Nuprl Lemma : fset-item-member
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. item(s) ∈ s supposing ||s|| = 1 ∈ ℤ
Proof
Definitions occuring in Statement :
fset-item: item(s),
fset-size: ||s||,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fset-item: item(s),
fset-member: a ∈ s,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
fset-size: ||s||,
fset: fset(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
cons: [a / b],
top: Top,
bfalse: ff,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
fset-item_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert_of_tt,
fset-member_witness,
equal-wf-T-base,
fset-size_wf,
nat_wf,
fset_wf,
deq_wf,
list_wf,
set-equal_wf,
set-equal-reflex,
length-remove-repeats-le,
iff_imp_equal_bool,
deq-member_wf,
hd_wf,
btrue_wf,
l_member_wf,
hd_member,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
true_wf,
assert-deq-member,
assert_wf,
iff_wf,
le_wf,
length_wf,
equal-wf-base,
equal_wf,
squash_wf,
iff_weakening_equal,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
remove-repeats-set-equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
isect_memberFormation,
intEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
baseClosed,
isect_memberEquality,
universeEquality,
promote_hyp,
lambdaFormation,
pointwiseFunctionality,
pertypeElimination,
productElimination,
independent_pairFormation,
natural_numberEquality,
unionElimination,
voidElimination,
hypothesis_subsumption,
voidEquality,
addLevel,
impliesFunctionality,
productEquality,
imageElimination,
imageMemberEquality,
dependent_pairFormation,
int_eqEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. item(s) \mmember{} s supposing ||s|| = 1
Date html generated:
2017_04_17-AM-09_23_12
Last ObjectModification:
2017_02_27-PM-05_24_53
Theory : finite!sets
Home
Index