Nuprl Lemma : prec-sub-size
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[j:P]. ∀[x:prec(lbl,p.a[lbl;p];j)]. ∀[i:P].
∀[y:prec(lbl,p.a[lbl;p];i)].
||j;x|| < ||i;y|| supposing prec-sub(P;lbl,p.a[lbl;p];j;x;i;y)
Proof
Definitions occuring in Statement :
prec-sub: prec-sub(P;lbl,p.a[lbl; p];j;x;i;y),
prec-size: ||i;x||,
prec: prec(lbl,p.a[lbl; p];i),
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ext-eq: A ≡ B,
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
prec: prec(lbl,p.a[lbl; p];i),
list: T List,
ge: i ≥ j ,
le: A ≤ B,
prec-sub: prec-sub(P;lbl,p.a[lbl; p];j;x;i;y),
dest-prec: dest-prec(x),
let: let,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
l_all: (∀x∈L.P[x]),
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
less_than': less_than'(a;b),
cand: A c∧ B,
l_member: (x ∈ l),
so_lambda: λ2x.t[x],
so_apply: x[s],
respects-equality: respects-equality(S;T),
prec-size: ||i;x||,
pcorec-size: pcorec-size(lbl,p.a[lbl; p]),
sq_type: SQType(T)
Lemmas referenced :
less_than_wf,
squash_wf,
true_wf,
istype-int,
prec-size_wf,
istype-atom,
prec-size-unfold,
subtype_rel_self,
iff_weakening_equal,
prec-ext,
tuple-sum_wf,
prec_wf,
subtype_rel_universe1,
list_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
nat_properties,
decidable__lt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
prec-sub_wf,
member-less_than,
istype-universe,
le-tuple-sum,
int_seg_properties,
l_sum_wf,
map_wf,
l_sum_nonneg,
non_neg_length,
nat_wf,
map_length,
select_wf,
length_wf,
int_seg_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bfalse_wf,
btrue_wf,
btrue_neq_bfalse,
select-tuple_wf,
int_seg_subtype_nat,
istype-false,
map-length,
select-map,
subtype_rel_list,
top_wf,
equal_wf,
subtype_rel-equal,
respects-equality-set,
pcorec_wf,
has-value_wf-partial,
set-value-type,
le_wf,
int-value-type,
pcorec-size_wf,
subtype-respects-equality,
subtype_rel_set,
change-equality-type,
istype-less_than,
le-l_sum,
subtype_base_sq,
set_subtype_base,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
sqequalRule,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
productElimination,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
unionEquality,
cumulativity,
closedConclusion,
universeEquality,
setElimination,
rename,
unionElimination,
Error :unionIsType,
Error :dependent_set_memberEquality_alt,
dependent_functionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
Error :lambdaFormation_alt,
applyLambdaEquality,
addEquality,
int_eqEquality,
independent_pairFormation,
Error :equalityIstype,
Error :isectIsTypeImplies,
Error :functionIsType,
intEquality,
functionExtensionality,
Error :inlEquality_alt,
Error :productIsType,
Error :inrEquality_alt,
hyp_replacement
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[j:P]. \mforall{}[x:prec(lbl,p.a[lbl;p];j)]. \mforall{}[i:P].
\mforall{}[y:prec(lbl,p.a[lbl;p];i)].
||j;x|| < ||i;y|| supposing prec-sub(P;lbl,p.a[lbl;p];j;x;i;y)
Date html generated:
2019_06_20-PM-02_05_47
Last ObjectModification:
2019_02_23-PM-01_13_41
Theory : tuples
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