Nuprl Lemma : fset-max_property
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[f:T ⟶ ℕ]. ∀[s:fset(T)].
((∀[x:T]. (f x) ≤ fset-max(f;s) supposing x ∈ s)
∧ (¬((∀x:T. (x ∈ s ⇒ f x < fset-max(f;s))) ∧ (¬(s = {} ∈ fset(T))))))
Proof
Definitions occuring in Statement :
empty-fset: {},
fset-max: fset-max(f;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
fset: fset(T),
prop: ℙ,
quotient: x,y:A//B[x; y],
fset-max: fset-max(f;s),
fset-member: a ∈ s,
sq_type: SQType(T),
guard: {T},
true: True,
false: False,
le: A ≤ B,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
top: Top,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
cons: [a / b],
rev_implies: P ⇐ Q,
subtract: n - m,
fset-null: fset-null(s),
less_than: a < b,
squash: ↓T
Lemmas referenced :
decidable-equal-deq,
decidable__le,
fset-max_wf,
list_wf,
set-equal_wf,
set-equal-reflex,
equal-wf-base,
equal_wf,
subtype_base_sq,
int_subtype_base,
less_than'_wf,
fset-member_wf,
all_wf,
less_than_wf,
not_wf,
equal-wf-T-base,
fset_wf,
nat_wf,
deq_wf,
imax-list-lb,
cons_wf,
map_wf,
imax-list_wf,
length_of_cons_lemma,
non_neg_length,
map_length,
decidable__lt,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
l_all_iff,
false_wf,
le_wf,
l_member_wf,
list-cases,
length_of_nil_lemma,
nil_member,
product_subtype_list,
length_wf_nat,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
assert-deq-member,
cons_member,
member_map,
imax-list-ub,
l_exists_iff,
set_subtype_base,
assert-fset-null,
list_subtype_fset,
assert_wf,
null_wf,
pos_length3,
hd_wf,
hd_member,
nat_properties,
decidable__equal_int,
pos_length2,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
sqequalRule,
independent_isectElimination,
unionElimination,
promote_hyp,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
pertypeElimination,
productElimination,
productEquality,
intEquality,
natural_numberEquality,
instantiate,
voidElimination,
independent_pairEquality,
lambdaEquality,
axiomEquality,
isect_memberEquality,
independent_pairFormation,
functionEquality,
baseClosed,
universeEquality,
setElimination,
rename,
voidEquality,
addEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
dependent_set_memberEquality,
hypothesis_subsumption,
minusEquality,
setEquality,
inrFormation,
addLevel,
impliesFunctionality,
levelHypothesis,
impliesLevelFunctionality,
applyLambdaEquality,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[f:T {}\mrightarrow{} \mBbbN{}]. \mforall{}[s:fset(T)].
((\mforall{}[x:T]. (f x) \mleq{} fset-max(f;s) supposing x \mmember{} s)
\mwedge{} (\mneg{}((\mforall{}x:T. (x \mmember{} s {}\mRightarrow{} f x < fset-max(f;s))) \mwedge{} (\mneg{}(s = \{\})))))
Date html generated:
2017_04_17-AM-09_20_46
Last ObjectModification:
2017_02_27-PM-05_24_32
Theory : finite!sets
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