Nuprl Lemma : mrec-label-cases
∀s:MutualRectypeSpec. ∀n:Atom. ∀lbl:{lbl:Atom| 0 < ||mrec-spec(s;lbl;n)||} .
((↑isl(apply-alist(AtomDeq;s;n))) ∧ (lbl ∈ map(λp.(fst(p));outl(apply-alist(AtomDeq;s;n)))))
Proof
Definitions occuring in Statement :
mrec-spec: mrec-spec(L;lbl;p),
mrec_spec: MutualRectypeSpec,
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
length: ||as||,
map: map(f;as),
atom-deq: AtomDeq,
outl: outl(x),
assert: ↑b,
isl: isl(x),
less_than: a < b,
pi1: fst(t),
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
natural_number: $n,
atom: Atom
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
mrec_spec: MutualRectypeSpec,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
isl: isl(x),
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
outl: outl(x),
uimplies: b supposing a,
not: ¬A,
false: False,
mrec-spec: mrec-spec(L;lbl;p),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
less_than: a < b,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
sq_stable_from_decidable,
assert_wf,
apply-alist_wf,
atom-deq_wf,
list_wf,
btrue_wf,
bfalse_wf,
decidable__assert,
sq_stable__l_member,
decidable__atom_equal,
map_wf,
pi1_wf,
assert_elim,
btrue_neq_bfalse,
istype-less_than,
length_wf,
mrec-spec_wf,
istype-atom,
mrec_spec_wf,
nil_wf,
length_of_nil_lemma,
isl-apply-alist,
subtype_base_sq,
bool_wf,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
isectElimination,
instantiate,
cumulativity,
atomEquality,
hypothesis,
hypothesisEquality,
productEquality,
unionEquality,
universeEquality,
Error :inhabitedIsType,
unionElimination,
sqequalRule,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
Error :lambdaEquality_alt,
Error :productIsType,
Error :universeIsType,
independent_isectElimination,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
productElimination,
voidElimination,
Error :setIsType,
natural_numberEquality
Latex:
\mforall{}s:MutualRectypeSpec. \mforall{}n:Atom. \mforall{}lbl:\{lbl:Atom| 0 < ||mrec-spec(s;lbl;n)||\} .
((\muparrow{}isl(apply-alist(AtomDeq;s;n))) \mwedge{} (lbl \mmember{} map(\mlambda{}p.(fst(p));outl(apply-alist(AtomDeq;s;n)))))
Date html generated:
2019_06_20-PM-02_14_58
Last ObjectModification:
2019_02_27-PM-02_09_06
Theory : tuples
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