Ordset

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
H₂ : 1 ≤ t.size → 1 ≤ rr.size → rr.size ≤ delta * t.size
H₁ : t.size = 0 → rr.size ≤ 1
⊢ 2 * (ll.size + lr.size + 1 + rl.size) ≤ 9 * rr.size + 5 ∨ ll.size + lr.size + 1 + rl.size ≤ 3

Tactic:
;

Goal state after:
case refine_1
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
H₂ : 1 ≤ t.size → 1 ≤ rr.size → rr.size ≤ delta * t.size
H₁ : t.size = 0 → rr.size ≤ 1
⊢ 2 * (ll.size + lr.size + 1 + rl.size) ≤ 9 * rr.size + 5 ∨ ll.size + lr.size + 1 + rl.size ≤ 3
Module: Mathlib.Data.PNat.Xgcd

Goal state before:
pp: case mp
u : XgcdType
wp x y zp ap bp : ℕ
⊢ (wp + 1) * (zp + 1) = wp + zp + wp * zp + 1

Tactic:
;

Goal state after:
case mp
u : XgcdType
wp x y zp ap bp : ℕ
⊢ (wp + 1) * (zp + 1) = wp + zp + wp * zp + 1
Module: Mathlib.Data.PNat.Xgcd

Goal state before:
pp: case mpr
u : XgcdType
wp x y zp ap bp : ℕ
h : wp * zp + zp + wp = x * y
⊢ wp + zp + wp * zp = wp * zp + zp + wp

Tactic:
;

Goal state after:
case mpr
u : XgcdType
wp x y zp ap bp : ℕ
h : wp * zp + zp + wp = x * y
⊢ wp + zp + wp * zp = wp * zp + zp + wp
Module: Mathlib.Data.PNat.Xgcd

Goal state before:
pp: u : XgcdType
⊢ u.flip.IsReduced ↔ u.IsReduced

Tactic:
dsimp [IsReduced, flip]

Goal state after:
u : XgcdType
⊢ u.bp = u.ap ↔ u.ap = u.bp
Module: Mathlib.Data.Ordmap.Ordset

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
H₂ : 1 ≤ t.size → 1 ≤ rr.size → rr.size ≤ delta * t.size
⊢ ll.size + lr.size + rl.size + 1 = 0 → rr.size ≤ 1

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
H₂ : 1 ≤ t.size → 1 ≤ rr.size → rr.size ≤ delta * t.size
⊢ ll.size + lr.size + rl.size + 1 = 0 → rr.size ≤ 1
Module: Mathlib.Data.Ordmap.Ordset

Goal state before:
pp: case inr
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
a✝¹ : 1 ≤ t.size
a✝ : 1 ≤ rr.size
⊢ rr.size ≤ delta * t.size

Tactic:
;

Goal state after:
case inr
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
a✝¹ : 1 ≤ t.size
a✝ : 1 ≤ rr.size
⊢ rr.size ≤ delta * t.size
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case pos.node.node
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rx : α
h : delta * ls < rs
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
rrs : ℕ
rrl : Ordnode α
rrx : α
rrr : Ordnode α
hr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Balanced
sr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Sized
rd : delta ≤ (node rls rll rlx rlr).size + (node rrs rrl rrx rrr).size
⊢ (if rls < ratio * rrs then (node ls ll lx lr).node3L x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)
    else (node ls ll lx lr).node4L x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)) =
    if rls < ratio * rrs then
      node (ls + rs + 1) (node (ls + rls + 1) (node ls ll lx lr) x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
    else
      node (ls + rs + 1) (node (ls + rll.size + 1) (node ls ll lx lr) x rll) rlx
        (node (rlr.size + rrs + 1) rlr rx (node rrs rrl rrx rrr))

Tactic:
;

Goal state after:
case pos.node.node
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rx : α
h : delta * ls < rs
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
rrs : ℕ
rrl : Ordnode α
rrx : α
rrr : Ordnode α
hr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Balanced
sr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Sized
rd : delta ≤ (node rls rll rlx rlr).size + (node rrs rrl rrx rrr).size
⊢ (if rls < ratio * rrs then (node ls ll lx lr).node3L x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)
    else (node ls ll lx lr).node4L x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)) =
    if rls < ratio * rrs then
      node (ls + rs + 1) (node (ls + rls + 1) (node ls ll lx lr) x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
    else
      node (ls + rs + 1) (node (ls + rll.size + 1) (node ls ll lx lr) x rll) rlx
        (node (rlr.size + rrs + 1) rlr rx (node rrs rrl rrx rrr))
Module: Mathlib.Data.PSigma.Order

Goal state before:
pp: case left
ι : Type u_1
α : ι → Type u_2
inst✝⁴ : Preorder ι
inst✝³ : DenselyOrdered ι
inst✝² : ∀ (i : ι), Nonempty (α i)
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), DenselyOrdered (α i)
i : ι
a : α i
j : ι
b : α j
h : i < j
k : ι
hi : i < k
hj : k < j
c : α k
⊢ ∃ a_1, ⟨i, a⟩ < a_1 ∧ a_1 < ⟨j, b⟩

Tactic:
exact ⟨⟨k, c⟩, left _ _ hi, left _ _ hj⟩

Goal state after:
Module: Mathlib.Data.Ordmap.Ordset

Goal state before:
pp: case inr
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
a✝¹ : 1 ≤ t.size
a✝ : 1 ≤ rr.size
⊢ rr.size ≤ delta * (ll.size + lr.size + 1 + rl.size)

Tactic:
;

Goal state after:
case inr
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
a✝¹ : 1 ≤ t.size
a✝ : 1 ≤ rr.size
⊢ rr.size ≤ delta * (ll.size + lr.size + 1 + rl.size)
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case pos
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rx : α
h : delta * ls < rs
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
rrs : ℕ
rrl : Ordnode α
rrx : α
rrr : Ordnode α
hr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Balanced
sr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Sized
rd : delta ≤ (node rls rll rlx rlr).size + (node rrs rrl rrx rrr).size
h✝ : rls < ratio * rrs
⊢ ls + rls + 1 + rrs = ls + (rls + rrs + 1)

Tactic:
;

Goal state after:
case pos
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rx : α
h : delta * ls < rs
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
rrs : ℕ
rrl : Ordnode α
rrx : α
rrr : Ordnode α
hr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Balanced
sr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Sized
rd : delta ≤ (node rls rll rlx rlr).size + (node rrs rrl rrx rrr).size
h✝ : rls < ratio * rrs
⊢ ls + rls + 1 + rrs = ls + (rls + rrs + 1)
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case neg
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rx : α
h : delta * ls < rs
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
rrs : ℕ
rrl : Ordnode α
rrx : α
rrr : Ordnode α
hr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Balanced
sr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Sized
rd : delta ≤ (node rls rll rlx rlr).size + (node rrs rrl rrx rrr).size
h✝ : ¬rls < ratio * rrs
⊢ ls + rll.size + 1 + (rlr.size + rrs + 1) = ls + (rll.size + rlr.size + 1 + rrs + 1)

Tactic:
;

Goal state after:
case neg
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rx : α
h : delta * ls < rs
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
rrs : ℕ
rrl : Ordnode α
rrx : α
rrr : Ordnode α
hr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Balanced
sr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Sized
rd : delta ≤ (node rls rll rlx rlr).size + (node rrs rrl rrx rrr).size
h✝ : ¬rls < ratio * rrs
⊢ ls + rll.size + 1 + (rlr.size + rrs + 1) = ls + (rll.size + rlr.size + 1 + rrs + 1)
Module: Mathlib.Data.Ordmap.Ordset

Goal state before:
pp: case inr
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ 3 * rl.size
a✝¹ : 1 ≤ t.size
a✝ : 1 ≤ rr.size
⊢ rr.size ≤ 3 * (ll.size + lr.size + 1 + rl.size)

Tactic:
;

Goal state after:
case inr
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ 3 * rl.size
a✝¹ : 1 ≤ t.size
a✝ : 1 ≤ rr.size
⊢ rr.size ≤ 3 * (ll.size + lr.size + 1 + rl.size)
Module: Mathlib.Data.PSigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : Preorder ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), NoMaxOrder (α i)
⊢ ∀ (a : Σₗ' (i : ι), α i), ∃ b, a < b

Tactic:
by
    rintro ⟨i, a⟩
    obtain ⟨b, h⟩ := exists_gt a
    exact ⟨⟨i, b⟩, right _ h⟩

Goal state after:
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case pos.node.node
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
ld : delta ≤ (node lls lll llx llr).size + (node lrs lrl lrx lrr).size
⊢ (if lrs < ratio * lls then (node lls lll llx llr).node3R lx (node lrs lrl lrx lrr) x (node rs rl rx rr)
    else (node lls lll llx llr).node4R lx (node lrs lrl lrx lrr) x (node rs rl rx rr)) =
    if lrs < ratio * lls then
      node (ls + rs + 1) (node lls lll llx llr) lx (node (lrs + rs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
    else
      node (ls + rs + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
        (node (lrr.size + rs + 1) lrr x (node rs rl rx rr))

Tactic:
;

Goal state after:
case pos.node.node
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
ld : delta ≤ (node lls lll llx llr).size + (node lrs lrl lrx lrr).size
⊢ (if lrs < ratio * lls then (node lls lll llx llr).node3R lx (node lrs lrl lrx lrr) x (node rs rl rx rr)
    else (node lls lll llx llr).node4R lx (node lrs lrl lrx lrr) x (node rs rl rx rr)) =
    if lrs < ratio * lls then
      node (ls + rs + 1) (node lls lll llx llr) lx (node (lrs + rs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
    else
      node (ls + rs + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
        (node (lrr.size + rs + 1) lrr x (node rs rl rx rr))
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case pos
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
ld : delta ≤ (node lls lll llx llr).size + (node lrs lrl lrx lrr).size
h✝ : lrs < ratio * lls
⊢ lls + (lrs + rs + 1) = lls + lrs + 1 + rs

Tactic:
;

Goal state after:
case pos
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
ld : delta ≤ (node lls lll llx llr).size + (node lrs lrl lrx lrr).size
h✝ : lrs < ratio * lls
⊢ lls + (lrs + rs + 1) = lls + lrs + 1 + rs
Module: Mathlib.Data.QPF.Multivariate.Basic

Goal state before:
pp: case h.left
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
p : ⦃i : Fin2 n⦄ → α i → Prop
x : F α
y : F fun i ↦ Subtype p
hy : (fun i ↦ Subtype.val) <$$> y = x
a : (P F).A
f : (P F).B a ⟹ fun i ↦ Subtype p
h : repr y = ⟨a, f⟩
⊢ abs ((fun i ↦ Subtype.val) <$$> ⟨a, f⟩) = abs ⟨a, fun i j ↦ ↑(f i j)⟩

Tactic:
;

Goal state after:
case h.left
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
p : ⦃i : Fin2 n⦄ → α i → Prop
x : F α
y : F fun i ↦ Subtype p
hy : (fun i ↦ Subtype.val) <$$> y = x
a : (P F).A
f : (P F).B a ⟹ fun i ↦ Subtype p
h : repr y = ⟨a, f⟩
⊢ abs ((fun i ↦ Subtype.val) <$$> ⟨a, f⟩) = abs ⟨a, fun i j ↦ ↑(f i j)⟩
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case neg
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
ld : delta ≤ (node lls lll llx llr).size + (node lrs lrl lrx lrr).size
h✝ : ¬lrs < ratio * lls
⊢ lls + lrl.size + 1 + (lrr.size + rs + 1) = lls + (lrl.size + lrr.size + 1) + 1 + rs

Tactic:
;

Goal state after:
case neg
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Balanced
sl : (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)).Sized
ld : delta ≤ (node lls lll llx llr).size + (node lrs lrl lrx lrr).size
h✝ : ¬lrs < ratio * lls
⊢ lls + lrl.size + 1 + (lrr.size + rs + 1) = lls + (lrl.size + lrr.size + 1) + 1 + rs
Module: Mathlib.Data.QPF.Multivariate.Basic

Goal state before:
pp: case h
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
p : ⦃i : Fin2 n⦄ → α i → Prop
x : F α
a : (P F).A
f : (P F).B a ⟹ α
h₀ : x = abs ⟨a, f⟩
h₁ : ∀ (i : Fin2 n) (j : (P F).B a i), p (f i j)
⊢ abs ((fun i ↦ Subtype.val) <$$> ⟨a, fun i j ↦ ⟨f i j, ⋯⟩⟩) = abs ⟨a, f⟩

Tactic:
;

Goal state after:
case h
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
p : ⦃i : Fin2 n⦄ → α i → Prop
x : F α
a : (P F).A
f : (P F).B a ⟹ α
h₀ : x = abs ⟨a, f⟩
h₁ : ∀ (i : Fin2 n) (j : (P F).B a i), p (f i j)
⊢ abs ((fun i ↦ Subtype.val) <$$> ⟨a, fun i j ↦ ⟨f i j, ⋯⟩⟩) = abs ⟨a, f⟩
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case neg
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : ¬delta * rs < ls
⊢ (node ls ll lx lr).node' x (node rs rl rx rr) = node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)

Tactic:
· simp [node']

Goal state after:
Module: Mathlib.Data.QPF.Multivariate.Basic

Goal state before:
pp: case h.left
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
r : ⦃i : Fin2 n⦄ → α i → α i → Prop
x y : F α
u : F fun i ↦ { p // r p.1 p.2 }
xeq : (fun i t ↦ (↑t).1) <$$> u = x
yeq : (fun i t ↦ (↑t).2) <$$> u = y
a : (P F).A
f : (P F).B a ⟹ fun i ↦ { p // r p.1 p.2 }
h : repr u = ⟨a, f⟩
⊢ abs ((fun i t ↦ (↑t).1) <$$> ⟨a, f⟩) = abs ⟨a, fun i j ↦ (↑(f i j)).1⟩

Tactic:
;

Goal state after:
case h.left
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
r : ⦃i : Fin2 n⦄ → α i → α i → Prop
x y : F α
u : F fun i ↦ { p // r p.1 p.2 }
xeq : (fun i t ↦ (↑t).1) <$$> u = x
yeq : (fun i t ↦ (↑t).2) <$$> u = y
a : (P F).A
f : (P F).B a ⟹ fun i ↦ { p // r p.1 p.2 }
h : repr u = ⟨a, f⟩
⊢ abs ((fun i t ↦ (↑t).1) <$$> ⟨a, f⟩) = abs ⟨a, fun i j ↦ (↑(f i j)).1⟩
Module: Mathlib.Data.QPF.Multivariate.Basic

Goal state before:
pp: case h.right.left
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
r : ⦃i : Fin2 n⦄ → α i → α i → Prop
x y : F α
u : F fun i ↦ { p // r p.1 p.2 }
xeq : (fun i t ↦ (↑t).1) <$$> u = x
yeq : (fun i t ↦ (↑t).2) <$$> u = y
a : (P F).A
f : (P F).B a ⟹ fun i ↦ { p // r p.1 p.2 }
h : repr u = ⟨a, f⟩
⊢ abs ((fun i t ↦ (↑t).2) <$$> ⟨a, f⟩) = abs ⟨a, fun i j ↦ (↑(f i j)).2⟩

Tactic:
;

Goal state after:
case h.right.left
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
r : ⦃i : Fin2 n⦄ → α i → α i → Prop
x y : F α
u : F fun i ↦ { p // r p.1 p.2 }
xeq : (fun i t ↦ (↑t).1) <$$> u = x
yeq : (fun i t ↦ (↑t).2) <$$> u = y
a : (P F).A
f : (P F).B a ⟹ fun i ↦ { p // r p.1 p.2 }
h : repr u = ⟨a, f⟩
⊢ abs ((fun i t ↦ (↑t).2) <$$> ⟨a, f⟩) = abs ⟨a, fun i j ↦ (↑(f i j)).2⟩
Module: Mathlib.Data.Ordmap.Invariants

Goal state before:
pp: case node.nil.refl.refl
α : Type u_1
x : α
rs : ℕ
rx : α
sl : nil.Sized
sr : (node rs nil rx nil).Sized
H1 : nil.size = 0 → (node rs nil rx nil).size ≤ 1
H2 : 1 ≤ nil.size → 1 ≤ (node rs nil rx nil).size → (node rs nil rx nil).size ≤ delta * nil.size
this : nil.size = 0 ∧ nil.size = 0
⊢ nil.balanceL x (nil.node' rx nil) = nil.balance x (nil.node' rx nil)

Tactic:
;

Goal state after:
case node.nil.refl.refl
α : Type u_1
x : α
rs : ℕ
rx : α
sl : nil.Sized
sr : (node rs nil rx nil).Sized
H1 : nil.size = 0 → (node rs nil rx nil).size ≤ 1
H2 : 1 ≤ nil.size → 1 ≤ (node rs nil rx nil).size → (node rs nil rx nil).size ≤ delta * nil.size
this : nil.size = 0 ∧ nil.size = 0
⊢ nil.balanceL x (nil.node' rx nil) = nil.balance x (nil.node' rx nil)