feat(builtin/num): primitive recursion theorem
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
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Leonardo de Moura
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f28c56b188
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2 changed files with 114 additions and 8 deletions
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@ -67,13 +67,15 @@ theorem succ_inj {a b : num} : succ a = succ b → a = b
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show a = b,
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from rep_inj rep_eq
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theorem succ_nz (a : num) : succ a ≠ zero
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theorem succ_nz (a : num) : ¬ (succ a = zero)
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:= assume R : succ a = zero,
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have Heq1 : S (rep a) = Z,
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from abst_inj inhab (succ_pred a) zero_pred R,
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show false,
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from absurd Heq1 (S_ne_Z (rep a))
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add_rewrite num::succ_nz
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theorem induction {P : num → Bool} (H1 : P zero) (H2 : ∀ n, P n → P (succ n)) : ∀ a, P a
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:= take a,
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let Q := λ x, N x ∧ P (abst x inhab)
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@ -134,6 +136,8 @@ theorem lt_nrefl (n : num) : ¬ (n < n)
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(assume N : n < n,
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lt_elim N (λ P Pred Pn nPn, absurd Pn nPn))
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add_rewrite num::lt_nrefl
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theorem lt_succ {m n : num} : succ m < n → m < n
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:= assume H : succ m < n,
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lt_elim H
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@ -235,6 +239,8 @@ theorem zero_lt_succ_n {n : num} : zero < succ n
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(λ (n : num) (iH : zero < succ n),
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lt_to_lt_succ iH)
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add_rewrite num::zero_lt_succ_n
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theorem lt_succ_to_disj {m n : num} : m < succ n → m = n ∨ m < n
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:= assume H : m < succ n,
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lt_elim H
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@ -290,7 +296,7 @@ definition simp_rec_fun {A : (Type U)} (x : A) (f : A → A) (n : num) : num →
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definition simp_rec {A : (Type U)} (x : A) (f : A → A) (n : num) : A
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:= simp_rec_fun x f (succ n) n
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theorem simp_rec_lemma1 {A : (Type U)} (x : A) (f : A → A) (n : num)
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theorem simp_rec_def {A : (Type U)} (x : A) (f : A → A) (n : num)
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: (∃ fn, simp_rec_rel fn x f n)
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↔
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(simp_rec_fun x f n zero = x ∧
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@ -309,7 +315,7 @@ theorem simp_rec_lemma1 {A : (Type U)} (x : A) (f : A → A) (n : num)
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show (∃ fn, simp_rec_rel fn x f n),
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from exists_intro (simp_rec_fun x f n) H1)
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theorem simp_rec_lemma2 {A : (Type U)} (x : A) (f : A → A) (n : num)
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theorem simp_rec_ex {A : (Type U)} (x : A) (f : A → A) (n : num)
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: ∃ fn, simp_rec_rel fn x f n
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:= induction_on n
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(let fn : num → A := λ n, x in
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@ -372,9 +378,109 @@ theorem simp_rec_lemma2 {A : (Type U)} (x : A) (f : A → A) (n : num)
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show ∃ fn, simp_rec_rel fn x f (succ n),
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from exists_intro fn1 (and_intro f1z f1s))
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theorem simp_rec_lemma1 {A : (Type U)} (x : A) (f : A → A) (n : num)
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: simp_rec_fun x f n zero = x ∧
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∀ m, m < n → simp_rec_fun x f n (succ m) = f (simp_rec_fun x f n m)
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:= (simp_rec_def x f n) ◂ (simp_rec_ex x f n)
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theorem simp_rec_lemma2 {A : (Type U)} (x : A) (f : A → A) (n m1 m2 : num)
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: n < m1 → n < m2 → simp_rec_fun x f m1 n = simp_rec_fun x f m2 n
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:= induction_on n
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(assume H1 H2,
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calc simp_rec_fun x f m1 zero = x : and_eliml (simp_rec_lemma1 x f m1)
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... = simp_rec_fun x f m2 zero : symm (and_eliml (simp_rec_lemma1 x f m2)))
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(λ (n : num) (iH : n < m1 → n < m2 → simp_rec_fun x f m1 n = simp_rec_fun x f m2 n),
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assume (Hs1 : succ n < m1) (Hs2 : succ n < m2),
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have H1 : n < m1,
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from lt_succ Hs1,
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have H2 : n < m2,
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from lt_succ Hs2,
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have Heq1 : simp_rec_fun x f m1 (succ n) = f (simp_rec_fun x f m1 n),
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from and_elimr (simp_rec_lemma1 x f m1) n H1,
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have Heq2 : simp_rec_fun x f m1 n = simp_rec_fun x f m2 n,
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from iH H1 H2,
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have Heq3 : simp_rec_fun x f m2 (succ n) = f (simp_rec_fun x f m2 n),
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from and_elimr (simp_rec_lemma1 x f m2) n H2,
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calc simp_rec_fun x f m1 (succ n) = f (simp_rec_fun x f m1 n) : Heq1
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... = f (simp_rec_fun x f m2 n) : congr2 f Heq2
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... = simp_rec_fun x f m2 (succ n) : symm Heq3)
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theorem simp_rec_thm {A : (Type U)} (x : A) (f : A → A)
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: simp_rec x f zero = x ∧
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∀ m, simp_rec x f (succ m) = f (simp_rec x f m)
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:= have Heqz : simp_rec_fun x f (succ zero) zero = x,
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from and_eliml (simp_rec_lemma1 x f (succ zero)),
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have Hz : simp_rec x f zero = x,
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from calc simp_rec x f zero = simp_rec_fun x f (succ zero) zero : refl _
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... = x : Heqz,
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have Hs : ∀ m, simp_rec x f (succ m) = f (simp_rec x f m),
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from take m,
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have Hlt1 : m < succ (succ m),
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from lt_to_lt_succ (n_lt_succ_n m),
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have Hlt2 : m < succ m,
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from n_lt_succ_n m,
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have Heq1 : simp_rec_fun x f (succ (succ m)) (succ m) = f (simp_rec_fun x f (succ (succ m)) m),
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from and_elimr (simp_rec_lemma1 x f (succ (succ m))) m Hlt1,
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have Heq2 : simp_rec_fun x f (succ (succ m)) m = simp_rec_fun x f (succ m) m,
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from simp_rec_lemma2 x f m (succ (succ m)) (succ m) Hlt1 Hlt2,
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calc simp_rec x f (succ m) = simp_rec_fun x f (succ (succ m)) (succ m) : refl _
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... = f (simp_rec_fun x f (succ (succ m)) m) : Heq1
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... = f (simp_rec_fun x f (succ m) m) : { Heq2 }
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... = f (simp_rec x f m) : refl _,
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show simp_rec x f zero = x ∧ ∀ m, simp_rec x f (succ m) = f (simp_rec x f m),
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from and_intro Hz Hs
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definition pre (m : num) := if m = zero then zero else ε inhab (λ n, succ n = m)
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set_option simplifier::unfold true
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theorem pre_zero : pre zero = zero
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:= by simp
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theorem pre_succ (m : num) : pre (succ m) = m
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:= have Heq : (λ n, succ n = succ m) = (λ n, n = m),
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from funext (λ n, iff_intro (assume Hl, succ_inj Hl)
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(assume Hr, congr2 succ Hr)),
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calc pre (succ m) = ε inhab (λ n, succ n = succ m) : by simp
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... = ε inhab (λ n, n = m) : { Heq }
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... = m : eps_singleton inhab m
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definition prim_rec_fun {A : (Type U)} (x : A) (f : A → num → A)
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:= simp_rec (λ n : num, x) (λ fn n, f (fn (pre n)) n)
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definition prim_rec {A : (Type U)} (x : A) (f : A → num → A) (m : num)
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:= prim_rec_fun x f m (pre m)
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theorem prim_rec_thm {A : (Type U)} (x : A) (f : A → num → A)
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: prim_rec x f zero = x ∧
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∀ m, prim_rec x f (succ m) = f (prim_rec x f m) m
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:= let faux := λ fn n, f (fn (pre n)) n in
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have Hz : prim_rec x f zero = x,
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from have Heq1 : simp_rec (λ n, x) faux zero = (λ n : num, x),
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from and_eliml (simp_rec_thm (λ n, x) faux),
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calc prim_rec x f zero = prim_rec_fun x f zero (pre zero) : refl _
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... = prim_rec_fun x f zero zero : { pre_zero }
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... = simp_rec (λ n, x) faux zero zero : refl _
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... = x : congr1 zero Heq1,
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have Hs : ∀ m, prim_rec x f (succ m) = f (prim_rec x f m) m,
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from take m,
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have Heq1 : pre (succ m) = m,
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from pre_succ m,
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have Heq2 : simp_rec (λ n, x) faux (succ m) = faux (simp_rec (λ n, x) faux m),
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from and_elimr (simp_rec_thm (λ n, x) faux) m,
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calc prim_rec x f (succ m) = prim_rec_fun x f (succ m) (pre (succ m)) : refl _
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... = prim_rec_fun x f (succ m) m : congr2 (prim_rec_fun x f (succ m)) Heq1
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... = simp_rec (λ n, x) faux (succ m) m : refl _
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... = faux (simp_rec (λ n, x) faux m) m : congr1 m Heq2
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... = f (prim_rec x f m) m : refl _,
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show prim_rec x f zero = x ∧ ∀ m, prim_rec x f (succ m) = f (prim_rec x f m) m,
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from and_intro Hz Hs
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set_opaque simp_rec_rel true
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set_opaque simp_rec_fun true
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set_opaque simp_rec true
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set_opaque prim_rec_fun true
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set_opaque prim_rec true
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end
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definition num := num::num
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