Nuprl Lemma : mu-dec-in-bar-nat
∀[A:Type]. ∀[P:A ⟶ ℕ ⟶ ℙ]. ∀[d:a:A ⟶ k:ℕ ⟶ Dec(P[a;k])]. ∀[a:A]. (mu-dec(d;a) ∈ partial(ℕ))
Proof
Definitions occuring in Statement :
mu-dec: mu-dec(d;a),
partial: partial(T),
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s1;s2],
prop: ℙ,
mu-dec: mu-dec(d;a),
mu: mu(f),
mu-ge: mu-ge(f;n),
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
has-value: (a)↓,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
isl: isl(x),
bool: 𝔹
Lemmas referenced :
istype-universe,
nat_wf,
decidable_wf,
fixpoint-induction-bottom,
partial_wf,
set-value-type,
le_wf,
istype-int,
int-value-type,
nat-mono,
bottom_wf_function,
ifthenelse_wf-partial,
inclusion-partial,
value-type-has-value,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
btrue_wf,
bool_wf,
union-value-type,
unit_wf2,
bfalse_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :functionIsType,
Error :universeIsType,
applyEquality,
universeEquality,
because_Cache,
Error :isect_memberFormation_alt,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
functionEquality,
independent_isectElimination,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
Error :lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
callbyvalueReduce,
addEquality,
setElimination,
rename,
Error :dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
Error :inhabitedIsType,
Error :equalityIsType1
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]. (mu-dec(d;a) \mmember{} partial(\mBbbN{}))
Date html generated:
2019_06_20-PM-01_17_31
Last ObjectModification:
2018_10_06-AM-11_21_35
Theory : int_2
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