Nuprl Lemma : iseg_filter
∀[T:Type]
∀P:T ⟶ 𝔹. ∀L_1,L_2:T List. (L_2 ≤ filter(P;L_1) ⇒ (∃L_3:T List. (L_3 ≤ L_1 ∧ (L_2 = filter(P;L_3) ∈ (T List)))))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
cand: A c∧ B,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
not: ¬A,
false: False
Lemmas referenced :
list_induction,
all_wf,
list_wf,
iseg_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
exists_wf,
equal_wf,
filter_nil_lemma,
filter_cons_lemma,
nil_wf,
iseg_weakening,
iseg_nil,
assert_of_null,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
cons_wf,
nil_iseg,
equal-wf-base-T,
cons_iseg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
applyEquality,
because_Cache,
setEquality,
independent_isectElimination,
setElimination,
rename,
productEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
dependent_pairFormation,
independent_pairFormation,
productElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
equalityElimination,
cumulativity,
imageElimination,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L$_{1}$,L$_{2}$:T List. (L$_{2\mbackslash{}ff\000C7d$ \mleq{} filter(P;L$_{1}$) {}\mRightarrow{} (\mexists{}L$_{3}$:T List. (L$\mbackslash{}ff5\000Cf{3}$ \mleq{} L$_{1}$ \mwedge{} (L$_{2}$ = filter(P;L$\mbackslash{}ff5\000Cf{3}$)))))
Date html generated:
2019_06_20-PM-01_29_18
Last ObjectModification:
2018_09_17-PM-06_52_30
Theory : list_1
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