Nuprl Lemma : compatible-half-cubes
∀k:ℕ. ∀a,b,c,d:ℚCube(k). ((↑is-half-cube(k;c;a)) ⇒ (↑is-half-cube(k;d;b)) ⇒ Compatible(a;b) ⇒ Compatible(c;d))
Proof
Definitions occuring in Statement :
compatible-rat-cubes: Compatible(c;d),
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
compatible-rat-cubes: Compatible(c;d),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rat-cube-intersection: c ⋂ d,
prop: ℙ,
nat: ℕ,
rational-cube: ℚCube(k),
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
rat-cube-face: c ≤ d,
cand: A c∧ B,
guard: {T},
rational-interval: ℚInterval,
rat-interval-intersection: I ⋂ J,
inhabited-rat-interval: Inhabited(I),
rat-interval-face: I ≤ J,
is-half-interval: is-half-interval(I;J),
or: P ∨ Q,
rat-point-interval: [a],
pi1: fst(t),
pi2: snd(t),
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
qge: a ≥ b,
sq_type: SQType(T),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi
Lemmas referenced :
assert-inhabited-rat-cube,
rat-cube-intersection_wf,
istype-assert,
inhabited-rat-cube_wf,
compatible-rat-cubes_wf,
int_seg_wf,
is-half-interval_wf,
iff_weakening_uiff,
assert_wf,
is-half-cube_wf,
assert-is-half-cube,
rational-cube_wf,
istype-nat,
inhabited-intersection-half-cubes,
qle_wf,
qavg_wf,
qle-qavg-iff-1,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rationals_wf,
subtype_rel_self,
iff_weakening_equal,
rational-interval_wf,
qmax_wf,
qmin_wf,
rat-point-interval_wf,
qavg-qle-iff-1,
qmin_ub,
qmax_lb,
qmin-eq-iff,
qmax-eq-iff,
qle_antisymmetry,
qle-qavg-iff-2,
qavg-eq-iff-1,
qavg-eq-iff-2,
qle-qavg-iff-4,
qle-qavg-iff-6,
qle-qavg-iff-5,
qavg-eq-iff-3,
qavg-eq-iff-4,
implies_weakening_uimplies,
qle_functionality_wrt_implies,
qle_weakening_eq_qorder,
qle-qavg-iff-3,
member_wf,
qavg-same,
rev_implies_wf,
qavg-qle-iff-2,
qavg-eq-iff-7,
qavg-eq-iff-5,
qavg-eq-iff-6,
qavg-eq-iff-8,
bor_wf,
qeq_wf2,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert-qeq,
bfalse_wf,
q_le_wf,
iff_transitivity,
assert_of_bor,
assert_of_band,
assert-q_le-eq,
qmax-eq-iff-2,
qmin-eq-iff-2,
qmax-eq-iff-1,
qmin-eq-iff-1,
qle_connex
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
introduction,
extract_by_obid,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
universeIsType,
functionIsType,
natural_numberEquality,
setElimination,
rename,
applyEquality,
independent_functionElimination,
functionEquality,
dependent_functionElimination,
independent_pairFormation,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
unionElimination,
hyp_replacement,
applyLambdaEquality,
promote_hyp,
dependent_set_memberEquality_alt,
productIsType,
independent_pairEquality,
lambdaEquality_alt,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
productEquality,
unionEquality,
unionIsType,
inlFormation_alt,
inrFormation_alt,
cumulativity,
isect_memberEquality_alt
Latex:
\mforall{}k:\mBbbN{}. \mforall{}a,b,c,d:\mBbbQ{}Cube(k).
((\muparrow{}is-half-cube(k;c;a)) {}\mRightarrow{} (\muparrow{}is-half-cube(k;d;b)) {}\mRightarrow{} Compatible(a;b) {}\mRightarrow{} Compatible(c;d))
Date html generated:
2020_05_20-AM-09_20_37
Last ObjectModification:
2020_01_04-PM-10_55_23
Theory : rationals
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