Nuprl Lemma : oal_grp_wf1
∀s:LOSet. ∀g:OGrp. (oal_grp(s;g) ∈ OMon)
Proof
Definitions occuring in Statement :
oal_grp: oal_grp(s;g),
all: ∀x:A. B[x],
member: t ∈ T,
ocgrp: OGrp,
omon: OMon,
loset: LOSet
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
abdmonoid: AbDMon,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
dmon: DMon,
mon: Mon,
prop: ℙ,
oal_grp: oal_grp(s;g),
grp_car: |g|,
pi1: fst(t),
grp_le: ≤b,
pi2: snd(t),
infix_ap: x f y,
oal_le: ps ≤{s,g} qs,
ulinorder: UniformLinorder(T;x,y.R[x; y]),
and: P ∧ Q,
uorder: UniformOrder(T;x,y.R[x; y]),
order: Order(T;x,y.R[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
implies: P ⇒ Q,
cand: A c∧ B,
utrans: UniformlyTrans(T;x,y.E[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
dset: DSet,
refl: Refl(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
omon: OMon,
abdgrp: AbDGrp,
abgrp: AbGrp,
grp: Group{i},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
grp_eq: =b,
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,
set_eq: =b,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
label: ...$L... t,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ocgrp_abdgrp,
omon_inc,
ocmon_subtype_omon,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
omon_wf,
abdmonoid_dmon,
ocgrp_properties,
ocmon_properties,
abmonoid_properties,
comm_wf,
grp_car_wf,
grp_op_wf,
oal_le_is_order,
assert_witness,
oal_ble_wf,
oal_le_wf,
oal_le_connex,
set_car_wf,
oalist_wf,
loset_wf,
oal_grp_wf,
subtype_rel_sets,
mon_wf,
inverse_wf,
grp_id_wf,
grp_inv_wf,
eqfun_p_wf,
grp_eq_wf,
sq_stable__comm,
ulinorder_wf,
assert_wf,
grp_le_wf,
bool_wf,
infix_ap_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
bfalse_wf,
list_wf,
istype-assert,
sd_ordered_wf,
map_wf,
mon_subtype_grp_sig,
dmon_subtype_mon,
ocmon_subtype_abdmonoid,
abdmonoid_wf,
dmon_wf,
grp_sig_wf,
mem_wf,
dset_of_mon_wf,
dset_of_mon_wf0,
istype-void,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
eq_list_wf,
set_prod_wf,
assert_of_eq_list,
equal_functionality_wrt_subtype_rel2,
not_wf,
subtype_rel_self,
assert_elim,
squash_wf,
true_wf,
abdgrp_wf,
member_wf,
not_assert_elim,
btrue_neq_bfalse,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
universeIsType,
because_Cache,
independent_pairFormation,
productElimination,
isect_memberFormation_alt,
independent_functionElimination,
lambdaEquality_alt,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
cumulativity,
setIsType,
imageMemberEquality,
baseClosed,
imageElimination,
productIsType,
equalityIstype,
functionIsType,
unionElimination,
productEquality,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
applyLambdaEquality,
hyp_replacement,
natural_numberEquality
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. (oal\_grp(s;g) \mmember{} OMon)
Date html generated:
2019_10_16-PM-01_08_41
Last ObjectModification:
2018_11_27-AM-10_42_22
Theory : polynom_2
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