Nuprl Lemma : select-filter-from-upto-order-preserving
∀[n,m:ℤ]. ∀[P:{n..m-} ⟶ 𝔹]. ∀[i,j:ℕ||filter(P;[n, m))||]. uiff(i < j;filter(P;[n, m))[i] < filter(P;[n, m))[j])
Proof
Definitions occuring in Statement :
from-upto: [n, m),
select: L[n],
length: ||as||,
filter: filter(P;l),
int_seg: {i..j-},
bool: 𝔹,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
istype: istype(T),
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
from-upto: [n, m),
cand: A c∧ B,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
has-value: (a)↓,
sq_type: SQType(T)
Lemmas referenced :
complete-nat-induction,
all_wf,
le_wf,
subtract_wf,
int_seg_wf,
bool_wf,
length_wf,
filter_wf5,
from-upto_wf,
subtype_rel_dep_function,
l_member_wf,
iff_wf,
less_than_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
istype-nat,
istype-le,
istype-less_than,
lt_int_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
filter_cons_lemma,
value-type-has-value,
int-value-type,
itermSubtract_wf,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
le_reflexive,
subtype_rel_self,
equal-wf-T-base,
bnot_wf,
not_wf,
le_int_wf,
filter_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
assert_of_bnot,
subtype_rel_set,
length_of_cons_lemma,
set_subtype_base,
lelt_wf,
select-cons,
bnot_of_le_int,
add-is-int-iff,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
list_wf,
set_wf,
satisfiable-full-omega-tt,
member-less_than,
subtype_base_sq,
list_subtype_base,
from-upto-nil,
nil_wf,
less_than_transitivity1,
less_than_irreflexivity
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
Error :lambdaEquality_alt,
closedConclusion,
intEquality,
because_Cache,
functionEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
applyEquality,
Error :universeIsType,
setEquality,
Error :setIsType,
independent_isectElimination,
Error :lambdaFormation_alt,
productElimination,
dependent_functionElimination,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
unionElimination,
imageElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :functionIsType,
Error :productIsType,
baseApply,
baseClosed,
Error :dependent_set_memberEquality_alt,
addEquality,
minusEquality,
multiplyEquality,
equalityElimination,
Error :equalityIstype,
callbyvalueReduce,
pointwiseFunctionality,
promote_hyp,
applyLambdaEquality,
lambdaEquality,
productEquality,
lambdaFormation,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
computeAll,
isect_memberFormation,
independent_pairEquality,
dependent_set_memberEquality,
instantiate,
cumulativity
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[P:\{n..m\msupminus{}\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[i,j:\mBbbN{}||filter(P;[n, m))||].
uiff(i < j;filter(P;[n, m))[i] < filter(P;[n, m))[j])
Date html generated:
2019_06_20-PM-01_34_24
Last ObjectModification:
2019_01_21-PM-01_56_58
Theory : list_1
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