agda-spa/lean/Spa/Lattice/Prod.lean

import Spa.Lattice

/-!

# Product Lattice

This file provides a proof that, in addition to being a lattice,
the product of two types $\alpha \times \beta$ forms a `Spa.FiniteHeightLattice`
if both $\alpha$ and $\beta$ have a finite height.

The proof proceeds by "unzipping" a chain:

$$
(a_1, b_1) < (a_1, b_2) < \ldots < (a_n, b_m)
$$

In which, at each step, either an $\alpha$ or $\beta$ element
might ratchet up, into two chains:

$$
\begin{aligned}
a_1 < \ldots < a_n \\
b_1 < \ldots < b_m
\end{aligned}
$$

Because at least one of the two "unzipped" chains grows with
each element of the product chain, the full chain length
can't exceed the sum of the two components. By the definition
of finite height, these two chains are bounded, and therefore,
the product chain is bounded too.

-/

namespace Spa

section Unzip

variable {α β : Type*} [PartialOrder α] [PartialOrder β]

/-- The unzipping lemma: any chain (`LTSeries`) of product
    elements can be decomposed into chains of components,
    whose lengths bound the chain. -/
lemma LTSeries.exists_unzip (c : LTSeries (α × β)) :
    ∃ (c₁ : LTSeries α) (c₂ : LTSeries β),
      c₁.head = c.head.1 ∧ c₁.last = c.last.1 ∧
      c₂.head = c.head.2 ∧ c₂.last = c.last.2 ∧
      c.length ≤ c₁.length + c₂.length := by
  suffices H : ∀ (n : ℕ) (c : LTSeries (α × β)), c.length = n →
      ∃ (c₁ : LTSeries α) (c₂ : LTSeries β),
        c₁.head = c.head.1 ∧ c₁.last = c.last.1 ∧
        c₂.head = c.head.2 ∧ c₂.last = c.last.2 ∧
        c.length ≤ c₁.length + c₂.length from H c.length c rfl
  intro n
  induction n with
  | zero =>
    intro c hn
    refine ⟨RelSeries.singleton _ c.head.1, RelSeries.singleton _ c.head.2,
      rfl, ?_, rfl, ?_, by simp [hn]⟩ <;>
    · have hlast : Fin.last c.length = 0 := by ext; simp [hn]
      simp [RelSeries.last, RelSeries.head, hlast]
  | succ n ih =>
    intro c hn
    have h0 : c.length ≠ 0 := by omega
    haveI : NeZero c.length := ⟨h0⟩
    obtain ⟨c₁, c₂, hh₁, hl₁, hh₂, hl₂, hlen⟩ :=
      ih (c.tail h0) (by simp [RelSeries.tail_length, hn])
    rw [RelSeries.last_tail] at hl₁ hl₂
    rw [RelSeries.head_tail] at hh₁ hh₂
    rw [RelSeries.tail_length] at hlen
    have hstep : c.head < c 1 := c.strictMono Fin.one_pos'
    obtain ⟨hle1, hle2⟩ := Prod.le_def.mp hstep.le
    rcases eq_or_lt_of_le hle1 with heq1 | hlt1 <;>
      rcases eq_or_lt_of_le hle2 with heq2 | hlt2
    · exact absurd (Prod.ext heq1 heq2) hstep.ne
    · refine ⟨c₁, c₂.cons c.head.2 (hh₂ ▸ hlt2),
        hh₁.trans heq1.symm, hl₁, RelSeries.head_cons .., by
          rw [RelSeries.last_cons]; exact hl₂, by
          simp only [RelSeries.cons_length]; omega⟩
    · refine ⟨c₁.cons c.head.1 (hh₁ ▸ hlt1), c₂,
        RelSeries.head_cons .., by
          rw [RelSeries.last_cons]; exact hl₁,
        hh₂.trans heq2.symm, hl₂, by
          simp only [RelSeries.cons_length]; omega⟩
    · refine ⟨c₁.cons c.head.1 (hh₁ ▸ hlt1), c₂.cons c.head.2 (hh₂ ▸ hlt2),
        RelSeries.head_cons .., by
          rw [RelSeries.last_cons]; exact hl₁,
        RelSeries.head_cons .., by
          rw [RelSeries.last_cons]; exact hl₂, by
          simp only [RelSeries.cons_length]; omega⟩

end Unzip

section FixedHeight

variable {α β : Type*}

/-- The longest possible chain is one in which only one of the components grows
    at a time, making the maximum height of $\alpha \times \beta$ be
    $\text{height}_\alpha + \text{height}_\beta$. -/
instance prod [A : FiniteHeightLattice α] [B : FiniteHeightLattice β] :
    FiniteHeightLattice (α × β) where
  toLattice := inferInstance
  longestChain :=
    RelSeries.smash
      (A.longestChain.map (fun a => (a, (⊥ : β)))
        (fun _ _ h => Prod.mk_lt_mk_iff_left.mpr h))
      (B.longestChain.map (fun b => ((⊤ : α), b))
        (fun _ _ h => Prod.mk_lt_mk_iff_right.mpr h))
      rfl
  chains_bounded := fun c => by
    obtain ⟨c₁, c₂, -, -, -, -, hlen⟩ := LTSeries.exists_unzip c
    have h₁ := A.chains_bounded c₁
    have h₂ := B.chains_bounded c₂
    show c.length ≤ A.longestChain.length + B.longestChain.length
    omega

end FixedHeight

end Spa