Nuprl Lemma : prec-ext
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[i:P].
prec(lbl,p.a[lbl;p];i) ≡ labl:{lbl:Atom| 0 < ||a[lbl;i]||} × tuple-type(map(λx.case x
of inl(y) =>
case y
of inl(p) =>
prec(lbl,p.a[lbl;p];p)
| inr(p) =>
prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E;a[labl;i]))
Proof
Definitions occuring in Statement :
prec: prec(lbl,p.a[lbl; p];i),
tuple-type: tuple-type(L),
length: ||as||,
map: map(f;as),
list: T List,
ext-eq: A ≡ B,
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
natural_number: $n,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-family: F ≡ G,
all: ∀x:A. B[x],
ptuple: ptuple(lbl,p.a[lbl; p];X),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
pi2: snd(t),
pi1: fst(t),
prec: prec(lbl,p.a[lbl; p];i),
guard: {T},
nat: ℕ,
uimplies: b supposing a,
top: Top,
has-value: (a)↓,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
unit: Unit,
bool: 𝔹,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
it: ⋅,
nil: [],
colength: colength(L),
decidable: Dec(P),
cons: [a / b],
or: P ∨ Q,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
false: False,
tuple-sum: tuple-sum(f;L;x),
add-sz: add-sz(sz;L;x),
map: map(f;as),
list_ind: list_ind,
le: A ≤ B,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
null: null(as),
istype: istype(T),
uiff: uiff(P;Q),
reduce: reduce(f;k;as),
l_sum: l_sum(L),
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
true: True
Lemmas referenced :
pcorec-ext,
prec_wf,
istype-atom,
istype-less_than,
length_wf,
tuple-type_wf,
map_wf,
list_wf,
istype-universe,
pi1_wf,
pcorec_wf,
less_than_wf,
pi2_wf,
subtype_rel_weakening,
ext-eq_inversion,
pcorec-size_wf,
int-value-type,
istype-int,
le_wf,
set-value-type,
nat_wf,
has-value_wf-partial,
istype-void,
unroll-pcorec-size,
add-sz_wf,
termination,
has-value_wf_base,
value-type-has-value,
int_subtype_base,
set_subtype_base,
partial_subtype_base,
istype-nat,
null_wf,
null-map,
tupletype_cons_lemma,
map_cons_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
itermAdd_wf,
itermSubtract_wf,
subtract_wf,
decidable__equal_int,
spread_cons_lemma,
int_formula_prop_eq_lemma,
intformeq_wf,
subtype_base_sq,
subtract-1-ge-0,
istype-le,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
colength_wf_list,
colength-cons-not-zero,
product_subtype_list,
unit_wf2,
nil_wf,
tupletype_nil_lemma,
map_nil_lemma,
list-cases,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
bool_subtype_base,
bool_wf,
null_cons_lemma,
subtype_partial_sqtype_base,
add-zero,
l_sum_cons_lemma,
l_sum_nil_lemma,
cons_wf,
partial_wf,
l_sum-wf-partial-nat,
null_nil_lemma,
subtype_rel-equal,
btrue_neq_bfalse,
nat-partial-nat,
istype-false,
add-wf-partial-nat,
select-map,
subtype_rel_tuple-type,
map-length,
length_wf_nat,
int_seg_wf,
subtype_rel_list,
top_wf,
select_wf,
int_seg_properties,
decidable__lt,
is-exception_wf,
subtype_rel_dep_function,
add-has-value-iff,
sq_stable__has-value,
iff_weakening_equal,
subtype_rel_self,
add_functionality_wrt_eq,
istype-base,
true_wf,
squash_wf,
zero-add
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
independent_pairFormation,
Error :lambdaEquality_alt,
Error :universeIsType,
applyEquality,
Error :inhabitedIsType,
Error :productIsType,
Error :setIsType,
natural_numberEquality,
instantiate,
unionEquality,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
Error :lambdaFormation_alt,
unionElimination,
Error :equalityIstype,
independent_functionElimination,
Error :unionIsType,
setElimination,
rename,
productElimination,
independent_pairEquality,
axiomEquality,
Error :functionIsType,
Error :dependent_pairEquality_alt,
atomEquality,
setEquality,
applyLambdaEquality,
productEquality,
because_Cache,
intEquality,
independent_isectElimination,
Error :dependent_set_memberEquality_alt,
voidElimination,
Error :isect_memberEquality_alt,
closedConclusion,
baseApply,
baseClosed,
callbyvalueAdd,
equalityElimination,
imageElimination,
hypothesis_subsumption,
promote_hyp,
sqequalBase,
Error :functionIsTypeImplies,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
intWeakElimination,
addEquality,
sqleReflexivity,
divergentSqle,
axiomSqleEquality,
imageMemberEquality
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[i:P].
prec(lbl,p.a[lbl;p];i) \mequiv{} labl:\{lbl:Atom| 0 < ||a[lbl;i]||\} \mtimes{} tuple-type(map(\mlambda{}x.case x
of inl(y) =>
case y
of inl(p) =>
prec(lbl,p.a[lbl;p];p)
| inr(p) =>
prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E;a[labl;i]))
Date html generated:
2019_06_20-PM-02_04_53
Last ObjectModification:
2019_03_28-PM-02_51_54
Theory : tuples
Home
Index