Nuprl Lemma : half-cubes-listable
∀k:ℕ. ∀c:{c:ℚCube(k)| ↑Inhabited(c)} .
(∃L:ℚCube(k) List [(no_repeats(ℚCube(k);L) ∧ (∀h:ℚCube(k). ((h ∈ L) ⇐⇒ ↑is-half-cube(k;h;c))))])
Proof
Definitions occuring in Statement :
inhabited-rat-cube: Inhabited(c),
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
rational-cube: ℚCube(k),
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
is-half-cube: is-half-cube(k;h;c),
bdd-all: bdd-all(n;i.P[i]),
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
btrue: tt,
true: True,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
rational-interval: ℚInterval,
rat-interval-dimension: dim(I),
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
istype: istype(T),
inject: Inj(A;B;f),
l_disjoint: l_disjoint(T;l1;l2),
pi2: snd(t),
pi1: fst(t),
qavg: qavg(a;b),
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
eq_int: (i =z j),
qadd: r + s,
is-half-interval: is-half-interval(I;J),
band: p ∧b q,
inhabited-rat-interval: Inhabited(I)
Lemmas referenced :
rational-cube_wf,
istype-void,
istype-le,
istype-assert,
inhabited-rat-cube_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
list_wf,
no_repeats_wf,
l_member_wf,
is-half-cube_wf,
istype-less_than,
primrec-wf2,
assert_wf,
sq_exists_wf,
iff_wf,
istype-nat,
cons_wf,
int_seg_properties,
int_seg_wf,
nil_wf,
no_repeats_singleton,
member_singleton,
subtype_rel_function,
rational-interval_wf,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
assert-inhabited-rat-cube,
decidable__equal_int,
rat-interval-dimension_wf,
decidable__lt,
q_less_wf,
eqtt_to_assert,
assert-q_less-eq,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
qless_wf,
int_subtype_base,
lt_int_wf,
assert_of_lt_int,
qavg_wf,
append_wf,
map_wf,
no_repeats-append,
no_repeats_map,
subtype_rel_dep_function,
iff_weakening_uiff,
less_than_wf,
member-map,
qmul_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rationals_wf,
qmul-qdiv-cancel,
int-subtype-rationals,
qadd_wf,
assert-qeq,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
qadd_ac_1_q,
qadd_comm_q,
qinverse_q,
mon_ident_q,
q_distrib,
qmul_one_qrng,
member_append,
assert_functionality_wrt_uiff,
intformeq_wf,
int_formula_prop_eq_lemma,
assert-is-half-cube,
set_subtype_base,
lelt_wf,
is-half-interval_wf,
bor_wf,
qeq_wf2,
bool_cases,
band_wf,
btrue_wf,
bfalse_wf,
member_wf,
iff_transitivity,
assert_of_bor,
assert_of_band,
assert_elim,
ifthenelse_wf,
q_le_wf,
assert-q_le-eq,
qle-iff,
qle_wf,
bnot_wf,
not_wf,
qavg-same,
assert_of_bnot,
equal-wf-T-base,
uiff_transitivity2,
qmul-preserves-eq,
qdiv_wf,
qmul_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
setIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
voidElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
functionIsType,
productIsType,
because_Cache,
functionEquality,
setEquality,
productEquality,
inhabitedIsType,
dependent_set_memberFormation_alt,
productElimination,
functionExtensionality,
imageElimination,
addEquality,
minusEquality,
multiplyEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
intEquality,
baseClosed,
sqequalBase,
closedConclusion,
independent_pairEquality,
applyLambdaEquality,
universeEquality,
imageMemberEquality,
unionIsType,
inlFormation_alt,
inrFormation_alt,
hyp_replacement,
unionEquality
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\{c:\mBbbQ{}Cube(k)| \muparrow{}Inhabited(c)\} .
(\mexists{}L:\mBbbQ{}Cube(k) List [(no\_repeats(\mBbbQ{}Cube(k);L) \mwedge{} (\mforall{}h:\mBbbQ{}Cube(k). ((h \mmember{} L) \mLeftarrow{}{}\mRightarrow{} \muparrow{}is-half-cube(k;h;c))))])
Date html generated:
2020_05_20-AM-09_19_41
Last ObjectModification:
2019_11_02-PM-07_44_47
Theory : rationals
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