/-
Port of `Lattice/Prod.agda`.

The component-wise lattice structure on `α × β` is lifted into mathlib
(`Prod.instLattice`), as is decidability of equality. What remains custom is
the fixed-height content:

  unzip                      ↦ LTSeries.exists_unzip
  a,∙-Monotonic/∙,b-Monotonic ↦ Prod.mk_lt_mk_iff_right/left (strict mono of
                                the two injections, used to map the chains)
  fixedHeight (h₁ + h₂)      ↦ FixedHeight.prod
  isFiniteHeightLattice      ↦ instance FiniteHeightLattice (α × β)
-/
import Spa.Lattice

namespace Spa

section Unzip

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

/-- Agda: `unzip` — a chain in the product splits into chains of the
components whose lengths sum to at least the original length. -/
theorem 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
    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 := by
      have h := c.step ⟨0, by omega⟩
      have h1 : (⟨0, by omega⟩ : Fin c.length).succ = 1 := by
        ext; simp [Fin.val_one, Nat.mod_eq_of_lt (by omega : 1 < c.length + 1)]
      rwa [h1] at h
    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*} [Lattice α] [Lattice β]

/-- Agda: `Lattice/Prod.agda`'s `fixedHeight` — the product of lattices of
heights `h₁` and `h₂` has height `h₁ + h₂`. The longest chain climbs the first
component (at `⊥₂`), then the second component (at `⊤₁`). -/
def FixedHeight.prod {h₁ h₂ : ℕ} (fhA : FixedHeight α h₁) (fhB : FixedHeight β h₂) :
    FixedHeight (α × β) (h₁ + h₂) where
  bot := (fhA.bot, fhB.bot)
  top := (fhA.top, fhB.top)
  longestChain :=
    RelSeries.smash
      (fhA.longestChain.map (fun a => (a, fhB.bot))
        (fun _ _ h => Prod.mk_lt_mk_iff_left.mpr h))
      (fhB.longestChain.map (fun b => (fhA.top, b))
        (fun _ _ h => Prod.mk_lt_mk_iff_right.mpr h))
      (by simp [fhA.last_longestChain, fhB.head_longestChain])
  head_longestChain := by simp [fhA.head_longestChain]
  last_longestChain := by simp [fhB.last_longestChain]
  length_longestChain := by
    simp [fhA.length_longestChain, fhB.length_longestChain]
  bounded := fun c => by
    obtain ⟨c₁, c₂, -, -, -, -, hlen⟩ := LTSeries.exists_unzip c
    have h₁ := fhA.bounded c₁
    have h₂ := fhB.bounded c₂
    omega

/-- Agda: `Lattice/Prod.agda`'s `isFiniteHeightLattice`/`finiteHeightLattice`. -/
instance [IA : FiniteHeightLattice α] [IB : FiniteHeightLattice β] :
    FiniteHeightLattice (α × β) where
  height := IA.height + IB.height
  fixedHeight := IA.fixedHeight.prod IB.fixedHeight

end FixedHeight

end Spa
