Nuprl Lemma : assert-is-half-cube
∀[k:ℕ]. ∀[h,c:ℚCube(k)]. uiff(↑is-half-cube(k;h;c);∀i:ℕk. (↑is-half-interval(h i;c i)))
Proof
Definitions occuring in Statement :
is-half-cube: is-half-cube(k;h;c),
is-half-interval: is-half-interval(I;J),
rational-cube: ℚCube(k),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
rational-cube: ℚCube(k),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
is-half-cube: is-half-cube(k;h;c)
Lemmas referenced :
istype-nat,
rational-cube_wf,
is-half-cube_wf,
bdd-all_wf,
assert-bdd-all,
istype-assert,
is-half-interval_wf,
assert_witness,
int_seg_wf
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
promote_hyp,
independent_isectElimination,
productElimination,
because_Cache,
functionIsType,
inhabitedIsType,
functionIsTypeImplies,
independent_functionElimination,
applyEquality,
dependent_functionElimination,
lambdaEquality_alt,
sqequalRule,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
universeIsType,
lambdaFormation_alt,
introduction,
isect_memberFormation_alt,
independent_pairFormation,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[h,c:\mBbbQ{}Cube(k)]. uiff(\muparrow{}is-half-cube(k;h;c);\mforall{}i:\mBbbN{}k. (\muparrow{}is-half-interval(h i;c i)))
Date html generated:
2019_10_29-AM-07_50_59
Last ObjectModification:
2019_10_21-PM-01_14_18
Theory : rationals
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