Nuprl Lemma : map-simplify-test
∀[H,f,g:Base]. ∀[L:Atom List]. (map(λx.H[if x ∈b L then f[x] else g[x] fi ];L) ~ map(λx.H[f[x]];L))
Proof
Definitions occuring in Statement :
deq-member: x ∈b L,
map: map(f;as),
list: T List,
atom-deq: AtomDeq,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
base: Base,
atom: Atom,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q
Lemmas referenced :
map_functionality_wrt_sq,
atom_subtype_base,
list_subtype_base,
deq-member_wf,
atom-deq_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_member_wf,
list_wf,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
atomEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
sqequalAxiom,
isect_memberEquality
Latex:
\mforall{}[H,f,g:Base]. \mforall{}[L:Atom List]. (map(\mlambda{}x.H[if x \mmember{}\msubb{} L then f[x] else g[x] fi ];L) \msim{} map(\mlambda{}x.H[f[x]];L))
Date html generated:
2017_04_17-AM-09_08_50
Last ObjectModification:
2017_02_27-PM-05_17_07
Theory : decidable!equality
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