Nuprl Lemma : select-shorten-tuple
∀[n,m:ℕ]. ∀[L:Type List]. ∀[x:tuple-type(L)]. (shorten-tuple(x;n).m ~ x.n + m) supposing n + m < ||L||
Proof
Definitions occuring in Statement :
shorten-tuple: shorten-tuple(x;n),
select-tuple: x.n,
tuple-type: tuple-type(L),
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
shorten-tuple: shorten-tuple(x;n),
select-tuple: x.n,
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtract: n - m,
btrue: tt,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
true: True,
sq_type: SQType(T),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
assert: ↑b,
nequal: a ≠ b ∈ T ,
less_than: a < b,
le: A ≤ B,
cons: [a / b],
pi2: snd(t)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eq_int_wf,
squash_wf,
true_wf,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
subtract_wf,
length_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
minus-zero,
add-zero,
equal_wf,
istype-universe,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
iff_weakening_equal,
int_subtype_base,
length_wf_nat,
set_subtype_base,
le_wf,
bfalse_wf,
bnot_wf,
assert_elim,
btrue_neq_bfalse,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
non_neg_length,
itermSubtract_wf,
int_term_value_subtract_lemma,
tuple-type_wf,
list_wf,
istype-nat,
subtract-1-ge-0,
equal-wf-base,
assert_wf,
equal-wf-T-base,
le_int_wf,
lt_int_wf,
less_than_wf,
not_wf,
istype-assert,
list-cases,
length_of_nil_lemma,
tupletype_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
tupletype_cons_lemma,
null_nil_lemma,
null_cons_lemma,
decidable__lt,
add-is-int-iff,
false_wf,
add-subtract-cancel,
general_arith_equation1,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-commutes,
add-associates,
subtype_rel-equal,
nat_wf,
base_wf,
sqeq-copath5,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_le_int,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomSqEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
instantiate,
cumulativity,
applyEquality,
imageElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
unionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
equalityElimination,
productElimination,
Error :equalityIsType3,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :equalityIsType4,
baseApply,
closedConclusion,
intEquality,
applyLambdaEquality,
Error :equalityIsType2,
promote_hyp,
Error :equalityIsType1,
addEquality,
Error :equalityIstype,
sqequalBase,
Error :functionIsType,
hypothesis_subsumption,
pointwiseFunctionality,
multiplyEquality,
minusEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[L:Type List].
\mforall{}[x:tuple-type(L)]. (shorten-tuple(x;n).m \msim{} x.n + m) supposing n + m < ||L||
Date html generated:
2019_06_20-PM-02_03_46
Last ObjectModification:
2018_11_22-AM-11_23_35
Theory : tuples
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