Nuprl Lemma : implies-member-fset-minimals
∀[T:Type]
∀eq:EqDecider(T). ∀s:fset(fset(T)). ∀a:fset(T).
(a ∈ s
⇒ (¬(∃z:fset(T). (z ∈ fset-minimals(x,y.f-proper-subset-dec(eq;x;y); s) ∧ z ⊆≠ a)))
⇒ a ∈ fset-minimals(x,y.f-proper-subset-dec(eq;x;y); s))
Proof
Definitions occuring in Statement :
fset-minimals: fset-minimals(x,y.less[x; y]; s),
f-proper-subset-dec: f-proper-subset-dec(eq;xs;ys),
f-proper-subset: xs ⊆≠ ys,
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
fset-member_witness,
deq-fset_wf,
fset-minimals_wf,
fset_wf,
f-proper-subset-dec_wf,
not_wf,
exists_wf,
fset-member_wf,
f-proper-subset_wf,
fset-size_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
add_nat_wf,
false_wf,
le_wf,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
equal_wf,
decidable__lt,
deq_wf,
member-fset-minimals,
fset-all-iff,
bnot_wf,
iff_transitivity,
assert_wf,
iff_weakening_uiff,
assert_of_bnot,
assert-f-proper-subset-dec,
assert_witness,
fset-size-proper-subset,
f-proper-subset_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
cumulativity,
productEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
dependent_set_memberEquality,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
universeEquality,
impliesFunctionality,
imageElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}s:fset(fset(T)). \mforall{}a:fset(T).
(a \mmember{} s
{}\mRightarrow{} (\mneg{}(\mexists{}z:fset(T). (z \mmember{} fset-minimals(x,y.f-proper-subset-dec(eq;x;y); s) \mwedge{} z \msubseteq{}\mneq{} a)))
{}\mRightarrow{} a \mmember{} fset-minimals(x,y.f-proper-subset-dec(eq;x;y); s))
Date html generated:
2017_04_17-AM-09_23_30
Last ObjectModification:
2017_02_27-PM-05_25_29
Theory : finite!sets
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