Nuprl Lemma : apply-alist-inl
∀[A,T:Type]. ∀eq:EqDecider(T). ∀x:T. ∀L:(T × A) List. ∀z:A. ((apply-alist(eq;L;x) = (inl z) ∈ (A?)) ⇒ (<x, z> ∈ L))
Proof
Definitions occuring in Statement :
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
pair: <a, b>,
product: x:A × B[x],
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
isl: isl(x),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
outl: outl(x)
Lemmas referenced :
isl-apply-alist,
btrue_wf,
bfalse_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
unit_wf2,
apply-alist_wf,
list_wf,
deq_wf,
istype-universe,
l_member_wf,
squash_wf,
true_wf,
outl_wf,
assert_wf,
isl_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
productElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
unionElimination,
instantiate,
cumulativity,
independent_functionElimination,
natural_numberEquality,
unionIsType,
universeIsType,
inlEquality_alt,
productEquality,
universeEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
independent_pairEquality,
because_Cache,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,T:Type].
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}L:(T \mtimes{} A) List. \mforall{}z:A. ((apply-alist(eq;L;x) = (inl z)) {}\mRightarrow{} (<x, z> \mmember{} L))
Date html generated:
2020_05_19-PM-09_41_51
Last ObjectModification:
2020_01_26-PM-10_42_18
Theory : list_1
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