Nuprl Lemma : half-cubes_wf
∀[k:ℕ]
(half-cubes(k) ∈ c:{c:ℚCube(k)| ↑Inhabited(c)} ⟶ {L:ℚCube(k) List|
no_repeats(ℚCube(k);L) ∧ (∀h:ℚCube(k). ((h ∈ L) ⇐⇒ ↑is-half-cube(k;h;c)))} )
Proof
Definitions occuring in Statement :
half-cubes: half-cubes(k),
inhabited-rat-cube: Inhabited(c),
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
nat: ℕ,
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
so_apply: x[s],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
half-cubes-listable-ext,
primtailrec: primtailrec(n;i;b;f),
primrec: primrec(n;b;c),
half-cubes: half-cubes(k),
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_exists: ∃x:A [B[x]],
all: ∀x:A. B[x]
Lemmas referenced :
istype-nat,
is-half-cube_wf,
l_member_wf,
iff_wf,
no_repeats_wf,
list_wf,
sq_exists_wf,
inhabited-rat-cube_wf,
assert_wf,
rational-cube_wf,
nat_wf,
subtype_rel_self,
half-cubes-listable-ext
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
universeIsType,
rename,
setElimination,
productEquality,
lambdaEquality_alt,
hypothesisEquality,
setEquality,
functionEquality,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]
(half-cubes(k) \mmember{} c:\{c:\mBbbQ{}Cube(k)| \muparrow{}Inhabited(c)\} {}\mrightarrow{} \{L:\mBbbQ{}Cube(k) List|
no\_repeats(\mBbbQ{}Cube(k);L)
\mwedge{} (\mforall{}h:\mBbbQ{}Cube(k). ((h \mmember{} L) \mLeftarrow{}{}\mRightarrow{} \muparrow{}is-half-cube(k;h;c)))\} )
Date html generated:
2019_10_29-AM-07_53_14
Last ObjectModification:
2019_10_21-PM-02_59_47
Theory : rationals
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