Nuprl Lemma : bag-lub-property
∀[T:Type]
∀eq:EqDecider(T). ∀as,bs:bag(T).
(sub-bag(T;as;bag-lub(eq;as;bs))
∧ sub-bag(T;bs;bag-lub(eq;as;bs))
∧ (∀cs:bag(T). (sub-bag(T;as;cs) ⇒ sub-bag(T;bs;cs) ⇒ sub-bag(T;bag-lub(eq;as;bs);cs))))
Proof
Definitions occuring in Statement :
bag-lub: bag-lub(eq;b1;b2),
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
nat: ℕ,
uiff: uiff(P;Q)
Lemmas referenced :
nat_wf,
imax_lb,
deq_wf,
bag_wf,
sub-bag_wf,
less_than'_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
imax_ub,
iff_weakening_equal,
bag-count-bag-lub,
bag-count_wf,
true_wf,
squash_wf,
le_wf,
bag-lub_wf,
sub-bag-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
productElimination,
independent_functionElimination,
introduction,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
inlFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_pairEquality,
axiomEquality,
independent_pairFormation,
inrFormation,
setElimination,
rename
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}as,bs:bag(T).
(sub-bag(T;as;bag-lub(eq;as;bs))
\mwedge{} sub-bag(T;bs;bag-lub(eq;as;bs))
\mwedge{} (\mforall{}cs:bag(T). (sub-bag(T;as;cs) {}\mRightarrow{} sub-bag(T;bs;cs) {}\mRightarrow{} sub-bag(T;bag-lub(eq;as;bs);cs))))
Date html generated:
2016_05_15-PM-08_09_36
Last ObjectModification:
2016_01_16-PM-01_27_46
Theory : bags_2
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