Nuprl Lemma : pW-rec_wf
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P]. ∀[Q:par:P ⟶ (pW par) ⟶ ℙ].
∀[ind:par:P
⟶ a:A[par]
⟶ f:(b:B[par;a] ⟶ (pW C[par;a;b]))
⟶ (b:B[par;a] ⟶ Q[C[par;a;b];f b])
⟶ Q[par;pW-sup(a;f)]]. ∀[par:P]. ∀[w:pW par].
(let ind(p,a,f,G) = ind[p;a;f;G] in
letrec F(p,w) = let a,f=w in
ind(p,a,f,λb.F(C[p;a;b],f(b)) in
F(par;w) ∈ Q[par;w])
Proof
Definitions occuring in Statement :
pW-rec: pW-rec,
pW-sup: pW-sup(a;f),
param-W: pW,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
decidable: Dec(P),
all: ∀x:A. B[x],
nat: ℕ,
so_apply: x[s],
so_apply: x[s1;s2],
so_apply: x[s1;s2;s3],
prop: ℙ,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
param-W: pW,
pcw-partial: pcw-partial(path;n),
implies: P ⇒ Q,
squash: ↓T,
pcw-path: Path,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pi2: snd(t),
isl: isl(x),
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
lelt: i ≤ j < k,
int_seg: {i..j-},
pcw-pp-barred: Barred(pp),
spreadn: spread3,
isr: isr(x),
exposed-it: exposed-it,
ext-family: F ≡ G,
ext-eq: A ≡ B,
pi1: fst(t),
pW-rec: pW-rec,
pW-sup: pW-sup(a;f),
so_apply: x[s1;s2;s3;s4],
pcw-steprel: StepRel(s1;s2),
pcw-step-agree: StepAgree(s;p1;w),
cand: A c∧ B
Lemmas referenced :
pcw-step_wf,
param-W_wf,
param-W-rel_wf,
pcw-pp-barred_wf0,
decidable__pcw-pp-barred,
int_seg_wf,
istype-universe,
subtype_rel_self,
pW-sup_wf,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-void,
istype-int,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
add-subtract-cancel,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
assert_wf,
btrue_wf,
bfalse_wf,
pcw-steprel_wf,
le_weakening2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
less_than_wf,
not-lt-2,
pcw-step-agree_wf,
nat_wf,
false_wf,
true_wf,
equal_wf,
lelt_wf,
le-add-cancel-alt,
zero-mul,
add-mul-special,
decidable__lt,
minus-minus,
less-iff-le,
subtract_wf,
top_wf,
squash_wf,
equal-wf-base,
member_wf,
param-co-W-ext,
param-co-W_wf,
subtype_rel-equal,
param-W-ext,
pcw-pp-lemma,
equal-implies-member-param-W,
subtype_rel_weakening,
unit_wf2,
iff_weakening_equal,
set_subtype_base,
int_subtype_base,
decidable__int_equal,
not-equal-2,
minus-zero,
subtype_rel_function,
int_seg_subtype
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
strong_bar_Induction,
equalityTransitivity,
equalitySymmetry,
applyEquality,
sqequalRule,
dependent_functionElimination,
Error :dependent_pairEquality_alt,
Error :functionIsType,
Error :universeIsType,
natural_numberEquality,
setElimination,
rename,
axiomEquality,
Error :isect_memberEquality_alt,
instantiate,
universeEquality,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
addEquality,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_isectElimination,
minusEquality,
equalityElimination,
lessCases,
axiomSqEquality,
Error :productIsType,
Error :equalityIsType1,
Error :dependent_pairFormation_alt,
promote_hyp,
cumulativity,
functionExtensionality,
lambdaEquality,
lambdaFormation,
dependent_set_memberEquality,
intEquality,
voidEquality,
isect_memberEquality,
isect_memberFormation,
productEquality,
Error :setIsType,
int_eqReduceTrueSq,
hypothesis_subsumption,
applyLambdaEquality,
functionEquality,
Error :inlEquality_alt,
Error :unionIsType,
hyp_replacement
Latex:
\mforall{}[P:Type]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type]. \mforall{}[C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P]. \mforall{}[Q:par:P
{}\mrightarrow{} (pW
par)
{}\mrightarrow{} \mBbbP{}].
\mforall{}[ind:par:P
{}\mrightarrow{} a:A[par]
{}\mrightarrow{} f:(b:B[par;a] {}\mrightarrow{} (pW C[par;a;b]))
{}\mrightarrow{} (b:B[par;a] {}\mrightarrow{} Q[C[par;a;b];f b])
{}\mrightarrow{} Q[par;pW-sup(a;f)]]. \mforall{}[par:P]. \mforall{}[w:pW par].
(let ind(p,a,f,G) = ind[p;a;f;G] in
letrec F(p,w) = let a,f=w in
ind(p,a,f,\mlambda{}b.F(C[p;a;b],f(b)) in
F(par;w) \mmember{} Q[par;w])
Date html generated:
2019_06_20-PM-00_36_17
Last ObjectModification:
2018_10_06-AM-11_20_30
Theory : co-recursion
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