Nuprl Lemma : lookup_fails
∀a:DSet. ∀B:Type. ∀z:B. ∀k:|a|. ∀ps:(|a| × B) List. ((¬↑(k ∈b map(λx.(fst(x));ps))) ⇒ ((ps[k]) = z ∈ B))
Proof
Definitions occuring in Statement :
lookup: as[k],
mem: a ∈b as,
map: map(f;as),
list: T List,
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
lambda: λx.A[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
dset: DSet,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
infix_ap: x f y,
not: ¬A,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_implies: P ⇐ Q,
or: P ∨ Q,
false: False,
pi1: fst(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt
Lemmas referenced :
list_induction,
set_car_wf,
not_wf,
assert_wf,
mem_wf,
map_wf,
pi1_wf,
equal_wf,
lookup_wf,
list_wf,
map_nil_lemma,
lookup_nil_lemma,
mem_nil_lemma,
false_wf,
map_cons_lemma,
mem_cons_lemma,
bor_wf,
set_eq_wf,
dset_wf,
iff_transitivity,
or_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq,
not_over_or,
lookup_cons_pr_lemma,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
eqtt_to_assert,
bnot_wf,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
setElimination,
rename,
because_Cache,
hypothesis,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
universeEquality,
addLevel,
impliesFunctionality,
independent_pairFormation,
orFunctionality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
levelHypothesis,
promote_hyp,
impliesLevelFunctionality,
unionElimination,
equalityElimination,
baseClosed
Latex:
\mforall{}a:DSet. \mforall{}B:Type. \mforall{}z:B. \mforall{}k:|a|. \mforall{}ps:(|a| \mtimes{} B) List.
((\mneg{}\muparrow{}(k \mmember{}\msubb{} map(\mlambda{}x.(fst(x));ps))) {}\mRightarrow{} ((ps[k]) = z))
Date html generated:
2017_10_01-AM-10_02_05
Last ObjectModification:
2017_03_03-PM-01_04_24
Theory : polynom_2
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