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feat(builtin/num): primitive recursion theorem

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2014-02-08 22:05:57 -08:00
parent f28c56b188
commit cc4148a98d
2 changed files with 114 additions and 8 deletions
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122
src/builtin/num.lean
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@ -67,13 +67,15 @@ theorem succ_inj {a b : num} : succ a = succ b → a = b
show a = b, show a = b,
from rep_inj rep_eq from rep_inj rep_eq
theorem succ_nz (a : num) : succ a ≠ zero theorem succ_nz (a : num) : ¬ (succ a = zero)
:= assume R : succ a = zero, := assume R : succ a = zero,
have Heq1 : S (rep a) = Z, have Heq1 : S (rep a) = Z,
from abst_inj inhab (succ_pred a) zero_pred R, from abst_inj inhab (succ_pred a) zero_pred R,
show false, show false,
from absurd Heq1 (S_ne_Z (rep a)) from absurd Heq1 (S_ne_Z (rep a))
add_rewrite num::succ_nz
theorem induction {P : num → Bool} (H1 : P zero) (H2 : ∀ n, P n → P (succ n)) : ∀ a, P a theorem induction {P : num → Bool} (H1 : P zero) (H2 : ∀ n, P n → P (succ n)) : ∀ a, P a
:= take a, := take a,
let Q := λ x, N x ∧ P (abst x inhab) let Q := λ x, N x ∧ P (abst x inhab)
@ -134,6 +136,8 @@ theorem lt_nrefl (n : num) : ¬ (n < n)
(assume N : n < n, (assume N : n < n,
lt_elim N (λ P Pred Pn nPn, absurd Pn nPn)) lt_elim N (λ P Pred Pn nPn, absurd Pn nPn))
add_rewrite num::lt_nrefl
theorem lt_succ {m n : num} : succ m < n → m < n theorem lt_succ {m n : num} : succ m < n → m < n
:= assume H : succ m < n, := assume H : succ m < n,
lt_elim H lt_elim H
@ -235,6 +239,8 @@ theorem zero_lt_succ_n {n : num} : zero < succ n
(λ (n : num) (iH : zero < succ n), (λ (n : num) (iH : zero < succ n),
lt_to_lt_succ iH) lt_to_lt_succ iH)
add_rewrite num::zero_lt_succ_n
theorem lt_succ_to_disj {m n : num} : m < succ n → m = n ∨ m < n theorem lt_succ_to_disj {m n : num} : m < succ n → m = n ∨ m < n
:= assume H : m < succ n, := assume H : m < succ n,
lt_elim H lt_elim H
@ -290,11 +296,11 @@ definition simp_rec_fun {A : (Type U)} (x : A) (f : A → A) (n : num) : num →
definition simp_rec {A : (Type U)} (x : A) (f : A → A) (n : num) : A definition simp_rec {A : (Type U)} (x : A) (f : A → A) (n : num) : A
:= simp_rec_fun x f (succ n) n := simp_rec_fun x f (succ n) n
theorem simp_rec_lemma1 {A : (Type U)} (x : A) (f : A → A) (n : num) theorem simp_rec_def {A : (Type U)} (x : A) (f : A → A) (n : num)
: (∃ fn, simp_rec_rel fn x f n) : (∃ fn, simp_rec_rel fn x f n)
↔ ↔
(simp_rec_fun x f n zero = x ∧ (simp_rec_fun x f n zero = x ∧
∀ m, m < n → simp_rec_fun x f n (succ m) = f (simp_rec_fun x f n m)) ∀ m, m < n → simp_rec_fun x f n (succ m) = f (simp_rec_fun x f n m))
:= iff_intro := iff_intro
(assume Hl : (∃ fn, simp_rec_rel fn x f n), (assume Hl : (∃ fn, simp_rec_rel fn x f n),
obtain (fn : num → A) (Hfn : simp_rec_rel fn x f n), obtain (fn : num → A) (Hfn : simp_rec_rel fn x f n),
@ -309,8 +315,8 @@ theorem simp_rec_lemma1 {A : (Type U)} (x : A) (f : A → A) (n : num)
show (∃ fn, simp_rec_rel fn x f n), show (∃ fn, simp_rec_rel fn x f n),
from exists_intro (simp_rec_fun x f n) H1) from exists_intro (simp_rec_fun x f n) H1)
theorem simp_rec_lemma2 {A : (Type U)} (x : A) (f : A → A) (n : num) theorem simp_rec_ex {A : (Type U)} (x : A) (f : A → A) (n : num)
: ∃ fn, simp_rec_rel fn x f n : ∃ fn, simp_rec_rel fn x f n
:= induction_on n := induction_on n
(let fn : num → A := λ n, x in (let fn : num → A := λ n, x in
have fz : fn zero = x, have fz : fn zero = x,
@ -372,9 +378,109 @@ theorem simp_rec_lemma2 {A : (Type U)} (x : A) (f : A → A) (n : num)
show ∃ fn, simp_rec_rel fn x f (succ n), show ∃ fn, simp_rec_rel fn x f (succ n),
from exists_intro fn1 (and_intro f1z f1s)) from exists_intro fn1 (and_intro f1z f1s))
theorem simp_rec_lemma1 {A : (Type U)} (x : A) (f : A → A) (n : num)
: simp_rec_fun x f n zero = x ∧
∀ m, m < n → simp_rec_fun x f n (succ m) = f (simp_rec_fun x f n m)
:= (simp_rec_def x f n) ◂ (simp_rec_ex x f n)
theorem simp_rec_lemma2 {A : (Type U)} (x : A) (f : A → A) (n m1 m2 : num)
: n < m1 → n < m2 → simp_rec_fun x f m1 n = simp_rec_fun x f m2 n
:= induction_on n
(assume H1 H2,
calc simp_rec_fun x f m1 zero = x : and_eliml (simp_rec_lemma1 x f m1)
... = simp_rec_fun x f m2 zero : symm (and_eliml (simp_rec_lemma1 x f m2)))
(λ (n : num) (iH : n < m1 → n < m2 → simp_rec_fun x f m1 n = simp_rec_fun x f m2 n),
assume (Hs1 : succ n < m1) (Hs2 : succ n < m2),
have H1 : n < m1,
from lt_succ Hs1,
have H2 : n < m2,
from lt_succ Hs2,
have Heq1 : simp_rec_fun x f m1 (succ n) = f (simp_rec_fun x f m1 n),
from and_elimr (simp_rec_lemma1 x f m1) n H1,
have Heq2 : simp_rec_fun x f m1 n = simp_rec_fun x f m2 n,
from iH H1 H2,
have Heq3 : simp_rec_fun x f m2 (succ n) = f (simp_rec_fun x f m2 n),
from and_elimr (simp_rec_lemma1 x f m2) n H2,
calc simp_rec_fun x f m1 (succ n) = f (simp_rec_fun x f m1 n) : Heq1
... = f (simp_rec_fun x f m2 n) : congr2 f Heq2
... = simp_rec_fun x f m2 (succ n) : symm Heq3)
theorem simp_rec_thm {A : (Type U)} (x : A) (f : A → A)
: simp_rec x f zero = x ∧
∀ m, simp_rec x f (succ m) = f (simp_rec x f m)
:= have Heqz : simp_rec_fun x f (succ zero) zero = x,
from and_eliml (simp_rec_lemma1 x f (succ zero)),
have Hz : simp_rec x f zero = x,
from calc simp_rec x f zero = simp_rec_fun x f (succ zero) zero : refl _
... = x : Heqz,
have Hs : ∀ m, simp_rec x f (succ m) = f (simp_rec x f m),
from take m,
have Hlt1 : m < succ (succ m),
from lt_to_lt_succ (n_lt_succ_n m),
have Hlt2 : m < succ m,
from n_lt_succ_n m,
have Heq1 : simp_rec_fun x f (succ (succ m)) (succ m) = f (simp_rec_fun x f (succ (succ m)) m),
from and_elimr (simp_rec_lemma1 x f (succ (succ m))) m Hlt1,
have Heq2 : simp_rec_fun x f (succ (succ m)) m = simp_rec_fun x f (succ m) m,
from simp_rec_lemma2 x f m (succ (succ m)) (succ m) Hlt1 Hlt2,
calc simp_rec x f (succ m) = simp_rec_fun x f (succ (succ m)) (succ m) : refl _
... = f (simp_rec_fun x f (succ (succ m)) m) : Heq1
... = f (simp_rec_fun x f (succ m) m) : { Heq2 }
... = f (simp_rec x f m) : refl _,
show simp_rec x f zero = x ∧ ∀ m, simp_rec x f (succ m) = f (simp_rec x f m),
from and_intro Hz Hs
definition pre (m : num) := if m = zero then zero else ε inhab (λ n, succ n = m)
set_option simplifier::unfold true
theorem pre_zero : pre zero = zero
:= by simp
theorem pre_succ (m : num) : pre (succ m) = m
:= have Heq : (λ n, succ n = succ m) = (λ n, n = m),
from funext (λ n, iff_intro (assume Hl, succ_inj Hl)
(assume Hr, congr2 succ Hr)),
calc pre (succ m) = ε inhab (λ n, succ n = succ m) : by simp
... = ε inhab (λ n, n = m) : { Heq }
... = m : eps_singleton inhab m
definition prim_rec_fun {A : (Type U)} (x : A) (f : A → num → A)
:= simp_rec (λ n : num, x) (λ fn n, f (fn (pre n)) n)
definition prim_rec {A : (Type U)} (x : A) (f : A → num → A) (m : num)
:= prim_rec_fun x f m (pre m)
theorem prim_rec_thm {A : (Type U)} (x : A) (f : A → num → A)
: prim_rec x f zero = x ∧
∀ m, prim_rec x f (succ m) = f (prim_rec x f m) m
:= let faux := λ fn n, f (fn (pre n)) n in
have Hz : prim_rec x f zero = x,
from have Heq1 : simp_rec (λ n, x) faux zero = (λ n : num, x),
from and_eliml (simp_rec_thm (λ n, x) faux),
calc prim_rec x f zero = prim_rec_fun x f zero (pre zero) : refl _
... = prim_rec_fun x f zero zero : { pre_zero }
... = simp_rec (λ n, x) faux zero zero : refl _
... = x : congr1 zero Heq1,
have Hs : ∀ m, prim_rec x f (succ m) = f (prim_rec x f m) m,
from take m,
have Heq1 : pre (succ m) = m,
from pre_succ m,
have Heq2 : simp_rec (λ n, x) faux (succ m) = faux (simp_rec (λ n, x) faux m),
from and_elimr (simp_rec_thm (λ n, x) faux) m,
calc prim_rec x f (succ m) = prim_rec_fun x f (succ m) (pre (succ m)) : refl _
... = prim_rec_fun x f (succ m) m : congr2 (prim_rec_fun x f (succ m)) Heq1
... = simp_rec (λ n, x) faux (succ m) m : refl _
... = faux (simp_rec (λ n, x) faux m) m : congr1 m Heq2
... = f (prim_rec x f m) m : refl _,
show prim_rec x f zero = x ∧ ∀ m, prim_rec x f (succ m) = f (prim_rec x f m) m,
from and_intro Hz Hs
set_opaque simp_rec_rel true set_opaque simp_rec_rel true
set_opaque simp_rec_fun true set_opaque simp_rec_fun true
set_opaque simp_rec true set_opaque simp_rec true
set_opaque prim_rec_fun true
set_opaque prim_rec true
end end
definition num := num::num definition num := num::num

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src/builtin/obj/num.olean
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