Nuprl Lemma : bar_recursion_wf1
∀[T:Type]. ∀[R,A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ]. ∀[d:∀n:ℕ. ∀s:ℕn ⟶ T. Dec(R[n;s])].
∀[b:∀n:ℕ. ∀s:ℕn ⟶ T. (R[n;s] ⇒ A[n;s])]. ∀[i:∀n:ℕ. ∀s:ℕn ⟶ T. ((∀t:T. A[n + 1;seq-append(n;1;s;λi.t)]) ⇒ A[n;s])].
((∀alpha:ℕ ⟶ T. (↓∃m:ℕ. R[m;alpha])) ⇒ (∀c:Top. (bar_recursion(d;b;i;0;c) ∈ A[0;c])))
Proof
Definitions occuring in Statement :
bar_recursion: bar_recursion,
seq-append: seq-append(n;m;s1;s2),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
label: ...$L... t,
lelt: i ≤ j < k,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
guard: {T},
less_than: a < b,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
seq-append: seq-append(n;m;s1;s2),
bar_recursion: bar_recursion,
true: True,
top: Top,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
exists: ∃x:A. B[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
ext-eq_weakening,
subtype_rel_weakening,
le-add-cancel2,
less-iff-le,
not-lt-2,
not-equal-2,
and_wf,
assert_of_bnot,
iff_weakening_uiff,
not_wf,
bnot_wf,
assert_wf,
iff_transitivity,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
less_than_wf,
assert_of_lt_int,
lt_int_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
equal_wf,
decidable_wf,
seq-append_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
decidable__le,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_dep_function,
exists_wf,
squash_wf,
nat_wf,
all_wf,
top_wf
Rules used in proof :
int_eqReduceFalseSq,
impliesFunctionality,
instantiate,
promote_hyp,
dependent_pairFormation,
sqequalAxiom,
lessCases,
int_eqReduceTrueSq,
equalityElimination,
bar_Induction,
universeEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
productElimination,
voidElimination,
unionElimination,
addEquality,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
independent_pairFormation,
independent_isectElimination,
rename,
setElimination,
natural_numberEquality,
functionExtensionality,
applyEquality,
because_Cache,
lambdaEquality,
sqequalRule,
hypothesisEquality,
cumulativity,
functionEquality,
thin,
isectElimination,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[R,A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. Dec(R[n;s])]. \mforall{}[b:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T.
(R[n;s]
{}\mRightarrow{} A[n;s])].
\mforall{}[i:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. ((\mforall{}t:T. A[n + 1;seq-append(n;1;s;\mlambda{}i.t)]) {}\mRightarrow{} A[n;s])].
((\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}m:\mBbbN{}. R[m;alpha])) {}\mRightarrow{} (\mforall{}c:Top. (bar\_recursion(d;b;i;0;c) \mmember{} A[0;c])))
Date html generated:
2017_09_29-PM-05_47_33
Last ObjectModification:
2017_09_01-PM-11_41_37
Theory : bar-induction
Home
Index