Nuprl Lemma : mu-dec-property
∀[A:Type]. ∀[P:A ⟶ ℕ ⟶ ℙ].
∀d:a:A ⟶ k:ℕ ⟶ Dec(P[a;k]). ∀a:A. ((↓∃k:ℕ. P[a;k]) ⇒ {P[a;mu-dec(d;a)] ∧ (∀i:ℕmu-dec(d;a). (¬P[a;i]))})
Proof
Definitions occuring in Statement :
mu-dec: mu-dec(d;a),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
so_apply: x[s1;s2],
guard: {T},
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
cand: A c∧ B,
mu-dec: mu-dec(d;a),
decidable: Dec(P),
or: P ∨ Q,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_stable: SqStable(P)
Lemmas referenced :
true_wf,
sq_stable__not,
sq_stable__all,
sq_stable_from_decidable,
sq_stable__and,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
le_wf,
nat_properties,
int_seg_properties,
assert_wf,
equal_wf,
isl_wf,
mu-property,
false_wf,
int_seg_subtype_nat,
not_wf,
int_seg_wf,
all_wf,
decidable_wf,
nat_wf,
exists_wf,
squash_wf,
mu-dec_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_isectElimination,
introduction,
applyEquality,
imageElimination,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
because_Cache,
independent_pairFormation,
productElimination,
dependent_pairFormation,
unionEquality,
unionElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
setEquality,
intEquality,
int_eqEquality,
voidEquality,
computeAll,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}].
\mforall{}d:a:A {}\mrightarrow{} k:\mBbbN{} {}\mrightarrow{} Dec(P[a;k]). \mforall{}a:A.
((\mdownarrow{}\mexists{}k:\mBbbN{}. P[a;k]) {}\mRightarrow{} \{P[a;mu-dec(d;a)] \mwedge{} (\mforall{}i:\mBbbN{}mu-dec(d;a). (\mneg{}P[a;i]))\})
Date html generated:
2016_05_14-AM-07_30_20
Last ObjectModification:
2016_01_14-PM-09_58_47
Theory : int_2
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