Nuprl Lemma : oalist_subtype
∀[a:LOSet]. ∀[b1,b2:AbDMon]. |oal(a;b1)| ⊆r |oal(a;b2)| supposing strong-subtype(|b1|;|b2|) ∧ (e = e ∈ |b2|)
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
member: t ∈ T,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
guard: {T},
pi2: snd(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T
Lemmas referenced :
subtype_rel_list,
set_car_wf,
grp_car_wf,
subtype_rel_product,
assert_wf,
mem_wf,
dset_of_mon_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf0,
sd_ordered_wf,
not_wf,
oalist_wf,
dset_wf,
strong-subtype_wf,
equal_wf,
grp_id_wf,
abdmonoid_wf,
loset_wf,
mem_iff_exists,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
pi2_wf,
intformless_wf,
int_formula_prop_less_lemma,
strong-subtype-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
setElimination,
thin,
rename,
dependent_set_memberEquality,
productElimination,
cut,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
productEquality,
hypothesis,
because_Cache,
independent_isectElimination,
lambdaFormation,
independent_pairFormation,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberFormation,
axiomEquality,
isect_memberEquality,
dependent_pairFormation,
unionElimination,
natural_numberEquality,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
independent_pairEquality,
imageElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[a:LOSet]. \mforall{}[b1,b2:AbDMon].
|oal(a;b1)| \msubseteq{}r |oal(a;b2)| supposing strong-subtype(|b1|;|b2|) \mwedge{} (e = e)
Date html generated:
2017_10_01-AM-10_01_38
Last ObjectModification:
2017_03_03-PM-01_05_09
Theory : polynom_2
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