Nuprl Lemma : member-face-complex
∀k:ℕ. ∀K:ℚCube(k) List. ∀f:ℚCube(k).
((f ∈ face-complex(k;K)) ⇐⇒ ∃c:ℚCube(k). ((c ∈ K) ∧ (↑Inhabited(c)) ∧ (f ∈ rat-cube-faces(k;c))))
Proof
Definitions occuring in Statement :
face-complex: face-complex(k;L),
rat-cube-faces: rat-cube-faces(k;c),
inhabited-rat-cube: Inhabited(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
Definitions unfolded in proof :
assert: ↑b,
bnot: ¬bb,
false: False,
not: ¬A,
true: True,
squash: ↓T,
cand: A c∧ B,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
nat: ℕ,
rev_implies: P ⇐ Q,
respects-equality: respects-equality(S;T),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
bfalse: ff,
uimplies: b supposing a,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
face-complex: face-complex(k;L),
all: ∀x:A. B[x]
Lemmas referenced :
bool_cases_sqequal,
ifthenelse_wf,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
le_wf,
length_wf_nat,
iff_weakening_equal,
subtype_rel_self,
istype-universe,
true_wf,
squash_wf,
assert_of_bnot,
eqff_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
istype-nat,
remove-repeats_wf,
rc-deq_wf,
member-remove-repeats,
list_wf,
concat_wf,
member-map,
map_wf,
member-concat,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
subtype_rel_list,
inhabited-rat-cube_wf,
istype-assert,
respects-equality-set-trivial2,
equal-wf-base,
respects-equality-list,
subtract_wf,
rat-cube-dimension_wf,
equal_wf,
rat-cube-face_wf,
nil_wf,
rat-cube-faces_wf,
eqtt_to_assert,
rational-cube_wf,
l_member_wf
Rules used in proof :
voidElimination,
applyLambdaEquality,
hyp_replacement,
dependent_set_memberEquality_alt,
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation_alt,
sqequalBase,
setIsType,
addEquality,
minusEquality,
independent_functionElimination,
dependent_functionElimination,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
rename,
setElimination,
lambdaEquality_alt,
applyEquality,
intEquality,
productEquality,
setEquality,
independent_isectElimination,
equalityElimination,
unionElimination,
inhabitedIsType,
equalityIstype,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
universeIsType,
because_Cache,
productIsType,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
sqequalRule,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}K:\mBbbQ{}Cube(k) List. \mforall{}f:\mBbbQ{}Cube(k).
((f \mmember{} face-complex(k;K)) \mLeftarrow{}{}\mRightarrow{} \mexists{}c:\mBbbQ{}Cube(k). ((c \mmember{} K) \mwedge{} (\muparrow{}Inhabited(c)) \mwedge{} (f \mmember{} rat-cube-faces(k;c))))
Date html generated:
2019_10_29-AM-07_57_48
Last ObjectModification:
2019_10_19-PM-10_12_33
Theory : rationals
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