Nuprl Lemma : fixpoint-induction-bottom
∀[E,S:Type].
(∀[G:S ⟶ partial(E)]. ∀[g:S ⟶ S]. (G fix(g) ∈ partial(E))) supposing ((⊥ ∈ S) and mono(E) and value-type(E))
Proof
Definitions occuring in Statement :
partial: partial(T),
mono: mono(T),
value-type: value-type(T),
bottom: ⊥,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
fix: fix(F),
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
top: Top,
guard: {T},
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
rev_implies: P ⇐ Q,
nat: ℕ,
mono: mono(T),
is-above: is-above(T;a;z),
exists: ∃x:A. B[x],
false: False,
not: ¬A
Lemmas referenced :
partial_wf,
equal-wf-base,
mono_wf,
value-type_wf,
pi2_wf,
pi1_wf,
equal_wf,
fixpoint-induction-bottom-base,
top_wf,
subtype_rel_product,
pair-eta,
member_wf,
base-equal-partial,
sq_stable__has-value,
has-value_wf_base,
has-value_wf-partial,
fun_exp_wf,
is-exception_wf,
fixpoint_ub,
has-value-monotonic,
set_subtype_base,
le_wf,
int_subtype_base,
and_wf,
termination,
fixpoint-upper-bound,
equal-wf-base-T,
sqle_wf_base,
termination-equality-base,
nat_wf,
partial-not-exception,
fix-not-exception
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
Error :inhabitedIsType,
hypothesisEquality,
isect_memberEquality,
isectElimination,
thin,
functionEquality,
Error :universeIsType,
extract_by_obid,
because_Cache,
baseClosed,
universeEquality,
independent_pairEquality,
productEquality,
lambdaFormation,
pointwiseFunctionality,
applyLambdaEquality,
lambdaEquality,
independent_pairFormation,
productElimination,
dependent_functionElimination,
independent_functionElimination,
closedConclusion,
baseApply,
voidEquality,
voidElimination,
independent_isectElimination,
cumulativity,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
compactness,
hyp_replacement,
sqleRule,
divergentSqle,
sqleReflexivity,
intEquality,
dependent_set_memberEquality,
setElimination,
rename,
dependent_pairFormation
Latex:
\mforall{}[E,S:Type].
(\mforall{}[G:S {}\mrightarrow{} partial(E)]. \mforall{}[g:S {}\mrightarrow{} S]. (G fix(g) \mmember{} partial(E))) supposing
((\mbot{} \mmember{} S) and
mono(E) and
value-type(E))
Date html generated:
2019_06_20-PM-00_34_08
Last ObjectModification:
2018_09_26-PM-01_21_20
Theory : partial_1
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