Nuprl Lemma : oal_merge_preserves_le
∀s:LOSet. ∀g:OGrp. ∀ps,qs,rs:|oal(s;g)|. ((qs ≤{s,g} rs) ⇒ ((ps ++ qs) ≤{s,g} (ps ++ rs)))
Proof
Definitions occuring in Statement :
oal_le: ps ≤{s,g} qs,
oal_merge: ps ++ qs,
oalist: oal(a;b),
all: ∀x:A. B[x],
implies: P ⇒ Q,
ocgrp: OGrp,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
grp_leq: a ≤ b,
oal_grp: oal_grp(s;g),
grp_le: ≤b,
pi2: snd(t),
pi1: fst(t),
infix_ap: x f y,
oal_le: ps ≤{s,g} qs,
grp_op: *,
ocgrp: OGrp,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set,
grp_car: |g|
Lemmas referenced :
oal_le_wf,
ocgrp_subtype_abdgrp,
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
loset_wf,
grp_op_preserves_le,
oal_grp_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
isectElimination,
instantiate,
independent_isectElimination,
because_Cache,
lambdaEquality,
setElimination,
rename
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. \mforall{}ps,qs,rs:|oal(s;g)|. ((qs \mleq{}\{s,g\} rs) {}\mRightarrow{} ((ps ++ qs) \mleq{}\{s,g\} (ps ++ rs)))
Date html generated:
2016_05_16-AM-08_22_12
Last ObjectModification:
2015_12_28-PM-06_28_17
Theory : polynom_2
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