56 lines
1.7 KiB
Text
56 lines
1.7 KiB
Text
open nat well_founded decidable prod
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set_option pp.all true
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-- Auxiliary lemma used to justify recursive call
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private definition lt_aux {x y : nat} (H : 0 < y ∧ y ≤ x) : x - y < x :=
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and.rec_on H (λ ypos ylex,
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sub_lt (nat.lt_of_lt_of_le ypos ylex) ypos)
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definition wdiv.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=
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if H : 0 < y ∧ y ≤ x then f (x - y) (lt_aux H) y + 1 else 0
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definition wdiv (x y : nat) :=
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fix lt_wf wdiv.F x y
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theorem wdiv_def (x y : nat) : wdiv x y = if H : 0 < y ∧ y ≤ x then wdiv (x - y) y + 1 else 0 :=
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congr_fun (well_founded.fix_eq lt_wf wdiv.F x) y
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/-
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See comment at fib_wrec.
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example : wdiv 5 2 = 2 :=
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rfl
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example : wdiv 9 3 = 3 :=
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rfl
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-/
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-- There is a little bit of cheating in the definition above.
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-- I avoid the packing/unpacking into tuples.
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-- The actual definitional package would not do that.
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-- It will always pack things.
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definition pair_nat.lt := lex nat.lt nat.lt -- Could also be (lex lt empty_rel)
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definition pair_nat.lt.wf : well_founded pair_nat.lt :=
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prod.lex_wf lt_wf lt_wf
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infixl `≺`:50 := pair_nat.lt
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-- Recursive lemma used to justify recursive call
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definition plt_aux (x y : nat) (H : 0 < y ∧ y ≤ x) : (x - y, y) ≺ (x, y) :=
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lex.left _ _ _ (lt_aux H)
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definition pdiv.F (p₁ : nat × nat) : (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat :=
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prod.cases_on p₁ (λ x y f,
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if H : 0 < y ∧ y ≤ x then f (x - y, y) (plt_aux x y H) + 1 else 0)
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definition pdiv (x y : nat) :=
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fix pair_nat.lt.wf pdiv.F (x, y)
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theorem pdiv_def (x y : nat) : pdiv x y = if H : 0 < y ∧ y ≤ x then pdiv (x - y) y + 1 else 0 :=
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well_founded.fix_eq pair_nat.lt.wf pdiv.F (x, y)
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/-
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See comment at fib_wrec.
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example : pdiv 17 2 = 8 :=
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rfl
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-/
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