Nuprl Lemma : list-index-cmp_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. ∀[A:Type]. ∀[f:A ⟶ T].
(list-index-cmp(eq;L;f) ∈ comparison({x:A| (f x ∈ L)} ))
Proof
Definitions occuring in Statement :
list-index-cmp: list-index-cmp(eq;L;f),
comparison: comparison(T),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
list-index-cmp: list-index-cmp(eq;L;f),
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
int_seg: {i..j-},
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
int-minus-comparison_wf,
l_member_wf,
outl_wf,
top_wf,
list-index_wf,
subtype_rel_union,
int_seg_wf,
length_wf,
isl-list-index,
list_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
hypothesis,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
intEquality,
natural_numberEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List]. \mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} T].
(list-index-cmp(eq;L;f) \mmember{} comparison(\{x:A| (f x \mmember{} L)\} ))
Date html generated:
2016_05_14-PM-03_30_38
Last ObjectModification:
2015_12_26-PM-06_02_33
Theory : decidable!equality
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