| import OAI.Computability.RationalHitting.TaylorComputation |
|
|
| namespace OAI |
|
|
| namespace RationalHitting.StackLanguage |
| open scoped BigOperators |
| namespace Runtime |
|
|
| def signed (q z i j : ℕ) : ℤ := |
| if (z+j)%q=i then (-1)^((z+j)/q) else 0 |
|
|
| def liftTerm (m N q : ℕ) (a t p r : ℤ) (u v i j x y k : ℕ) : ℤ := |
| if u=(v+y)%m then |
| padded m N a t k ((k+(m-y)%m)%m)* |
| signed q (((m-(x*k)%m)%m+(x*u)%m)%m) i j*p^x*r^y |
| else 0 |
|
|
| def lift (m N q : ℕ) (a t p r : ℤ) (u v i j : ℕ) : ℤ := |
| ((List.range m).map (fun x => |
| ((List.range m).map (fun y => |
| ((List.range m).map (fun k => liftTerm m N q a t p r u v i j x y k)).sum)).sum)).sum |
|
|
| def entry (m N q : ℕ) (a t p r : ℤ) (I J : ℕ) : ℤ := |
| lift m N q a t p r (I/q) (J/q) (I%q) (J%q) |
|
|
| abbrev LiftArgs := (Unary×Unary×Unary)×(ℤ×ℤ×ℤ×ℤ)×(Unary×Unary×Unary×Unary) |
| abbrev LiftTermArgs := ((LiftArgs×Unary)×Unary)×Unary |
|
|
| @[fun_prop] theorem signed_poly : Poly (fun a : (Unary×Unary)×(Unary×Unary) => |
| signed a.1.1.val a.1.2.val a.2.1.val a.2.2.val) := by |
| have hc : Poly (fun a : (Unary×Unary)×(Unary×Unary) => |
| Unary.eqb ((a.1.2+a.2.2)%a.1.1) a.2.1) := by fun_prop |
| have hf : Poly (fun a : (Unary×Unary)×(Unary×Unary) => |
| Unary.powInt (-1) ((a.1.2+a.2.2)/a.1.1)) := by fun_prop |
| exact (hc.cond hf (Poly.const 0)).of_eq (by intro a;simp only [Unary.eqb,decide_eq_true_eq];rfl) |
|
|
| @[fun_prop] theorem liftTerm_poly : Poly (fun a : LiftTermArgs => |
| liftTerm a.1.1.1.1.1.val a.1.1.1.1.2.1.val a.1.1.1.1.2.2.val |
| a.1.1.1.2.1.1 a.1.1.1.2.1.2.1 a.1.1.1.2.1.2.2.1 a.1.1.1.2.1.2.2.2 |
| a.1.1.1.2.2.1.val a.1.1.1.2.2.2.1.val a.1.1.1.2.2.2.2.1.val a.1.1.1.2.2.2.2.2.val |
| a.1.1.2.val a.1.2.val a.2.val) := by |
| let pars (a : LiftTermArgs) : Unary×TaylorArgs := |
| (a.1.1.1.1.2.1,a.1.1.1.1.1,a.1.1.1.2.1.1,a.1.1.1.2.1.2.1,a.2, |
| (a.2+(a.1.1.1.1.1-a.1.2)%a.1.1.1.1.1)%a.1.1.1.1.1) |
| have hpars : Poly pars := by unfold pars;fun_prop |
| have ht := padded_poly.comp hpars |
| let sp (a : LiftTermArgs) : (Unary×Unary)×(Unary×Unary) := |
| ((a.1.1.1.1.2.2,((a.1.1.1.1.1-(a.1.1.2*a.2)%a.1.1.1.1.1)%a.1.1.1.1.1+ |
| (a.1.1.2*a.1.1.1.2.2.1)%a.1.1.1.1.1)%a.1.1.1.1.1), |
| (a.1.1.1.2.2.2.2.1,a.1.1.1.2.2.2.2.2)) |
| have hsp : Poly sp := by unfold sp;fun_prop |
| have hs := signed_poly.comp hsp |
| have hp : Poly (fun a : LiftTermArgs => Unary.powInt a.1.1.1.2.1.2.2.1 a.1.1.2) := by fun_prop |
| have hr : Poly (fun a : LiftTermArgs => Unary.powInt a.1.1.1.2.1.2.2.2 a.1.2) := by fun_prop |
| have hv := Poly.intMul.comp ((Poly.intMul.comp ((Poly.intMul.comp (ht.pair hs)).pair hp)).pair hr) |
| have hc : Poly (fun a : LiftTermArgs => |
| Unary.eqb a.1.1.1.2.2.1 ((a.1.1.1.2.2.2.1+a.1.2)%a.1.1.1.1.1)) := by fun_prop |
| exact (hc.cond hv (Poly.const 0)).of_eq (by intro a;simp only [Unary.eqb,decide_eq_true_eq];rfl) |
|
|
| @[fun_prop] theorem lift_poly : Poly (fun a : LiftArgs => |
| lift a.1.1.val a.1.2.1.val a.1.2.2.val |
| a.2.1.1 a.2.1.2.1 a.2.1.2.2.1 a.2.1.2.2.2 |
| a.2.2.1.val a.2.2.2.1.val a.2.2.2.2.1.val a.2.2.2.2.2.val) := by |
| have h1 := Poly.sumUp (show Poly (fun a : (LiftArgs×Unary)×Unary => a.1.1.1.1) by fun_prop) liftTerm_poly |
| have h2 := Poly.sumUp (show Poly (fun a : LiftArgs×Unary => a.1.1.1) by fun_prop) h1 |
| exact Poly.sumUp (show Poly (fun a : LiftArgs => a.1.1) by fun_prop) h2 |
|
|
| abbrev EntryArgs := (Unary×Unary×Unary)×(ℤ×ℤ×ℤ×ℤ)×(Unary×Unary) |
| @[fun_prop] theorem entry_poly : Poly (fun a : EntryArgs => |
| entry a.1.1.val a.1.2.1.val a.1.2.2.val |
| a.2.1.1 a.2.1.2.1 a.2.1.2.2.1 a.2.1.2.2.2 a.2.2.1.val a.2.2.2.val) := by |
| exact lift_poly.comp (show Poly (fun a : EntryArgs => |
| (a.1,a.2.1,(a.2.2.1/a.1.2.2,a.2.2.2/a.1.2.2,a.2.2.1%a.1.2.2,a.2.2.2%a.1.2.2))) by fun_prop) |
|
|
| end Runtime |
| end RationalHitting.StackLanguage |
|
|
|
|
| namespace RationalHitting.StackLanguage |
| open scoped BigOperators |
|
|
| lemma range_sum_zmod_val {A : Type} [AddCommMonoid A] (m : ℕ) [NeZero m] (f : ℕ → A) : |
| ((List.range m).map f).sum=∑ z : ZMod m, f z.val := by |
| rw [range_sum_fin] |
| exact Fintype.sum_equiv (ZMod.finEquiv m).toEquiv _ _ (fun i => by |
| change f i.val=f ((ZMod.finEquiv m) i).val |
| rw [finEquiv_val]) |
|
|
| namespace Runtime |
| lemma signed_eq (e : ℕ) (z : ZMod (CyclotomicRing.dimension e)) |
| (i j : Fin (CyclotomicRing.degree e)) : |
| signed (CyclotomicRing.degree e) z.val i.val j.val = CyclotomicRing.signedEntry e z i j := rfl |
|
|
| lemma integerTest_apply (N e : ℕ) (a t : ℤ) (i j : ZMod (CyclotomicRing.dimension e)) : |
| Generator.integerTest N e a t i j= |
| padded (CyclotomicRing.dimension e) N a t i.val j.val := by |
| change IntegralTest.padded _ _ _ _ ((ZMod.finEquiv _).symm i) ((ZMod.finEquiv _).symm j)=_ |
| rw [←padded_eq,finEquiv_symm_val,finEquiv_symm_val] |
|
|
| lemma liftTerm_eq (N e : ℕ) (a t p r : ℤ) |
| (u v x y k : ZMod (CyclotomicRing.dimension e)) (i j : Fin (CyclotomicRing.degree e)) : |
| liftTerm (CyclotomicRing.dimension e) N (CyclotomicRing.degree e) a t p r |
| u.val v.val i.val j.val x.val y.val k.val = |
| if u=v+y then Generator.integerTest N e a t k (k-y)* |
| CyclotomicRing.signedEntry e (-x*k+x*u) i j*p^x.val*r^y.val else 0 := by |
| have hz : (-x*k+x*u).val= |
| (((CyclotomicRing.dimension e-(x.val*k.val)%CyclotomicRing.dimension e)%CyclotomicRing.dimension e+ |
| (x.val*u.val)%CyclotomicRing.dimension e)%CyclotomicRing.dimension e) := by |
| rw [neg_mul,ZMod.val_add,ZMod.neg_val',ZMod.val_mul,ZMod.val_mul] |
| unfold liftTerm |
| simp only [←ZMod.val_add, (ZMod.val_injective _).eq_iff,←hz,←zmod_val_sub,signed_eq,←integerTest_apply] |
| |
| lemma lift_eq (N e : ℕ) (a t p r : ℤ) |
| (u v : ZMod (CyclotomicRing.dimension e)) (i j : Fin (CyclotomicRing.degree e)) : |
| lift (CyclotomicRing.dimension e) N (CyclotomicRing.degree e) a t p r |
| u.val v.val i.val j.val= |
| IntegralLift.entry e (Generator.integerTest N e a t) p r u v i j := by |
| unfold lift IntegralLift.entry |
| rw [range_sum_zmod_val] |
| apply Finset.sum_congr rfl |
| intro x _ |
| rw [range_sum_zmod_val] |
| apply Finset.sum_congr rfl |
| intro y _ |
| rw [range_sum_zmod_val] |
| apply Finset.sum_congr rfl |
| intro k _ |
| exact liftTerm_eq N e a t p r u v x y k i j |
| |
| lemma entry_eq (n N e : ℕ) (x : Fin 3 → ℕ) (a : Fin (n+1)) (i j : Fin (Generator.D e)) : |
| entry (CyclotomicRing.dimension e) N (CyclotomicRing.degree e) |
| a.val (x 0) (x 1) (x 2) i.val j.val = Generator.U n N e x a i j := by |
| have hi := coordinate_vals e i |
| have hj := coordinate_vals e j |
| have hi1 := congrArg Prod.fst hi |
| have hi2 := congrArg Prod.snd hi |
| have hj1 := congrArg Prod.fst hj |
| have hj2 := congrArg Prod.snd hj |
| simp only [] at hi1 hi2 hj1 hj2 |
| unfold entry Generator.U |
| rw [←hi1,←hi2,←hj1,←hj2] |
| exact lift_eq N e a.val (x 0) (x 1) (x 2) _ _ _ _ |
| |
| end Runtime |
| end RationalHitting.StackLanguage |
| |
| end OAI |
| |
