Nuprl Lemma : corec_subtype_base
∀[F:Type ⟶ Type]. corec(T.F[T]) ⊆r Base supposing ∀n:ℕ. ∀T:Type. (sqntype(n;T) ⇒ sqntype(n + 1;F T))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
sqntype: sqntype(n;T),
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
true: True,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
top: Top,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
all: ∀x:A. B[x],
corec: corec(T.F[T]),
sqntype: sqntype(n;T),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
nat_wf,
primrec-wf2,
set_wf,
sqntype_wf,
subtract-add-cancel,
le-add-cancel,
add-swap,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
not-le-2,
decidable__le,
subtract_wf,
less_than_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le-add-cancel2,
add-commutes,
add-zero,
add-associates,
add_functionality_wrt_le,
less-iff-le,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
primrec-unroll,
int_seg_wf,
top_wf,
le_wf,
false_wf,
primrec_wf,
sqntype0,
corec_wf,
equal-wf-base,
equal_functionality_wrt_subtype_rel2,
istype-false,
sq_stable__le,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
axiomEquality,
hypothesis,
sqequalHypSubstitution,
Error :isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
intEquality,
minusEquality,
promote_hyp,
dependent_pairFormation,
independent_functionElimination,
addEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
setElimination,
rename,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
universeEquality,
instantiate,
extract_by_obid,
lambdaFormation,
sqequalDefinition,
pointwiseFunctionality,
isectEquality,
Error :functionIsType,
Error :universeIsType,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
Error :lambdaEquality_alt
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
corec(T.F[T]) \msubseteq{}r Base supposing \mforall{}n:\mBbbN{}. \mforall{}T:Type. (sqntype(n;T) {}\mRightarrow{} sqntype(n + 1;F T))
Date html generated:
2019_06_20-PM-00_37_36
Last ObjectModification:
2018_10_15-PM-01_45_04
Theory : co-recursion
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