Nuprl Lemma : fixpoint-induction2
∀A:Type. ∀T:A ⟶ Type.
((A ⊆r Base)
⇒ (∀a:A. value-type(T[a]))
⇒ (∀a:A. mono(T[a]))
⇒ (∀f:(a:A ⟶ partial(T[a])) ⟶ a:A ⟶ partial(T[a]) ⋂ Base. (fix(f) ∈ a:A ⟶ partial(T[a]))))
Proof
Definitions occuring in Statement :
partial: partial(T),
mono: mono(T),
isect2: T1 ⋂ T2,
value-type: value-type(T),
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
fix: fix(F),
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
guard: {T},
mono: mono(T),
is-above: is-above(T;a;z),
exists: ∃x:A. B[x],
nat: ℕ,
not: ¬A,
is-exception: is-exception(t),
false: False
Lemmas referenced :
void_wf,
strictness-apply,
bottom_wf-partial,
base-member-partial,
isect2_subtype_rel2,
partial_wf,
base_wf,
isect2_wf,
all_wf,
mono_wf,
value-type_wf,
subtype_rel_wf,
has-value_wf_base,
isect2_decomp,
termination,
fun_exp_wf,
set_subtype_base,
le_wf,
int_subtype_base,
fixpoint-upper-bound,
is-exception_wf,
equal-wf-base-T,
sqle_wf_base,
partial-not-exception,
isect2_subtype_rel,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionExtensionality,
voidElimination,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
independent_pairFormation,
because_Cache,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
isect_memberEquality,
voidEquality,
productElimination,
applyEquality,
hypothesisEquality,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
functionEquality,
lambdaEquality,
universeEquality,
isect_memberFormation,
axiomEquality,
compactness,
independent_functionElimination,
dependent_pairFormation,
intEquality,
natural_numberEquality,
sqleRule,
divergentSqle,
sqleReflexivity,
productEquality,
exceptionCompactness
Latex:
\mforall{}A:Type. \mforall{}T:A {}\mrightarrow{} Type.
((A \msubseteq{}r Base)
{}\mRightarrow{} (\mforall{}a:A. value-type(T[a]))
{}\mRightarrow{} (\mforall{}a:A. mono(T[a]))
{}\mRightarrow{} (\mforall{}f:(a:A {}\mrightarrow{} partial(T[a])) {}\mrightarrow{} a:A {}\mrightarrow{} partial(T[a]) \mcap{} Base. (fix(f) \mmember{} a:A {}\mrightarrow{} partial(T[a]))))
Date html generated:
2017_04_14-AM-07_41_21
Last ObjectModification:
2017_02_27-PM-03_13_13
Theory : partial_1
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