Nuprl Lemma : do-apply-mu'
∀[A:Type]. ∀[P:A ⟶ ℕ ⟶ 𝔹]. ∀[d:∀x:A. Dec(∃n:ℕ. (↑(P x n)))]. ∀[x:A].
{(↑(P x do-apply(mu'(P);x))) ∧ (∀[i:ℕdo-apply(mu'(P);x)]. (¬↑(P x i)))} supposing ↑can-apply(mu'(P);x)
Proof
Definitions occuring in Statement :
mu': mu'(P),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
mu': mu'(P),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
p-mu-decider,
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
not: ¬A,
false: False,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
exists: ∃x:A. B[x],
pi1: fst(t),
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
p-mu: p-mu(P;x),
bfalse: ff
Lemmas referenced :
assert_witness,
do-apply_wf,
nat_wf,
mu'_wf,
assert_wf,
le_wf,
int_seg_wf,
subtype_rel_dep_function,
top_wf,
subtype_rel_union,
can-apply_wf,
all_wf,
decidable_wf,
exists_wf,
bool_wf,
p-mu-decider,
p-mu_wf,
true_wf,
false_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalRule,
sqequalHypSubstitution,
independent_functionElimination,
hypothesis,
productElimination,
independent_pairEquality,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
independent_isectElimination,
isect_memberEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
intEquality,
unionEquality,
lambdaFormation,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
instantiate,
isectEquality,
cumulativity,
functionExtensionality,
unionElimination,
independent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[d:\mforall{}x:A. Dec(\mexists{}n:\mBbbN{}. (\muparrow{}(P x n)))]. \mforall{}[x:A].
\{(\muparrow{}(P x do-apply(mu'(P);x))) \mwedge{} (\mforall{}[i:\mBbbN{}do-apply(mu'(P);x)]. (\mneg{}\muparrow{}(P x i)))\}
supposing \muparrow{}can-apply(mu'(P);x)
Date html generated:
2018_05_21-PM-06_29_55
Last ObjectModification:
2018_05_19-PM-04_40_50
Theory : general
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