import logic data.nat.sub struc.relation data.prod import tools.fake_simplifier open nat relation relation.iff_ops prod open fake_simplifier decidable open eq_ops namespace nat -- A general recursion principle -- ----------------------------- -- -- Data: -- -- dom, codom : Type -- default : codom -- measure : dom → ℕ -- rec_val : dom → (dom → codom) → codom -- -- and a proof -- -- rec_decreasing : ∀m, m ≥ measure x, rec_val x f = rec_val x (restrict f m) -- -- ... which says that the recursive call only depends on f at values with measure less than x, -- in the sense that changing other values to the default value doesn't change the result. -- -- The result is a function f = rec_measure, satisfying -- -- f x = rec_val x f definition restrict {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom) (m : ℕ) (x : dom) := if measure x < m then f x else default theorem restrict_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom) (m : ℕ) (x : dom) (H : measure x < m) : restrict default measure f m x = f x := if_pos H theorem restrict_not_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom) (m : ℕ) (x : dom) (H : ¬ measure x < m) : restrict default measure f m x = default := if_neg H definition rec_measure_aux {dom codom : Type} (default : codom) (measure : dom → ℕ) (rec_val : dom → (dom → codom) → codom) : ℕ → dom → codom := rec (λx, default) (λm g x, if measure x < succ m then rec_val x g else default) definition rec_measure {dom codom : Type} (default : codom) (measure : dom → ℕ) (rec_val : dom → (dom → codom) → codom) (x : dom) : codom := rec_measure_aux default measure rec_val (succ (measure x)) x theorem rec_measure_aux_spec {dom codom : Type} (default : codom) (measure : dom → ℕ) (rec_val : dom → (dom → codom) → codom) (rec_decreasing : ∀g1 g2 x, (∀z, measure z < measure x → g1 z = g2 z) → rec_val x g1 = rec_val x g2) (m : ℕ) : let f' := rec_measure_aux default measure rec_val in let f := rec_measure default measure rec_val in ∀x, f' m x = restrict default measure f m x := let f' := rec_measure_aux default measure rec_val in let f := rec_measure default measure rec_val in case_strong_induction_on m (take x, have H1 : f' 0 x = default, from rfl, have H2 : ¬ measure x < 0, from not_lt_zero, have H3 : restrict default measure f 0 x = default, from if_neg H2, show f' 0 x = restrict default measure f 0 x, from H1 ⬝ H3⁻¹) (take m, assume IH: ∀n, n ≤ m → ∀x, f' n x = restrict default measure f n x, take x : dom, show f' (succ m) x = restrict default measure f (succ m) x, from by_cases -- (measure x < succ m) (assume H1 : measure x < succ m, have H2a : ∀z, measure z < measure x → f' m z = f z, proof take z, assume Hzx : measure z < measure x, calc f' m z = restrict default measure f m z : IH m le_refl z ... = f z : restrict_lt_eq _ _ _ _ _ (lt_le_trans Hzx (lt_succ_imp_le H1)) ∎, have H2 : f' (succ m) x = rec_val x f, proof calc f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl ... = rec_val x (f' m) : if_pos H1 ... = rec_val x f : rec_decreasing (f' m) f x H2a ∎, let m' := measure x in have H3a : ∀z, measure z < m' → f' m' z = f z, proof take z, assume Hzx : measure z < measure x, calc f' m' z = restrict default measure f m' z : IH _ (lt_succ_imp_le H1) _ ... = f z : restrict_lt_eq _ _ _ _ _ Hzx qed, have H3 : restrict default measure f (succ m) x = rec_val x f, proof calc restrict default measure f (succ m) x = f x : if_pos H1 ... = f' (succ m') x : eq.refl _ ... = if measure x < succ m' then rec_val x (f' m') else default : rfl ... = rec_val x (f' m') : if_pos self_lt_succ ... = rec_val x f : rec_decreasing _ _ _ H3a qed, show f' (succ m) x = restrict default measure f (succ m) x, from H2 ⬝ H3⁻¹) (assume H1 : ¬ measure x < succ m, have H2 : f' (succ m) x = default, from calc f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl ... = default : if_neg H1, have H3 : restrict default measure f (succ m) x = default, from if_neg H1, show f' (succ m) x = restrict default measure f (succ m) x, from H2 ⬝ H3⁻¹)) end nat