Nuprl Lemma : fset-size-empty
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. uiff(||s|| ≤ 0;s = {} ∈ fset(T))
Proof
Definitions occuring in Statement :
fset-size: ||s||,
empty-fset: {},
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
fset: fset(T),
all: ∀x:A. B[x],
prop: ℙ,
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
empty-fset: {},
nil: [],
it: ⋅,
implies: P ⇒ Q,
fset-size: ||s||,
or: P ∨ Q,
top: Top,
cons: [a / b],
deq: EqDecider(T),
ge: i ≥ j ,
false: False,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
nat: ℕ,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
remove-repeats: remove-repeats(eq;L)
Lemmas referenced :
list_wf,
set-equal_wf,
set-equal-reflex,
quotient-member-eq,
set-equal-equiv,
nil_wf,
list-cases,
remove_repeats_nil_lemma,
istype-void,
length_of_nil_lemma,
istype-le,
product_subtype_list,
remove_repeats_cons_lemma,
length_of_cons_lemma,
non_neg_length,
filter_wf5,
remove-repeats_wf,
l_member_wf,
bnot_wf,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
length_wf,
equal_wf,
squash_wf,
true_wf,
empty-fset_wf,
subtype_rel_self,
fset-size_wf,
istype-false,
le_wf,
le_witness_for_triv,
fset_wf,
deq_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
Error :universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
promote_hyp,
Error :lambdaFormation_alt,
Error :inhabitedIsType,
pointwiseFunctionality,
sqequalRule,
pertypeElimination,
productElimination,
Error :lambdaEquality_alt,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
unionElimination,
Error :isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
because_Cache,
hypothesis_subsumption,
rename,
setElimination,
applyEquality,
Error :setIsType,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
addEquality,
closedConclusion,
Error :equalityIstype,
Error :productIsType,
sqequalBase,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
hyp_replacement,
applyLambdaEquality,
independent_pairEquality,
axiomEquality,
Error :isectIsTypeImplies
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. uiff(||s|| \mleq{} 0;s = \{\})
Date html generated:
2019_06_20-PM-02_13_48
Last ObjectModification:
2019_06_20-PM-02_07_41
Theory : finite!sets
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