Nuprl Lemma : half-cube-of-face
∀k:ℕ. ∀a,b,c,d,e:ℚCube(k).
(Compatible(a;e)
⇒ b ≤ e
⇒ d ≤ c
⇒ (b ≤ a) supposing ((↑Inhabited(e)) and (↑Inhabited(a)) and (↑is-half-cube(k;c;a)) and (↑is-half-cube(k;d;b))))
Proof
Definitions occuring in Statement :
compatible-rat-cubes: Compatible(c;d),
inhabited-rat-cube: Inhabited(c),
is-half-cube: is-half-cube(k;h;c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
qge: a ≥ b,
true: True,
squash: ↓T,
cand: A c∧ B,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
guard: {T},
sq_type: SQType(T),
subtype_rel: A ⊆r B,
or: P ∨ Q,
rev_implies: P ⇐ Q,
pi2: snd(t),
pi1: fst(t),
iff: P ⇐⇒ Q,
nat: ℕ,
rat-point-interval: [a],
rat-interval-intersection: I ⋂ J,
rat-interval-face: I ≤ J,
is-half-interval: is-half-interval(I;J),
inhabited-rat-interval: Inhabited(I),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
rat-cube-intersection: c ⋂ d,
rat-cube-face: c ≤ d,
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
prop: ℙ,
compatible-rat-cubes: Compatible(c;d),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
uiff_transitivity3,
uiff_transitivity,
qle_connex,
qmin-eq-iff-1,
qmax-eq-iff-1,
qmin-eq-iff-2,
qmax-eq-iff-2,
qle-qavg-iff-4,
qle_antisymmetry,
qavg-eq-iff-4,
qavg-qle-iff-2,
qmax-eq-iff,
qmin-eq-iff,
qavg-eq-iff-2,
qle_functionality_wrt_implies,
qavg-eq-iff-8,
qavg-eq-iff-3,
qle_weakening_eq_qorder,
qle-qavg-iff-2,
qle-qavg-iff-1,
qavg-qle-iff-1,
qavg-eq-iff-7,
qavg-eq-iff-1,
qle_transitivity_qorder,
qle_reflexivity,
member_wf,
qavg-same,
istype-universe,
true_wf,
squash_wf,
qmin_ub,
qmax_lb,
iff_weakening_equal,
assert-q_le-eq,
assert_of_band,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
subtype_rel_self,
q_le_wf,
qmin_wf,
qmax_wf,
rationals_wf,
equal_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,
assert_wf,
rational-interval_wf,
qavg_wf,
qle_wf,
int_seg_wf,
rat-cube-intersection_wf,
assert-is-half-cube,
assert-inhabited-rat-cube,
istype-nat,
rational-cube_wf,
compatible-rat-cubes_wf,
rat-cube-face_wf,
istype-assert,
inhabited-rat-cube_wf,
is-half-cube_wf,
assert_witness
Rules used in proof :
functionIsType,
functionEquality,
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination,
lambdaEquality_alt,
hyp_replacement,
inrFormation_alt,
promote_hyp,
inlFormation_alt,
isect_memberEquality_alt,
productEquality,
unionEquality,
cumulativity,
instantiate,
unionElimination,
unionIsType,
independent_pairEquality,
because_Cache,
applyLambdaEquality,
productIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
setElimination,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
applyEquality,
dependent_functionElimination,
sqequalRule,
independent_isectElimination,
productElimination,
inhabitedIsType,
universeIsType,
rename,
independent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}a,b,c,d,e:\mBbbQ{}Cube(k).
(Compatible(a;e)
{}\mRightarrow{} b \mleq{} e
{}\mRightarrow{} d \mleq{} c
{}\mRightarrow{} (b \mleq{} a) supposing
((\muparrow{}Inhabited(e)) and
(\muparrow{}Inhabited(a)) and
(\muparrow{}is-half-cube(k;c;a)) and
(\muparrow{}is-half-cube(k;d;b))))
Date html generated:
2019_10_29-AM-07_55_14
Last ObjectModification:
2019_10_24-PM-05_39_25
Theory : rationals
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