Nuprl Lemma : finite-max
∀[T:Type]. (finite-type(T) ⇒ T ⇒ (∀g:T ⟶ ℤ. ∃x:T. ∀y:T. ((g y) ≤ (g x))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
cons: [a / b],
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
guard: {T},
decidable: Dec(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
l_all: (∀x∈L.P[x])
Lemmas referenced :
finite-type_wf,
finite-type-iff-list,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
product_subtype_list,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
equal_wf,
maximal-in-list,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
less_than'_wf,
all_wf,
le_wf,
l_all_iff,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
functionEquality,
cumulativity,
hypothesisEquality,
intEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
universeEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
sqequalRule,
rename,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
setElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
imageElimination,
independent_pairEquality,
applyEquality,
functionExtensionality,
axiomEquality,
setEquality
Latex:
\mforall{}[T:Type]. (finite-type(T) {}\mRightarrow{} T {}\mRightarrow{} (\mforall{}g:T {}\mrightarrow{} \mBbbZ{}. \mexists{}x:T. \mforall{}y:T. ((g y) \mleq{} (g x))))
Date html generated:
2017_04_17-AM-07_45_59
Last ObjectModification:
2017_02_27-PM-04_18_19
Theory : list_1
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