Nuprl Lemma : filter_map_upto2
∀t',m:ℕ.
∀[T:Type]
∀f:ℕ ⟶ T. ∀P:T ⟶ 𝔹.
∃t:ℕ. ((↑(P (f t))) ∧ (||filter(P;map(f;upto(t)))|| = m ∈ ℤ)) supposing (m + 1) ≤ ||filter(P;map(f;upto(t')))||
Proof
Definitions occuring in Statement :
upto: upto(n),
length: ||as||,
filter: filter(P;l),
map: map(f;as),
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
top: Top,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
guard: {T},
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
all_wf,
nat_wf,
uall_wf,
bool_wf,
isect_wf,
le_wf,
length_wf,
filter_wf5,
map_wf,
int_seg_wf,
subtract_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
upto_wf,
l_member_wf,
subtype_rel_self,
set_wf,
exists_wf,
assert_wf,
equal_wf,
less_than_wf,
primrec-wf2,
map_nil_lemma,
filter_nil_lemma,
length_of_nil_lemma,
less_than'_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_int_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
eqtt_to_assert,
assert_of_le_int,
intformless_wf,
int_formula_prop_less_lemma,
lt_int_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
upto_decomp1,
add-is-int-iff,
set_subtype_base,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
equal-wf-T-base,
not_wf,
map_append_sq,
filter_append_sq,
map_cons_lemma,
filter_cons_lemma,
assert_of_bnot,
squash_wf,
true_wf,
length_append,
subtype_rel_list,
top_wf,
cons_wf,
nil_wf,
iff_weakening_equal,
length_of_cons_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
append_nil_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
setElimination,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
applyEquality,
lambdaEquality,
cumulativity,
hypothesisEquality,
universeEquality,
sqequalRule,
functionEquality,
addEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
setEquality,
productEquality,
functionExtensionality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
isect_memberFormation,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
int_eqEquality,
computeAll,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}t',m:\mBbbN{}.
\mforall{}[T:Type]
\mforall{}f:\mBbbN{} {}\mrightarrow{} T. \mforall{}P:T {}\mrightarrow{} \mBbbB{}.
\mexists{}t:\mBbbN{}. ((\muparrow{}(P (f t))) \mwedge{} (||filter(P;map(f;upto(t)))|| = m))
supposing (m + 1) \mleq{} ||filter(P;map(f;upto(t')))||
Date html generated:
2017_04_17-AM-07_58_49
Last ObjectModification:
2017_02_27-PM-04_31_20
Theory : list_1
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