Nuprl Lemma : face-of-half-cube
∀k:ℕ. ∀a,b,c:ℚCube(k). (b ≤ c ⇒ (↑is-half-cube(k;a;b)) ⇒ (∃!d:ℚCube(k). ((↑is-half-cube(k;d;c)) ∧ a ≤ d)))
Proof
Definitions occuring in Statement :
is-half-cube: is-half-cube(k;h;c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
exists!: ∃!x:T. P[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
rev_uimplies: rev_uimplies(P;Q),
label: ...$L... t,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
is-half-interval: is-half-interval(I;J),
cand: A c∧ B,
exists: ∃x:A. B[x],
exists!: ∃!x:T. P[x],
pi1: fst(t),
pi2: snd(t),
top: Top,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
rat-point-interval: [a],
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
nat: ℕ,
rational-cube: ℚCube(k),
prop: ℙ,
rat-cube-face: c ≤ d,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
qavg-eq-iff-2,
qavg-eq-iff-7,
qavg-eq-iff-3,
rat-interval-face-self,
qavg-eq-iff-4,
qavg-same,
qavg-eq-iff-1,
istype-universe,
assert_functionality_wrt_uiff,
assert_of_band,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
equal_wf,
member_wf,
bfalse_wf,
assert-qeq,
btrue_wf,
band_wf,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
qeq_wf2,
bor_wf,
rat-point-interval_wf,
iff_weakening_equal,
subtype_rel_self,
rational-interval_wf,
true_wf,
squash_wf,
qavg_wf,
half-point-interval,
assert_wf,
istype-void,
pi1_wf_top,
rationals_wf,
pi2_wf,
int_seg_wf,
rat-interval-face_wf,
is-half-interval_wf,
istype-nat,
rational-cube_wf,
rat-cube-face_wf,
is-half-cube_wf,
istype-assert,
assert-is-half-cube
Rules used in proof :
functionExtensionality,
inrFormation_alt,
productEquality,
unionEquality,
cumulativity,
functionIsType,
productIsType,
unionIsType,
inlFormation_alt,
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
imageElimination,
independent_pairFormation,
dependent_pairFormation_alt,
because_Cache,
hyp_replacement,
promote_hyp,
voidElimination,
isect_memberEquality_alt,
independent_pairEquality,
lambdaEquality_alt,
applyLambdaEquality,
unionElimination,
rename,
setElimination,
natural_numberEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
applyEquality,
dependent_functionElimination,
inhabitedIsType,
universeIsType,
sqequalRule,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}a,b,c:\mBbbQ{}Cube(k).
(b \mleq{} c {}\mRightarrow{} (\muparrow{}is-half-cube(k;a;b)) {}\mRightarrow{} (\mexists{}!d:\mBbbQ{}Cube(k). ((\muparrow{}is-half-cube(k;d;c)) \mwedge{} a \mleq{} d)))
Date html generated:
2019_10_29-AM-07_55_22
Last ObjectModification:
2019_10_24-PM-06_04_49
Theory : rationals
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