Nuprl Lemma : rat-interval-face-dimension
∀J:ℚInterval. ((↑Inhabited(J)) ⇒ (∀I:ℚInterval. (I ≤ J ⇒ ((I = J ∈ ℚInterval) ∨ (dim(I) = (dim(J) - 1) ∈ ℤ)))))
Proof
Definitions occuring in Statement :
rat-interval-dimension: dim(I),
inhabited-rat-interval: Inhabited(I),
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rat-interval-dimension: dim(I),
false: False,
not: ¬A,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
rat-point-interval: [a],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
member: t ∈ T,
inhabited-rat-interval: Inhabited(I),
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
all: ∀x:A. B[x]
Lemmas referenced :
int_seg_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
rat-interval-dimension-single,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert-q_less-eq,
eqtt_to_assert,
q_less_wf,
qle-iff,
assert-q_le-eq,
subtract_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-interval-dimension_wf,
istype-int,
iff_weakening_equal,
istype-universe,
true_wf,
squash_wf,
equal_wf,
q_le_wf,
istype-assert,
rational-interval_wf,
subtype_rel_self,
rat-point-interval_wf,
decidable__equal_rationals
Rules used in proof :
independent_pairFormation,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
inrFormation_alt,
cumulativity,
promote_hyp,
dependent_pairFormation_alt,
equalityElimination,
voidElimination,
sqequalBase,
rename,
setElimination,
intEquality,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
instantiate,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality_alt,
inlFormation_alt,
universeIsType,
applyEquality,
independent_pairEquality,
because_Cache,
isectElimination,
inhabitedIsType,
equalityIstype,
unionIsType,
unionElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}J:\mBbbQ{}Interval. ((\muparrow{}Inhabited(J)) {}\mRightarrow{} (\mforall{}I:\mBbbQ{}Interval. (I \mleq{} J {}\mRightarrow{} ((I = J) \mvee{} (dim(I) = (dim(J) - 1))))))
Date html generated:
2019_10_29-AM-07_48_08
Last ObjectModification:
2019_10_18-AM-10_52_31
Theory : rationals
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