Nuprl Lemma : member-rat-cube-complex-inhabited
∀[k:ℕ]. ∀[c:ℚCube(k)]. ↑Inhabited(c) supposing ∃n:ℕ. ∃K:n-dim-complex. (c ∈ K)
Proof
Definitions occuring in Statement :
rational-cube-complex: n-dim-complex,
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
rational-cube-complex: n-dim-complex,
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
rat-cube-dimension: dim(c),
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top
Lemmas referenced :
sq_stable__assert,
inhabited-rat-cube_wf,
l_all_iff,
equal-wf-base,
rat-cube-dimension_wf,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
le_wf,
l_member_wf,
rational-cube_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
assert_witness,
rational-cube-complex_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
setElimination,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
intEquality,
applyEquality,
minusEquality,
natural_numberEquality,
addEquality,
independent_isectElimination,
setIsType,
inhabitedIsType,
universeIsType,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
unionElimination,
cumulativity,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productIsType,
isectIsTypeImplies
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. \muparrow{}Inhabited(c) supposing \mexists{}n:\mBbbN{}. \mexists{}K:n-dim-complex. (c \mmember{} K)
Date html generated:
2020_05_20-AM-09_21_39
Last ObjectModification:
2019_11_14-PM-02_46_39
Theory : rationals
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