Nuprl Lemma : mset_count_bound_for_union
∀s,s':DSet. ∀n:ℕ. ∀e:|s| ⟶ MSet{s'}. ∀x:|s'|.
((∀y:|s|. ((x #∈ e[y]) ≤ n)) ⇒ (∀a:MSet{s}. ((x #∈ (msFor{<MSet{s'},⋃,0>} y ∈ a. e[y])) ≤ n)))
Proof
Definitions occuring in Statement :
mset_union_mon: <MSet{s},⋃,0>,
mset_for: mset_for,
mset_count: x #∈ a,
mset: MSet{s},
nat: ℕ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
dset: DSet,
nat: ℕ,
prop: ℙ,
guard: {T},
top: Top,
mset_union_mon: <MSet{s},⋃,0>,
grp_id: e,
pi2: snd(t),
pi1: fst(t),
null_mset: 0{s},
mset_count: x #∈ a,
and: P ∧ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
quotient: x,y:A//B[x; y],
mset: MSet{s},
grp_car: |g|,
squash: ↓T,
infix_ap: x f y,
grp_op: *,
cand: A c∧ B,
uiff: uiff(P;Q)
Lemmas referenced :
mset_ind_a,
le_wf,
mset_count_wf,
mset_for_wf,
mset_union_mon_wf,
set_car_wf,
mset_wf,
sq_stable__le,
nat_wf,
dset_wf,
mset_for_null_lemma,
istype-void,
count_nil_lemma,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
iff_weakening_equal,
subtype_rel_self,
abmonoid_subtype_iabmonoid,
mset_for_mset_inj,
true_wf,
squash_wf,
mset_count_union,
mset_for_mset_sum,
imax_lb
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
isectElimination,
because_Cache,
hypothesis,
applyEquality,
universeIsType,
setElimination,
rename,
independent_functionElimination,
inhabitedIsType,
functionIsType,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
productElimination,
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
imageElimination
Latex:
\mforall{}s,s':DSet. \mforall{}n:\mBbbN{}. \mforall{}e:|s| {}\mrightarrow{} MSet\{s'\}. \mforall{}x:|s'|.
((\mforall{}y:|s|. ((x \#\mmember{} e[y]) \mleq{} n)) {}\mRightarrow{} (\mforall{}a:MSet\{s\}. ((x \#\mmember{} (msFor\{<MSet\{s'\},\mcup{},0>\} y \mmember{} a. e[y])) \mleq{} n)))
Date html generated:
2019_10_16-PM-01_06_41
Last ObjectModification:
2018_10_15-PM-08_51_16
Theory : mset
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