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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