Nuprl Lemma : filter_map_upto
∀[T:Type]. ∀[i,j:ℕ].
∀[f:ℕ ⟶ T]. ∀[P:T ⟶ 𝔹]. ||filter(P;map(f;upto(i)))|| < ||filter(P;map(f;upto(j)))|| supposing ↑(P (f i))
supposing i < j
Proof
Definitions occuring in Statement :
upto: upto(n),
length: ||as||,
filter: filter(P;l),
map: map(f;as),
nat: ℕ,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T}
Lemmas referenced :
less_than_wf,
bool_wf,
assert_wf,
equal_wf,
length-append,
nat_wf,
le_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
filter_append_sq,
map_append_sq,
lelt_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
upto_decomp,
set_wf,
l_member_wf,
subtype_rel_dep_function,
upto_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
subtype_rel_function,
subtract_wf,
int_seg_wf,
map_wf,
filter_wf5,
length_wf,
decidable__equal_int,
length_zero,
member_map,
member_filter,
int_term_value_subtract_lemma,
itermSubtract_wf,
member_upto,
int_formula_prop_eq_lemma,
intformeq_wf,
member-implies-null-eq-bfalse,
null_nil_lemma,
btrue_wf,
btrue_neq_bfalse,
equal-wf-T-base,
not_wf,
non_neg_length
Rules used in proof :
universeEquality,
applyEquality,
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
functionEquality,
because_Cache,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
addEquality,
dependent_functionElimination,
hypothesis,
independent_pairFormation,
dependent_set_memberEquality,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
setEquality,
productElimination,
productEquality,
functionExtensionality,
applyLambdaEquality,
hyp_replacement,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[i,j:\mBbbN{}].
\mforall{}[f:\mBbbN{} {}\mrightarrow{} T]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
||filter(P;map(f;upto(i)))|| < ||filter(P;map(f;upto(j)))|| supposing \muparrow{}(P (f i))
supposing i < j
Date html generated:
2019_06_20-PM-01_34_33
Last ObjectModification:
2019_01_10-PM-08_48_35
Theory : list_1
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