Nuprl Lemma : lower-rc-face-is-face
∀k:ℕ. ∀c:ℚCube(k). ∀j:ℕk. lower-rc-face(c;j) ≤ c
Proof
Definitions occuring in Statement :
lower-rc-face: lower-rc-face(c;j),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
nat: ℕ,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
exists: ∃x:A. B[x],
bfalse: ff,
or: P ∨ Q,
rat-interval-face: I ≤ J,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
pi1: fst(t),
rational-interval: ℚInterval,
implies: P ⇒ Q,
rational-cube: ℚCube(k),
member: t ∈ T,
lower-rc-face: lower-rc-face(c;j),
rat-cube-face: c ≤ d,
all: ∀x:A. B[x]
Lemmas referenced :
istype-nat,
rational-cube_wf,
int_seg_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
rat-point-interval_wf,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf
Rules used in proof :
natural_numberEquality,
universeIsType,
inrFormation_alt,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation_alt,
because_Cache,
equalityIstype,
unionIsType,
inlFormation_alt,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
rename,
setElimination,
isectElimination,
extract_by_obid,
introduction,
productElimination,
thin,
hypothesis,
inhabitedIsType,
hypothesisEquality,
sqequalHypSubstitution,
applyEquality,
cut,
sqequalRule,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k). \mforall{}j:\mBbbN{}k. lower-rc-face(c;j) \mleq{} c
Date html generated:
2019_10_29-AM-07_56_08
Last ObjectModification:
2019_10_17-PM-03_06_11
Theory : rationals
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