Nuprl Lemma : prec-size-unfold
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[i:P]. ∀[x:prec(lbl,p.a[lbl;p];i)].
(||i;x||
= let lbl,z = x
in 1
+ tuple-sum(λc,x. case c
of inl(p) =>
case p of inl(j) => ||j;x|| | inr(j) => l_sum(map(λz.||j;z||;x))
| inr(_) =>
0;a[lbl;i];z)
∈ ℤ)
Proof
Definitions occuring in Statement :
prec-size: ||i;x||,
prec: prec(lbl,p.a[lbl; p];i),
tuple-sum: tuple-sum(f;L;x),
l_sum: l_sum(L),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
lambda: λx.A[x],
function: x:A ⟶ B[x],
spread: spread def,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
add: n + m,
natural_number: $n,
int: ℤ,
atom: Atom,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prec-size: ||i;x||,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
add-sz: add-sz(sz;L;x),
squash: ↓T,
prop: ℙ,
nat: ℕ,
l_all: (∀x∈L.P[x]),
uimplies: b supposing a,
ge: i ≥ j ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
decidable: Dec(P),
or: P ∨ Q,
false: False,
guard: {T},
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
true: True,
prec: prec(lbl,p.a[lbl; p];i),
list: T List
Lemmas referenced :
prec-ext,
unroll-pcorec-size,
istype-void,
tuple-sum_wf,
squash_wf,
true_wf,
istype-nat,
tuple-type_wf,
map_wf,
list_wf,
istype-universe,
prec_wf,
istype-atom,
prec-size_wf,
l_sum_nonneg,
map-length,
select-map,
subtype_rel_list,
top_wf,
non_neg_length,
int_seg_properties,
decidable__le,
select_wf,
length_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
istype-le,
int_seg_wf,
nat_wf,
l_sum_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
applyEquality,
sqequalRule,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
setElimination,
rename,
addEquality,
natural_numberEquality,
instantiate,
Error :lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :functionIsType,
cumulativity,
universeEquality,
unionEquality,
unionElimination,
Error :equalityIstype,
dependent_functionElimination,
independent_functionElimination,
Error :unionIsType,
intEquality,
independent_isectElimination,
because_Cache,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
functionExtensionality,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[i:P]. \mforall{}[x:prec(lbl,p.a[lbl;p];i)].
(||i;x||
= let lbl,z = x
in 1
+ tuple-sum(\mlambda{}c,x. case c
of inl(p) =>
case p of inl(j) => ||j;x|| | inr(j) => l\_sum(map(\mlambda{}z.||j;z||;x))
| inr($_{}$) =>
0;a[lbl;i];z))
Date html generated:
2019_06_20-PM-02_05_07
Last ObjectModification:
2019_02_22-PM-06_23_02
Theory : tuples
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