Nuprl Lemma : max-map-exists
∀[T:Type]. ∀L:T List. ∀f:{x:T| (x ∈ L)} ⟶ ℤ. (∃x∈L. (∀y∈L.(f y) ≤ (f x))) supposing 0 < ||L||
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
l_all: (∀x∈L.P[x]),
le: A ≤ B,
int_seg: {i..j-},
uiff: uiff(P;Q),
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
subtract: n - m,
ge: i ≥ j ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
select: L[n],
cons: [a / b]
Lemmas referenced :
list_induction,
isect_wf,
less_than_wf,
length_wf,
l_exists_wf,
l_all_wf,
le_wf,
l_member_wf,
list_wf,
length_of_nil_lemma,
member-less_than,
length_of_cons_lemma,
decidable__lt,
decidable__le,
add-member-int_seg2,
cons_wf,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
non_neg_length,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
lelt_wf,
select-cons-tl,
int_seg_properties,
add-subtract-cancel,
l_all_cons,
select_wf,
false_wf,
int_seg_wf,
list-cases,
l_all_single,
equal_wf,
nil_wf,
product_subtype_list,
list-subtype
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
cumulativity,
hypothesis,
applyEquality,
functionExtensionality,
setElimination,
rename,
setEquality,
independent_functionElimination,
imageElimination,
productElimination,
voidElimination,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidEquality,
addEquality,
unionElimination,
functionEquality,
intEquality,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
int_eqEquality,
computeAll,
promote_hyp,
hypothesis_subsumption,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbZ{}. (\mexists{}x\mmember{}L. (\mforall{}y\mmember{}L.(f y) \mleq{} (f x))) supposing 0 < ||L||
Date html generated:
2017_04_17-AM-07_50_52
Last ObjectModification:
2017_02_27-PM-04_24_22
Theory : list_1
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