CAPACITACIÓN Y DESARROLLO TURÍSTICO APLICADO AL

We study application of a type-0 uniformly continuous function to a type-0 real number. We extract a program which takes a type-0 uniformly continuous function f and a type-0 real number x, and computes the application f(x) in type-0 real numbers.

For given w and ds of type W and LSD, the application builds another SDS to output.

In order to illustrate an intuition, we focus on Getand Putnodes of w, so that we regard

w simply as a tree with 1-branching nodes labelled with a signed digit and 3-branching nodes. The application procedure takes a look at the root of w. If it is a 1-branching node, it outputs the label of this node and goes on with the subtree. If it is a 3-branching node, it reads the head of the input SDS, say d, and chooses the left, the middle, or the right branch corresponding to d. The procedure goes on with the chosen subtree and the tail of the input SDS.

3.6.1

Definitions

The abstract theory of the function application is based on the abstract theory of real numbers in Section 3.3 and the abstract theory of uniformly continuous functions in Section 3.5. We define a constant Apply for the abstract function application additionally to the abstract theories from the previous sections.

Definition 3.6.1 (Apply). Recall thatρ andφ are used for abstract types of real numbers and uniformly continuous functions, respectively. Apply is a constant of type ρ→φ→ρ. Apply f x is abbreviated as f(x).

Then, we enrich the axiom to talk about the function application. Axiom 3.6.2.

∀x(Id(x) eqdx), (AppId)

∀f,x,d(f[I]⊆Id →Vad(f(x)) eqd (Outd◦f)(x)), (VaOut)

∀f,x,d(f(Avdx) eqd (f ◦Ind)(x)), (AvIn)

∀f,x,q,k(f[I]⊆Iq,k →f(x)∈Iq,k). (AppSubElem)

The constant Id works as we commonly expect due to (_AppId). The operations Vad

and Outd can replace each other by (VaOut), and the same happens to Avd and Ind by

(AvIn). The relation ⊆can be replaced by ∈ due to (AppSubElem).

The following equality between two real numbers is also used. Axiom 3.6.3.

∀x,d(xeqd Vad(Avdx)). (VaAvIdent)

We use the predicates coI, Read and coWrite which are defined in Definition 3.3.6, Definition 3.5.6 and Definition 3.5.7, respectively.

3.6.2

Proofs

We prove the following proposition in order to extract a program of the function application. Proposition 3.6.4. ∀nc

f,x(coWritef →coIx→coI(f(x))).

Proof. Letf and xbe given, and assume co_{Writef and}co_{Ix. We use}co_I+ _{with a competitor} predicate P = {z| ∃r

f,x(

co_Writef∧d co_Ix∧l_{zeqdf(x))} to prove}co_{I(f(x)). Since the first} premise P(f(x)) is clear from the assumptions, it suffices to prove the costep formula ∀nc

z (P z → zeqd Z∨r∃ry∃dd((coIy∨dP y)∧lzeqd Avd(y))). Let z be given and assume P z.

Unfolding P z, there are some f0 and x0 such thatcoWritef0, coIx0 and zeqdf0(x0) hold. Our goal is zeqd Z∨r_∃r

y∃dd((coIy∨dP y)∧lzeqd Avd(y)). Applying coWrite− to coWritef0,

f0eqd Id∨rReadco_Writef₀ makes two cases. Left case. Assume f₀eqd Id. ApplyingcoI− to

co_Ix

0,x0eqd Z∨r∃ry0∃

d

d0(

co_Iy

0∧lx0eqd Avd0(y0)) makes two subcases. Each side case comes

by (_AppId). Right case. Assume Readco_Writef₀. We also have coIx₀ and zeqdf₀(x₀), hence

Lemma 3.6.5 applies.

The Lemma 3.6.5 below is responsible for determining an output signed digit by recursion on an RW-total tree.

Lemma 3.6.5. Let P :={z|∃r

f,x(coWritef ∧d coIx∧lzeqdf(x))}, then

∀nc

f,x(ReadcoWritef →coIx→f(x) eqd Z∨r∃ry∃

d

d((

co_Iy∨d_{P y)∧}l_{f(x) eqd Av}

d(y))).

Proof. We prove the following formula which is same as the proposition. ∀nc

f (ReadcoWritef → ∀ncx (coIx→f(x) eqd Z∨r∃ry∃dd((coIy∨dP y)∧lf(x) eqd Avd(y)))).

Let f be given and assume Readco_Writef. By using Read− for Readco_Writef, it suffices to

prove the following two step cases ∀c

e∀ncf (f[I]⊆Ie →coWrite(Oute◦f)→Qf)

and

∀nc_f (Readco_Write(f ◦In_L)→Q(f◦In_L)→

ReadcoWrite(f ◦InM)→Q(f ◦InM)→

Readco_Write(f ◦In_R)→Q(f ◦In_R)→Qf),

where Q := {f| ∀nc

x (coIx → f(x) eqd Z∨r ∃ry∃dd((coIy∨d P y)∧lf(x) eqd Avd(y)))}. First

step case. Let e and f be given and assume f[I] ⊆ Ie and coWrite(Oute ◦f). Also let

x be given and assume coIx. We prove the right disjunct of the conclusion. Use (∃r₎+ and (∃d₎+ _{with y = (Out}

e ◦f)(x) and d = e, respectively. The equality is obvious

from (VaOut) and (AvVaIdent). In order to prove P y, we use (∃r)+ with Oute ◦f

and with x, then three conjuncts are trivial. Second step case. Let f be given and assume Readco_Write(f ◦In_e) and ∀_xnc(coIx→ (f ◦In_e)(x) eqd Z∨r∃r_y∃d_d((coIy∨dP y)∧l(f ◦

Ine)(x) eqd Avd(y))) for each e ∈ SD. Also let x be given and assume coIx. We prove

f(x) eqd Z∨r_∃r

y∃dd((

co_Iy∨d_{P y)∧}l_{f(x) eqd Av}

d(y)). Applying coI+ to coIx, there are two

cases from xeqd Z∨r_∃r

y0∃

d

d0(

co_Iy

0∧lxeqd Avd0(y0)). Left case. Assume xeqd Z. Applying

the induction hypothesis for e=M to co_{I(Z), (f◦In}

M)(Z) eqd Z∨r∃ry∃de((coIy∨dP y)∧l(f ◦

InM)(Z) eqd Ave(y)) holds, and moreover (f ◦InM)(Z) eqdf(Z) holds by means of (AvIn) and (_AvZero), hence the goal is implied. Right case. There existy0 andd0 such thatcoIy0 and xeqd Avd0(y0) hold. Applying the induction hypothesis corresponding to d0 to

co_Iy 0, (f◦Ind0)(y0) eqd Z∨

r_∃r

y∃de((coIy∨dP y)∧l(f◦Ind0)(y0) eqd Ave(y)) holds and we are done

because (f◦Ind0)(y0) eqdf(x) holds by the assumption xeqd Avd0(y0) and (AvIn).

3.6.3

Program Extraction

We extract a program from the proof of 3.6.4. Note that Remark 3.5.15 applies in this section as well. Suppose that uis a variable name ranging over W×I and q ranging over SD×I. The result isappwhich is shown in Figure 3.20. Intuitively, this function computes

λw,ds(coRW ×I I hw, dsi λu(Case DW(π0u) of λCase D(π1u) of λ InL U λqInRhπ0q,InL(π1q)i

λr(Case cApplyAux(r(π1u)) of λ(InL U)λv(InRv))))

Figure 3.20: appW→I→I

a trace from the root of the given type-0 function to the above by choosing at Geta branch indicated by the input type-0 real number. The output is an SDS which consists of signed digits which occur at Puton the computed trace.

Assume w and ds of types W and I, respectively, are given. The costep term first destructw. If Dw is InL U, the output is identical to the givends. Otherwise, it results in an RW-total ideal r. We apply cApplyAux to this r and ds, then it determines that the

result SDS is either the empty list or a pair of the head and the tail, i.e., a signed digit and an SDS, respectively.

We extract from the Lemma 3.6.5 a program cApplyAux shown in Figure 3.21. Let f

be a variable name ranging overI→U+ (SD×(I+W×I)). For the inputr and ds, this program computes by reading finite signed digits fromds a finite path in r, so that a signed digit is determined. There are two cases due to the recursion on r. Ifr is Putd w it does not touch ds and outputs InRhd,InRhw, dsii. It makes the corecursion operator in app to produce a stream with the head d and the rest appw ds. Otherwise, r is GetrLrMrR. If

λr(R

I→U+(SD×(I+W×I))

RW

r

λd,w,ds(InRhd,InRhw, dsii)

λ,fL, ,fM, ,fR,ds(Case DIds of

λ (fMds)

λq(Case π0q of λ(fL(π1q))λ (fM(π1q))λ (fR(π1q)))))

Figure 3.21: cApplyAuxRW→I→U+(SD×(I+W×I))

taken to reach the leaf. Otherwise if Dds isInRhd, ds0i, it takes a branch corresponding tod

and then the recursion carries on with ds0.

3.6.4

Experiments

Define a functionf(x) = 2x2−1 and consider its roots ±_√1

2. It is expected that the value

f(±_√1

2) is arbitrarily close to 0. We use app to compute the value of f( 1 √

2).

A type-1 uniformly continuous function for f is defined to be hh, α, ωi whereh a n = 2a2 − 1, α n = n + 1 and ω n = n + 1. Then by means of ucf1to0, we obtain the type-0 representation of hh, α, ωi. On the other hand, the type-0 representation of _√1

2 is computed by cauchysds and sqrt as in Section 3.3.4. Using app, we can compute app(ucf1to0f)(cauchysds(sqrt1₂)) in an SDS as follows:

M::M::M::M::M ::M ::M ::. . . .