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namespace OAI
/-!
# Localization of the four body modes
The arithmetic case analysis assumes the stated order inequalities and entry
values, independently of the host-position encoding.
-/
namespace Problem348.ModeLocalization
/-- The four entries of the body pattern, in row-major order. -/
def IsP (a b c d : Bool) : Prop :=
a = true ∧ b = false ∧ c = true ∧ d = true
/-- Within either parity class, taking the parent preserves and reflects order. -/
theorem le_iff_parent_le_of_same_parity {p q : ℕ}
(hparity : p % 2 = q % 2) :
p ≤ q ↔ p / 2 ≤ q / 2 := by
omega
theorem eq_of_parent_eq_of_same_parity {p q : ℕ}
(hparity : p % 2 = q % 2) (hparent : p / 2 = q / 2) : p = q := by
omega
/-- A `V_i^+` body cannot match `P`: its row order forces the zero to be one. -/
theorem vPlus_not_isP (p a b : ℕ) (horder : p / 2 ≤ a) :
¬ IsP true (decide (p / 2 ≤ b)) true (decide (a ≤ b)) := by
simp only [IsP, true_and, decide_eq_false_iff_not, decide_eq_true_eq]
omega
/-- A `V_i^-` body cannot match `P`: its row order forces the last one to be zero. -/
theorem vMinus_not_isP (p a b : ℕ) (horder : a ≤ p / 2) :
¬ IsP true (decide (a < b)) true (decide (p / 2 < b)) := by
simp only [IsP, true_and, decide_eq_false_iff_not, decide_eq_true_eq]
omega
/-- A `W_{i,d}^+` body cannot match `P`, including at the shared leaf level. -/
theorem wPlus_not_isP (p q b : ℕ) (horder : b ≤ q / 2)
(hparity : p % 2 = q % 2) :
¬ IsP (decide (p / 2 ≤ b)) (decide (p ≤ q)) true true := by
simp only [IsP, and_true, decide_eq_false_iff_not, decide_eq_true_eq]
omega
/-- In a minus `W` mode, a body match can occur only on a shared leaf diagonal. -/
theorem wMinus_isP_implies_leaf_diagonal
(h i p q b : ℕ) (horder : q / 2 ≤ b)
(hparity : p % 2 = q % 2)
(hbody : IsP (if i = h then decide (p ≤ q) else decide (p < q))
(decide (p / 2 < b)) true true) :
i = h ∧ p = q := by
by_cases hleaf : i = h
· simp only [hleaf, ite_true, IsP, and_true,
decide_eq_false_iff_not, decide_eq_true_eq] at hbody
constructor
· exact hleaf
· omega
· simp only [hleaf, ite_false, IsP, and_true,
decide_eq_false_iff_not, decide_eq_true_eq] at hbody
omega
/-- Conversely, the diagonal leaf choice with its parent column gives `P`. -/
theorem wMinus_leaf_diagonal_isP (h p : ℕ) :
IsP (if h = h then decide (p ≤ p) else decide (p < p))
(decide (p / 2 < p / 2)) true true := by
simp [IsP]
end Problem348.ModeLocalization
end OAI
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