Nuprl Lemma : oal_le_char
∀s:LOSet. ∀g:OGrp. ((ps,qs:|oal(s;g)|. ps ≤{s,g} qs) <≡>{|oal(s;g)|} ((ps,qs:|oal(s;g)|. ps << qs)o))
Proof
Definitions occuring in Statement :
oal_lt: ps << qs,
oal_le: ps ≤{s,g} qs,
oalist: oal(a;b),
binrel_eqv: E <≡>{T} E',
all: ∀x:A. B[x],
ocgrp: OGrp,
loset: LOSet,
set_car: |p|,
refl_cl: Eo,
ab_binrel: x,y:T. E[x; y]
Definitions unfolded in proof :
binrel_eqv: E <≡>{T} E',
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
refl_cl: Eo,
ab_binrel: x,y:T. E[x; y],
oal_le: ps ≤{s,g} qs,
oal_ble: ps ≤≤b qs,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
ocgrp: OGrp,
uiff: uiff(P;Q),
infix_ap: x f y
Lemmas referenced :
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
loset_wf,
ocgrp_abdgrp,
assert_wf,
bor_wf,
infix_ap_wf,
bool_wf,
set_eq_wf,
oal_blt_wf,
iff_transitivity,
or_wf,
equal_wf,
oal_lt_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq,
assert_of_oal_blt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
because_Cache,
independent_pairFormation,
lambdaEquality,
setElimination,
rename,
independent_functionElimination,
orFunctionality,
productElimination
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp.
((ps,qs:|oal(s;g)|. ps \mleq{}\{s,g\} qs) <\mequiv{}>\{|oal(s;g)|\} ((ps,qs:|oal(s;g)|. ps << qs)\msupzero{}))
Date html generated:
2017_10_01-AM-10_04_26
Last ObjectModification:
2017_03_03-PM-01_07_57
Theory : polynom_2
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