Nuprl Lemma : oal_lt_trans
∀s:LOSet. ∀g:OCMon. Trans(|oal(s;g)|;ps,qs.ps << qs)
Proof
Definitions occuring in Statement :
oal_lt: ps << qs
,
oalist: oal(a;b)
,
trans: Trans(T;x,y.E[x; y])
,
all: ∀x:A. B[x]
,
ocmon: OCMon
,
loset: LOSet
,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
trans: Trans(T;x,y.E[x; y])
,
implies: P
⇒ Q
,
oal_lt: ps << qs
,
exists: ∃x:A. B[x]
,
and: P ∧ Q
,
member: t ∈ T
,
prop: ℙ
,
uall: ∀[x:A]. B[x]
,
subtype_rel: A ⊆r B
,
dset: DSet
,
loset: LOSet
,
poset: POSet{i}
,
qoset: QOSet
,
decidable: Dec(P)
,
or: P ∨ Q
,
dset_of_mon: g↓set
,
set_prod: s × t
,
dset_list: s List
,
pi1: fst(t)
,
set_car: |p|
,
mk_dset: mk_dset(T, eq)
,
dset_set: dset_set,
oalist: oal(a;b)
,
mon: Mon
,
abmonoid: AbMon
,
ocmon: OCMon
,
uimplies: b supposing a
,
guard: {T}
,
iff: P
⇐⇒ Q
,
true: True
,
squash: ↓T
,
uiff: uiff(P;Q)
Lemmas referenced :
oal_lt_wf,
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
ocmon_wf,
loset_wf,
decidable__set_leq,
grp_lt_wf,
grp_id_wf,
lookup_wf,
grp_car_wf,
set_lt_wf,
set_lt_transitivity_1,
set_leq_iff_lt_or_eq,
ocmon_subtype_omon,
grp_leq_weakening_eq,
grp_lt_transitivity_2,
grp_leq_weakening_lt,
grp_lt_transitivity_1,
iff_weakening_equal,
subtype_rel_self,
poset_sig_wf,
istype-universe,
list_wf,
grp_sig_wf,
true_wf,
squash_wf,
set_leq_complement,
set_lt_transitivity_2,
set_leq_weakening_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
isectElimination,
applyEquality,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityIstype,
because_Cache,
functionIsType,
productIsType,
independent_pairFormation,
dependent_pairFormation_alt,
independent_isectElimination,
independent_functionElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
instantiate,
productEquality,
imageElimination
Latex:
\mforall{}s:LOSet. \mforall{}g:OCMon. Trans(|oal(s;g)|;ps,qs.ps << qs)
Date html generated:
2020_05_20-AM-09_36_13
Last ObjectModification:
2020_01_16-PM-04_55_24
Theory : polynom_2
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