Nuprl Lemma : inhabited-lower-rc-face
∀k:ℕ. ∀c:ℚCube(k). ∀j:ℕk. ((↑Inhabited(c j)) ⇒ Inhabited(lower-rc-face(c;j)) = Inhabited(c))
Proof
Definitions occuring in Statement :
lower-rc-face: lower-rc-face(c;j),
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k),
inhabited-rat-interval: Inhabited(I),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
false: False,
assert: ↑b,
bnot: ¬bb,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
bfalse: ff,
ifthenelse: if b then t else f fi ,
true: True,
nequal: a ≠ b ∈ T ,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
subtype_rel: A ⊆r B,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
prop: ℙ,
rev_implies: P ⇐ Q,
pi1: fst(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
int_seg: {i..j-},
and: P ∧ Q,
iff: P ⇐⇒ Q,
lower-rc-face: lower-rc-face(c;j),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
assert_of_tt,
inhabited-rat-point-interval,
assert_of_eq_int,
eqtt_to_assert,
iff_weakening_equal,
bfalse_wf,
eq_int_eq_false,
istype-universe,
true_wf,
squash_wf,
equal_wf,
bool_subtype_base,
bool_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
assert-inhabited-rat-cube,
subtype_rel_self,
int_seg_wf,
assert_wf,
iff_weakening_uiff,
rat-point-interval_wf,
rational-interval_wf,
eq_int_wf,
ifthenelse_wf,
inhabited-rat-interval_wf,
istype-assert,
lower-rc-face_wf,
inhabited-rat-cube_wf,
iff_imp_equal_bool
Rules used in proof :
voidElimination,
dependent_pairFormation_alt,
equalityElimination,
baseClosed,
imageMemberEquality,
universeEquality,
lambdaEquality_alt,
intEquality,
cumulativity,
instantiate,
unionElimination,
natural_numberEquality,
universeIsType,
promote_hyp,
imageElimination,
functionEquality,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
productElimination,
inhabitedIsType,
applyEquality,
rename,
setElimination,
functionIsType,
because_Cache,
independent_pairFormation,
sqequalRule,
independent_isectElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k). \mforall{}j:\mBbbN{}k. ((\muparrow{}Inhabited(c j)) {}\mRightarrow{} Inhabited(lower-rc-face(c;j)) = Inhabited(c))
Date html generated:
2019_10_29-AM-07_56_15
Last ObjectModification:
2019_10_17-PM-05_03_11
Theory : rationals
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