Nuprl Lemma : member-iter-subdiv-sub-cube
∀k,n,j:ℕ. ∀K:n-dim-complex. ∀c:ℚCube(k). ((c ∈ K'^(j)) ⇒ (∃a:ℚCube(k). ((a ∈ K) ∧ rat-sub-cube(k;c;a))))
Proof
Definitions occuring in Statement :
rat-complex-iter-subdiv: K'^(n),
rational-cube-complex: n-dim-complex,
rat-sub-cube: rat-sub-cube(k;a;b),
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
rational-cube-complex: n-dim-complex,
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
rat-complex-iter-subdiv: K'^(n),
cand: A c∧ B,
rat-sub-cube: rat-sub-cube(k;a;b),
rational-cube: ℚCube(k),
rational-interval: ℚInterval,
rat-sub-interval: rat-sub-interval(I;J),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
l_member_wf,
rational-cube_wf,
rat-complex-iter-subdiv_wf,
istype-void,
istype-le,
rational-cube-complex_wf,
nat_properties,
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,
rat-sub-cube_wf,
istype-less_than,
primrec-wf2,
istype-nat,
primrec0_lemma,
int_seg_wf,
qle_reflexivity,
qless_wf,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
member-rat-complex-subdiv-sub-cube,
rat-sub-cube_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
voidElimination,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
because_Cache,
functionIsType,
productIsType,
setIsType,
functionEquality,
productEquality,
productElimination,
equalityIstype,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}k,n,j:\mBbbN{}. \mforall{}K:n-dim-complex. \mforall{}c:\mBbbQ{}Cube(k).
((c \mmember{} K'\^{}(j)) {}\mRightarrow{} (\mexists{}a:\mBbbQ{}Cube(k). ((a \mmember{} K) \mwedge{} rat-sub-cube(k;c;a))))
Date html generated:
2020_05_20-AM-09_24_32
Last ObjectModification:
2019_11_14-PM-11_00_21
Theory : rationals
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