Nuprl Lemma : set_car_inc
∀s:LOSet. ∀g:OGrp. (|oal(s;g↓hgrp)| ⊆r |oal(s;g)|)
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
hgrp_of_ocgrp: g↓hgrp,
ocgrp: OGrp,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
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,
hgrp_of_ocgrp: g↓hgrp,
grp_id: e,
pi2: snd(t),
grp_car: |g|,
member: t ∈ T,
uall: ∀[x:A]. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
hgrp_car: |g|+,
implies: P ⇒ Q,
cand: A c∧ B,
mem: a ∈b as,
set_eq: =b,
grp_eq: =b,
mon_for: For{g} x ∈ as. f[x],
bor_mon: <𝔹,∨b>,
grp_op: *,
for: For{T,op,id} x ∈ as. f[x],
tlambda: λx:T. b[x]
Lemmas referenced :
ocgrp_wf,
loset_wf,
list_wf,
set_car_wf,
hgrp_car_wf,
grp_car_wf,
assert_wf,
sd_ordered_wf,
map_wf,
not_wf,
mem_wf,
dset_of_mon_wf,
hgrp_of_ocgrp_wf2,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
ocmon_wf,
abdmonoid_wf,
dmon_wf,
grp_id_wf2,
dset_of_mon_wf0,
hgrp_of_ocgrp_wf,
ocgrp_subtype_ocmon,
grp_id_wf,
subtype_rel_sets,
subtype_rel_set,
subtype_rel_list,
subtype_rel_product
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
setElimination,
rename,
hypothesisEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
productElimination,
applyEquality,
instantiate,
independent_isectElimination,
independent_pairFormation
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. (|oal(s;g\mdownarrow{}hgrp)| \msubseteq{}r |oal(s;g)|)
Date html generated:
2016_05_16-AM-08_22_16
Last ObjectModification:
2015_12_28-PM-06_25_07
Theory : polynom_2
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