Nuprl Lemma : map_reduce_spread2_to_reduce
∀[f,c,d,L:Top].
(map(f;reduce(λx,a. let y,z = x in let u,v = z in case d[y;u;v] of inl(x1) => a | inr(y1) => [c[y;u;v] / a];[];L))
~ reduce(λx,a. let y,z = x in let u,v = z in if d[y;u;v] then a else [f c[y;u;v] / a] fi ;[];L))
Proof
Definitions occuring in Statement :
map: map(f;as),
reduce: reduce(f;k;as),
cons: [a / b],
nil: [],
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2;s3],
apply: f a,
lambda: λx.A[x],
spread: spread def,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
reduce: reduce(f;k;as),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
strict1: strict1(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
map: map(f;as),
list_ind: list_ind,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
guard: {T},
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
strict4: strict4(F),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ifthenelse: if b then t else f fi ,
cons: [a / b]
Lemmas referenced :
top_wf,
map_nil_lemma,
sqle_wf_base,
lifting-strict-decide,
lifting-strict-spread,
is-exception_wf,
base_wf,
has-value_wf_base,
sqequal-list_ind
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueCallbyvalue,
hypothesis,
callbyvalueReduce,
callbyvalueExceptionCases,
inlFormation,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inrFormation,
isect_memberEquality,
voidElimination,
voidEquality,
sqleRule,
sqleReflexivity,
because_Cache,
dependent_functionElimination,
sqequalAxiom
Latex:
\mforall{}[f,c,d,L:Top].
(map(f;reduce(\mlambda{}x,a. let y,z = x
in let u,v = z
in case d[y;u;v] of inl(x1) => a | inr(y1) => [c[y;u;v] / a];[];L))
\msim{} reduce(\mlambda{}x,a. let y,z = x in let u,v = z in if d[y;u;v] then a else [f c[y;u;v] / a] fi ;[];L))
Date html generated:
2016_05_15-PM-02_08_24
Last ObjectModification:
2016_01_15-PM-10_23_56
Theory : untyped!computation
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