Nuprl Lemma : list-max-map
∀[T,A:Type]. ∀[g:A ⟶ T]. ∀[f:T ⟶ ℤ]. ∀[L:A List].
list-max(x.f[x];map(g;L)) = ((λp.<fst(p), g (snd(p))>) list-max(x.f[g x];L)) ∈ (i:ℤ × {x:T| f[x] = i ∈ ℤ} )
supposing 0 < ||L||
Proof
Definitions occuring in Statement :
list-max: list-max(x.f[x];L),
length: ||as||,
map: map(f;as),
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
list-max: list-max(x.f[x];L),
list-max-aux: list-max-aux(x.f[x];L),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
cons: [a / b],
so_apply: x[s],
outl: outl(x),
pi1: fst(t),
pi2: snd(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
isl: isl(x),
btrue: tt,
true: True,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
nat: ℕ,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
bfalse: ff,
colength: colength(L),
guard: {T},
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
bnot: ¬bb,
isr: isr(x)
Lemmas referenced :
less_than_wf,
length_wf,
list_wf,
list_accum-map,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
list_accum_cons_lemma,
value-type-has-value,
int-value-type,
top_wf,
assert_wf,
isl_wf,
set_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
int_subtype_base,
list_accum_nil_lemma,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
decidable__equal_int,
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,
ifthenelse_wf,
squash_wf,
true_wf,
list_accum_wf,
not-isr-isl,
isr_wf,
pi2_wf,
list-max_wf,
map_wf,
map-length,
list-max-property2
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
hypothesis,
because_Cache,
functionEquality,
intEquality,
universeEquality,
isect_memberFormation,
sqequalRule,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
dependent_functionElimination,
unionElimination,
imageElimination,
productElimination,
promote_hyp,
hypothesis_subsumption,
independent_isectElimination,
applyEquality,
dependent_set_memberEquality,
inlEquality,
independent_pairEquality,
productEquality,
unionEquality,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
setEquality,
dependent_pairEquality,
functionExtensionality,
cumulativity,
applyLambdaEquality,
addEquality,
baseClosed,
instantiate,
callbyvalueReduce,
equalityElimination,
hyp_replacement,
imageMemberEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[g:A {}\mrightarrow{} T]. \mforall{}[f:T {}\mrightarrow{} \mBbbZ{}]. \mforall{}[L:A List].
list-max(x.f[x];map(g;L)) = ((\mlambda{}p.<fst(p), g (snd(p))>) list-max(x.f[g x];L)) supposing 0 < ||L||
Date html generated:
2019_06_20-PM-01_30_49
Last ObjectModification:
2018_08_21-PM-01_55_35
Theory : list_1
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