Nuprl Lemma : rat-point-in-face
∀[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 d ≤ c and (↑Inhabited(c)))
Proof
Definitions occuring in Statement :
inhabited-rat-cube: Inhabited(c),
rat-point-in-cube: rat-point-in-cube(k;x;c),
rat-cube-face: c ≤ d,
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,
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
implies: P ⇒ Q,
rational-interval: ℚInterval,
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
nat: ℕ,
rat-interval-face: I ≤ J,
inhabited-rat-interval: Inhabited(I),
rat-point-interval: [a],
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
cand: A c∧ B,
guard: {T},
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q
Lemmas referenced :
assert-inhabited-rat-cube,
qle_witness,
rat-point-in-cube_wf,
rat-cube-face_wf,
istype-assert,
inhabited-rat-cube_wf,
rational-cube_wf,
int_seg_wf,
rationals_wf,
istype-nat,
pi2_wf,
pi1_wf_top,
istype-void,
qle_transitivity_qorder,
qle_weakening_eq_qorder,
rational-interval_wf,
subtype_rel_self,
qle_wf,
assert-q_le-eq,
iff_weakening_equal,
q_le_wf
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,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
independent_pairEquality,
functionIsTypeImplies,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
natural_numberEquality,
setElimination,
rename,
unionElimination,
applyLambdaEquality,
voidElimination,
independent_pairFormation,
unionIsType,
productIsType
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 d \mleq{} c and (\muparrow{}Inhabited(c)))
Date html generated:
2020_05_20-AM-09_18_38
Last ObjectModification:
2019_11_02-PM-05_00_04
Theory : rationals
Home
Index