Nuprl Lemma : oal_lt_iff_grp_lt
∀s:LOSet. ∀g:OGrp. ∀ps,qs:|oal(s;g)|. (ps << qs ⇐⇒ ps < qs)
Proof
Definitions occuring in Statement :
oal_lt: ps << qs,
oal_grp: oal_grp(s;g),
oalist: oal(a;b),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
ocgrp: OGrp,
grp_lt: a < b,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set,
oal_grp: oal_grp(s;g),
grp_car: |g|,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
prop: ℙ,
ocgrp: OGrp,
implies: P ⇒ Q,
ab_binrel: x,y:T. E[x; y],
binrel_ap: a [r] b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
abdgrp: AbDGrp,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
s_part: E\,
grp_leq: a ≤ b,
grp_le: ≤b,
pi2: snd(t),
infix_ap: x f y,
oal_le: ps ≤{s,g} qs
Lemmas referenced :
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
loset_wf,
grp_lt_is_sp_of_leq_a,
oal_grp_wf1,
grp_lt_wf,
oal_grp_wf,
ocgrp_subtype_abdgrp,
iff_wf,
oal_lt_wf,
grp_leq_wf,
not_wf,
binrel_ap_functionality_wrt_breqv,
ab_binrel_wf,
s_part_wf,
oal_le_wf,
oal_lt_char,
binrel_ap_wf,
abdgrp_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
setElimination,
rename,
productEquality,
lambdaEquality,
cumulativity,
universeEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. \mforall{}ps,qs:|oal(s;g)|. (ps << qs \mLeftarrow{}{}\mRightarrow{} ps < qs)
Date html generated:
2016_05_16-AM-08_22_01
Last ObjectModification:
2015_12_28-PM-06_25_38
Theory : polynom_2
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