Nuprl Lemma : iterate_functor_wf
∀[I:Type]. ∀[R:I ⟶ I ⟶ ℙ].
∀[F:Type ⟶ Type]. ∀[i:I]. (iterate_functor(I;x,y.R[x;y];T.F[T];i) ∈ Type) supposing tcWO(I;x,y.R[x;y])
Proof
Definitions occuring in Statement :
iterate_functor: iterate_functor(I;x,y.R[x; y];T.F[T];a),
tcWO: tcWO(T;x,y.>[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
decidable: Dec(P),
all: ∀x:A. B[x],
consistent-seq: R-consistent-seq(n),
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
guard: {T},
squash: ↓T,
tcWO: tcWO(T;x,y.>[x; y]),
exists: ∃x:A. B[x],
so_apply: x[s1;s2],
cWObar: cWObar(),
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
int_upper: {i...},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
top: Top,
true: True,
isr: isr(x),
isl: isl(x),
outl: outl(x),
eq_int: (i =z j),
cWO-rel: cWO-rel(R),
less_than: a < b,
cand: A c∧ B,
iterate_functor: iterate_functor(I;x,y.R[x; y];T.F[T];a),
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
unit_wf2,
cWO-rel_wf,
subtype_rel_self,
decidable__cWObar,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_function,
int_seg_subtype,
sq_stable__le,
le_weakening2,
cWO-rel-path-barred,
le_wf,
less_than_wf,
not_wf,
nat_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
set_subtype_base,
istype-int,
int_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
zero-add,
subtract_wf,
decidable__le,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
istype-void,
int_upper_wf,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
true_wf,
assert_wf,
isl_wf,
outl_wf,
bool_cases,
le_antisymmetry_iff,
iff_transitivity,
bnot_wf,
equal-wf-base,
iff_weakening_uiff,
assert_of_bnot,
add-subtract-cancel,
istype-universe,
tcWO_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
hypothesis,
strong_bar_Induction,
unionEquality,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
sqequalRule,
instantiate,
universeEquality,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
Error :functionIsType,
Error :universeIsType,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
Error :lambdaFormation_alt,
because_Cache,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
Error :dependent_pairFormation_alt,
Error :productIsType,
Error :inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
voidElimination,
Error :equalityIsType2,
baseApply,
closedConclusion,
intEquality,
Error :lambdaEquality_alt,
promote_hyp,
cumulativity,
hypothesis_subsumption,
addEquality,
Error :isect_memberEquality_alt,
minusEquality,
Error :equalityIsType1,
Error :inlEquality_alt,
functionEquality,
hyp_replacement,
applyLambdaEquality,
productEquality,
Error :equalityIsType4,
int_eqReduceTrueSq,
isectEquality,
setEquality
Latex:
\mforall{}[I:Type]. \mforall{}[R:I {}\mrightarrow{} I {}\mrightarrow{} \mBbbP{}].
\mforall{}[F:Type {}\mrightarrow{} Type]. \mforall{}[i:I]. (iterate\_functor(I;x,y.R[x;y];T.F[T];i) \mmember{} Type)
supposing tcWO(I;x,y.R[x;y])
Date html generated:
2019_06_20-PM-00_35_08
Last ObjectModification:
2018_10_07-AM-00_37_51
Theory : co-recursion
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