Nuprl Lemma : rat-point-in-half-cube
∀[k:ℕ]. ∀[x:ℕk ⟶ ℚ]. ∀[c,d:ℚCube(k)].
(rat-point-in-cube(k;x;c)) supposing (rat-point-in-cube(k;x;d) and (↑is-half-cube(k;d;c)))
Proof
Definitions occuring in Statement :
rat-point-in-cube: rat-point-in-cube(k;x;c),
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k),
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rat-point-in-cube: rat-point-in-cube(k;x;c),
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
rational-cube: ℚCube(k),
implies: P ⇒ Q,
rational-interval: ℚInterval,
is-half-interval: is-half-interval(I;J),
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
nat: ℕ,
cand: A c∧ B,
or: P ∨ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b
Lemmas referenced :
assert-is-half-cube,
qle_witness,
rat-point-in-cube_wf,
istype-assert,
is-half-cube_wf,
rational-cube_wf,
int_seg_wf,
rationals_wf,
istype-nat,
qle_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
qavg_wf,
assert_wf,
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,
equal_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_band,
qle_functionality_wrt_implies,
qle_weakening_eq_qorder,
qavg-qle-iff-1,
qle-qavg-iff-1,
qle_transitivity_qorder
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation_alt,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
productElimination,
independent_isectElimination,
applyEquality,
inhabitedIsType,
sqequalRule,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
lambdaEquality_alt,
independent_pairEquality,
functionIsTypeImplies,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
natural_numberEquality,
setElimination,
rename,
unionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_pairFormation,
unionIsType,
productIsType,
cumulativity,
unionEquality,
productEquality,
inlFormation_alt,
promote_hyp,
inrFormation_alt,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbN{}k {}\mrightarrow{} \mBbbQ{}]. \mforall{}[c,d:\mBbbQ{}Cube(k)].
(rat-point-in-cube(k;x;c)) supposing (rat-point-in-cube(k;x;d) and (\muparrow{}is-half-cube(k;d;c)))
Date html generated:
2020_05_20-AM-09_18_18
Last ObjectModification:
2020_01_04-PM-10_27_54
Theory : rationals
Home
Index