Nuprl Lemma : member-rat-complex-subdiv
∀k:ℕ. ∀K:{c:ℚCube(k)| ↑Inhabited(c)} List. ∀c:ℚCube(k).
((c ∈ (K)') ⇐⇒ ∃a:ℚCube(k). ((a ∈ K) ∧ (↑is-half-cube(k;c;a))))
Proof
Definitions occuring in Statement :
rat-complex-subdiv: (K)',
inhabited-rat-cube: Inhabited(c),
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k),
l_member: (x ∈ l),
list: T List,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
l_member: (x ∈ l),
guard: {T},
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
squash: ↓T,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rat-complex-subdiv: (K)',
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
l_member-set2,
istype-less_than,
istype-le,
select_member,
length_wf,
less_than_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
select_wf,
sq_stable__assert,
decidable__equal_rc,
sq_stable__l_member,
sq_stable__iff,
iff_wf,
sq_stable__all,
assert_functionality_wrt_uiff,
iff_weakening_equal,
subtype_rel_self,
istype-universe,
true_wf,
squash_wf,
l_member-settype,
istype-nat,
is-half-cube_wf,
subtype_rel_list,
concat_wf,
l_member_wf,
member_map,
istype-assert,
half-cubes-of_wf,
list_wf,
inhabited-rat-cube_wf,
assert_wf,
rational-cube_wf,
map_wf,
member-concat
Rules used in proof :
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
unionElimination,
natural_numberEquality,
universeEquality,
instantiate,
independent_isectElimination,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
equalityIstype,
universeIsType,
productIsType,
promote_hyp,
dependent_pairFormation_alt,
independent_functionElimination,
productElimination,
setIsType,
sqequalRule,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
rename,
setElimination,
applyEquality,
lambdaEquality_alt,
hypothesis,
hypothesisEquality,
setEquality,
dependent_functionElimination,
because_Cache,
thin,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}K:\{c:\mBbbQ{}Cube(k)| \muparrow{}Inhabited(c)\} List. \mforall{}c:\mBbbQ{}Cube(k).
((c \mmember{} (K)') \mLeftarrow{}{}\mRightarrow{} \mexists{}a:\mBbbQ{}Cube(k). ((a \mmember{} K) \mwedge{} (\muparrow{}is-half-cube(k;c;a))))
Date html generated:
2019_10_29-AM-07_59_21
Last ObjectModification:
2019_10_21-PM-03_47_01
Theory : rationals
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