Nuprl Lemma : partition-choice-subtype
∀[p:ℝ List]. ({f:ℕ||p|| - 1 ⟶ ℝ| is-partition-choice(p;f)} ⊆r partition-choice(p))
Proof
Definitions occuring in Statement :
partition-choice: partition-choice(p),
is-partition-choice: is-partition-choice(p;x),
real: ℝ,
length: ||as||,
list: T List,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
partition-choice: partition-choice(p),
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
is-partition-choice: is-partition-choice(p;x),
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q)
Lemmas referenced :
list_wf,
is-partition-choice_wf,
int_seg_wf,
int_term_value_add_lemma,
itermAdd_wf,
false_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
intformless_wf,
subtract-is-int-iff,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
subtract_wf,
int_seg_properties,
real_wf,
select_wf,
rleq_wf,
member_rccint_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
setElimination,
thin,
rename,
hypothesisEquality,
functionExtensionality,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
productElimination,
independent_pairFormation,
because_Cache,
dependent_set_memberEquality,
applyEquality,
productEquality,
isectElimination,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
addEquality,
setEquality,
functionEquality,
axiomEquality
Latex:
\mforall{}[p:\mBbbR{} List]. (\{f:\mBbbN{}||p|| - 1 {}\mrightarrow{} \mBbbR{}| is-partition-choice(p;f)\} \msubseteq{}r partition-choice(p))
Date html generated:
2016_05_18-AM-09_03_53
Last ObjectModification:
2016_01_17-AM-02_35_20
Theory : reals
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