Nuprl Lemma : decidable__q-constraints-opt
∀k:ℕ. ∀A:(ℕ ⟶ ℚ × ℤ) List. Dec(∃y:ℚ List [q-constraints(k;A;y)])
Proof
Definitions occuring in Statement :
q-constraints: q-constraints(k;A;y),
rationals: ℚ,
list: T List,
nat: ℕ,
decidable: Dec(P),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
function: x:A ⟶ B[x],
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
member: t ∈ T,
pi2: snd(t),
pi1: fst(t),
it: ⋅,
ifthenelse: if b then t else f fi ,
subtract: n - m,
qsub: r - s,
product-map: product-map(F;as;bs),
normalize-constraints: normalize-constraints(k;A),
callbyvalueall: callbyvalueall,
normalize-constraint: normalize-constraint(k;p),
upto: upto(n),
select?: as[i]?a,
lt_int: i <z j,
btrue: tt,
bfalse: ff,
mapfilter: mapfilter(f;P;L),
qmin-list: qmin-list(L),
combine-list: combine-list(x,y.f[x; y];L),
qmin: qmin(x;y),
q_le: q_le(r;s),
experimental: experimental{allFunctionality}(possibleextract),
eq_int: (i =z j),
experimental: experimental{orFunctionality}(possibleextract),
experimental: experimental{impliesFunctionality}(possibleextract),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
qmax: qmax(x;y),
natrec: natrec,
genrec: genrec,
genrec-ap: genrec-ap,
spreadn: spread4,
spreadn: spread3,
ifunion: ifunion(b; t),
q-constraints-decider: q-constraints-decider,
decidable__q-constraints,
decidable__equal_int,
iff_weakening_equal,
decidable__l_all-better-extract,
decidable__q-rel,
decidable__l_exists_better-extract,
decidable__cand,
decidable__not,
decidable__equal_rationals,
l_exists_iff,
product-map,
member_filter,
member_append,
decidable__qless,
sq_stable__and,
sq_stable__equal,
q-min-exists,
q-max-exists,
member_map_filter,
l_all_iff,
any: any x,
iff_transitivity,
iff_weakening_uiff,
eqtt_to_assert,
assert_of_band,
assert_of_eq_int,
qle_weakening_lt_qorder,
q-constraint-times,
bool_cases,
uiff_transitivity2,
qadd_preserves_qle,
qadd_preserves_qless,
decidable__int_equal,
decidable__l_all,
decidable__l_exists,
decidable__and2,
decidable__implies,
decidable__false,
select_member,
list-cases,
l_all_nil,
qmin-list-member,
qmin-list-bounds,
qle_reflexivity,
qmax-list-member,
qmax-list-bounds,
member-map,
l_member_set2,
bool_cases_sqequal,
grp_leq_weakening_lt,
decidable__and,
colist-cases,
combine-list-member,
combine-list-rel-and,
qle_transitivity_qorder,
member_map,
list_induction,
cons_member,
colist-ext,
member_singleton,
int_seg_properties,
l_all_single,
decidable__le,
grp_leq_transitivity,
l_all_cons,
decidable__assert,
decidable__less_than'
Lemmas referenced :
decidable__q-constraints,
decidable__equal_int,
iff_weakening_equal,
decidable__l_all-better-extract,
decidable__q-rel,
decidable__l_exists_better-extract,
decidable__cand,
decidable__not,
decidable__equal_rationals,
l_exists_iff,
product-map,
member_filter,
member_append,
decidable__qless,
sq_stable__and,
sq_stable__equal,
q-min-exists,
q-max-exists,
member_map_filter,
l_all_iff,
iff_transitivity,
iff_weakening_uiff,
eqtt_to_assert,
assert_of_band,
assert_of_eq_int,
qle_weakening_lt_qorder,
q-constraint-times,
bool_cases,
uiff_transitivity2,
qadd_preserves_qle,
qadd_preserves_qless,
decidable__int_equal,
decidable__l_all,
decidable__l_exists,
decidable__and2,
decidable__implies,
decidable__false,
select_member,
list-cases,
l_all_nil,
qmin-list-member,
qmin-list-bounds,
qle_reflexivity,
qmax-list-member,
qmax-list-bounds,
member-map,
l_member_set2,
bool_cases_sqequal,
grp_leq_weakening_lt,
decidable__and,
colist-cases,
combine-list-member,
combine-list-rel-and,
qle_transitivity_qorder,
member_map,
list_induction,
cons_member,
colist-ext,
member_singleton,
int_seg_properties,
l_all_single,
decidable__le,
grp_leq_transitivity,
l_all_cons,
decidable__assert,
decidable__less_than'
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}k:\mBbbN{}. \mforall{}A:(\mBbbN{} {}\mrightarrow{} \mBbbQ{} \mtimes{} \mBbbZ{}) List. Dec(\mexists{}y:\mBbbQ{} List [q-constraints(k;A;y)])
Date html generated:
2020_05_20-AM-09_30_43
Last ObjectModification:
2020_01_17-AM-09_48_53
Theory : rationals
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