Nuprl Lemma : omral_fact
∀g:OCMon. ∀r:CDRng. ∀ps:|omral(g;r)|. (ps = (msFor{oal_mon(g↓oset;r↓+gp)} k' ∈ dom(ps). inj(k',ps[k'])) ∈ |omral(g;r)|)
Proof
Definitions occuring in Statement :
omral_inj: inj(k,v),
omral_dom: dom(ps),
omralist: omral(g;r),
oal_mon: oal_mon(a;b),
lookup: as[k],
mset_for: mset_for,
all: ∀x:A. B[x],
equal: s = t ∈ T,
add_grp_of_rng: r↓+gp,
cdrng: CDRng,
rng_zero: 0,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
ocmon: OCMon,
omon: OMon,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
infix_ap: x f y,
so_apply: x[s],
cand: A c∧ B,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
pi1: fst(t),
omralist: omral(g;r),
omral_dom: dom(ps),
omral_inj: inj(k,v)
Lemmas referenced :
oalist_fact,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
grp_car_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
cdrng_is_abdmonoid,
cdrng_wf,
ocmon_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
hypothesis,
because_Cache,
lambdaEquality,
productEquality,
setElimination,
rename,
cumulativity,
universeEquality,
functionEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
independent_pairFormation
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}ps:|omral(g;r)|.
(ps = (msFor\{oal\_mon(g\mdownarrow{}oset;r\mdownarrow{}+gp)\} k' \mmember{} dom(ps). inj(k',ps[k'])))
Date html generated:
2017_10_01-AM-10_05_25
Last ObjectModification:
2017_03_03-PM-01_11_53
Theory : polynom_3
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