Nuprl Lemma : list-max-aux-property
∀[T:Type]
∀f:T ⟶ ℤ. ∀L:T List.
(↑isl(list-max-aux(x.f[x];L)))
∧ let n,x = outl(list-max-aux(x.f[x];L))
in (x ∈ L) ∧ (f[x] = n ∈ ℤ) ∧ (∀y∈L.f[y] ≤ n)
supposing 0 < ||L||
Proof
Definitions occuring in Statement :
list-max-aux: list-max-aux(x.f[x];L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
outl: outl(x),
assert: ↑b,
isl: isl(x),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
spread: spread def,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff,
subtract: n - m,
list-max-aux: list-max-aux(x.f[x];L),
outl: outl(x),
isl: isl(x),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
true: True,
bnot: ¬bb,
int_iseg: {i...j},
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi1: fst(t),
l_all: (∀x∈L.P[x]),
list_accum: list_accum,
firstn: firstn(n;as),
list_ind: list_ind,
nil: [],
lt_int: i <z j,
length: ||as||,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
last: last(L),
select: L[n]
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
lelt_wf,
subtype_rel_self,
le_wf,
length_wf,
non_neg_length,
nat_properties,
less_than_wf,
member-less_than,
last-lemma-sq,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
all_wf,
list_wf,
isect_wf,
assert_wf,
isl_wf,
equal-wf-T-base,
top_wf,
list-max-aux_wf,
l_member_wf,
l_all_wf,
assert_elim,
and_wf,
equal_wf,
bfalse_wf,
btrue_neq_bfalse,
set_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
firstn_wf,
subtype_rel_list,
list_accum_cons_lemma,
list_accum_nil_lemma,
value-type-has-value,
int-value-type,
last_wf,
list_accum_append,
lt_int_wf,
pi1_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
length_firstn_eq,
iff_weakening_equal,
member_append,
cons_member,
l_all_append,
cons_wf,
nil_wf,
l_all_cons,
l_all_nil,
select_wf,
list_ind_nil_lemma,
first0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
hypothesis_subsumption,
imageElimination,
promote_hyp,
functionEquality,
productEquality,
setEquality,
unionEquality,
cumulativity,
functionExtensionality,
addEquality,
universeEquality,
callbyvalueReduce,
equalityElimination,
imageMemberEquality,
baseClosed,
inrFormation,
inlFormation
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} \mBbbZ{}. \mforall{}L:T List.
(\muparrow{}isl(list-max-aux(x.f[x];L)))
\mwedge{} let n,x = outl(list-max-aux(x.f[x];L))
in (x \mmember{} L) \mwedge{} (f[x] = n) \mwedge{} (\mforall{}y\mmember{}L.f[y] \mleq{} n)
supposing 0 < ||L||
Date html generated:
2019_06_20-PM-01_30_41
Last ObjectModification:
2018_09_24-PM-00_40_42
Theory : list_1
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