Nuprl Lemma : decidable__q-constraints2
∀k:ℕ. ∀A:(ℚ List × ℤ × (ℚ List)) List. Dec(∃y:ℚ List [q-sat-constraints(k;A;y)])
Proof
Definitions occuring in Statement :
q-sat-constraints: q-sat-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]],
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
q-sat-constraints: q-sat-constraints(k;A;y),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
spreadn: spread3,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
q-constraints: q-constraints(k;A;y),
l_all: (∀x∈L.P[x]),
q-rel: q-rel(r;x),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
pi2: snd(t),
pi1: fst(t),
qsub: r - s,
rev_implies: P ⇐ Q,
label: ...$L... t,
le: A ≤ B
Lemmas referenced :
decidable__q-constraints-opt,
list_wf,
rationals_wf,
nat_wf,
qsub_wf,
select?_wf,
int-subtype-rationals,
normalize-constraints_wf,
map_wf,
decidable_wf,
squash_wf,
true_wf,
sq_exists_wf,
q-constraints_wf,
normalize-constraints-eq,
iff_weakening_equal,
equal_wf,
length_wf,
l_all_wf2,
l_member_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
q-linear_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
qle_wf,
qless_wf,
not_wf,
length-map,
int_seg_wf,
subtype_rel_list,
top_wf,
select_wf,
int_seg_properties,
itermConstant_wf,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
select-map,
equal-wf-base-T,
qadd_wf,
qmul_wf,
q-linear-times,
q-linear-sum,
equal-wf-base,
int_subtype_base,
assert_wf,
bnot_wf,
qadd_preserves_qle,
qadd_preserves_qless,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
qadd_comm_q,
qadd_inv_assoc_q,
mon_ident_q,
less_than_wf,
length-map-sq,
lelt_wf,
q-rel_wf,
uiff_transitivity2,
qinverse_q
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
productEquality,
hypothesis,
intEquality,
lambdaEquality,
productElimination,
independent_pairEquality,
natural_numberEquality,
applyEquality,
because_Cache,
functionEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
cumulativity,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
unionElimination,
inlFormation,
setElimination,
rename,
dependent_set_memberEquality,
equalityElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
instantiate,
setEquality,
inrFormation,
hyp_replacement,
applyLambdaEquality,
minusEquality,
baseApply,
closedConclusion,
impliesFunctionality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}k:\mBbbN{}. \mforall{}A:(\mBbbQ{} List \mtimes{} \mBbbZ{} \mtimes{} (\mBbbQ{} List)) List. Dec(\mexists{}y:\mBbbQ{} List [q-sat-constraints(k;A;y)])
Date html generated:
2018_05_22-AM-00_25_45
Last ObjectModification:
2017_07_26-PM-06_56_01
Theory : rationals
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