Nuprl Lemma : in-rat-cube-face
∀[k:ℕ]. ∀[p:ℝ^k]. ∀[c,f:ℚCube(k)]. (in-rat-cube(k;p;c)) supposing (in-rat-cube(k;p;f) and (↑Inhabited(c)) and f ≤ c)
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
real-vec: ℝ^n,
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
nat: ℕ,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
guard: {T},
cand: A c∧ B,
inhabited-rat-interval: Inhabited(I),
prop: ℙ,
top: Top,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
rat-point-interval: [a],
pi2: snd(t),
pi1: fst(t),
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
implies: P ⇒ Q,
real-vec: ℝ^n,
rat-cube-face: c ≤ d,
all: ∀x:A. B[x],
in-rat-cube: in-rat-cube(k;p;c),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq-rat2real,
istype-nat,
real-vec_wf,
rational-cube_wf,
rat-cube-face_wf,
inhabited-rat-cube_wf,
in-rat-cube_wf,
le_witness_for_triv,
int_seg_wf,
subtype_rel_self,
rational-interval_wf,
q_le_wf,
istype-assert,
iff_weakening_equal,
assert-q_le-eq,
qle_wf,
rat2real_wf,
rleq_wf,
istype-void,
pi1_wf_top,
rationals_wf,
pi2_wf,
assert-inhabited-rat-cube
Rules used in proof :
isectIsTypeImplies,
functionIsTypeImplies,
rename,
setElimination,
natural_numberEquality,
equalityIstype,
unionIsType,
productIsType,
because_Cache,
equalityTransitivity,
independent_functionElimination,
universeIsType,
independent_pairFormation,
productEquality,
hyp_replacement,
equalitySymmetry,
promote_hyp,
voidElimination,
isect_memberEquality_alt,
independent_pairEquality,
lambdaEquality_alt,
applyLambdaEquality,
unionElimination,
sqequalRule,
inhabitedIsType,
applyEquality,
dependent_functionElimination,
lambdaFormation_alt,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[p:\mBbbR{}\^{}k]. \mforall{}[c,f:\mBbbQ{}Cube(k)].
(in-rat-cube(k;p;c)) supposing (in-rat-cube(k;p;f) and (\muparrow{}Inhabited(c)) and f \mleq{} c)
Date html generated:
2019_10_30-AM-10_12_51
Last ObjectModification:
2019_10_27-AM-00_17_24
Theory : real!vectors
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