Nuprl Lemma : select-update-tuple
∀[m,n:ℕ]. ∀[L:Type List].
(∀[x:tuple-type(L)]. ∀[y:Top]. (update-tuple(||L||;x;n;y).m ~ if (n =z m) then y else x.m fi )) supposing
(m < ||L|| and
n < ||L||)
Proof
Definitions occuring in Statement :
select-tuple: x.n,
update-tuple: update-tuple(len;x;n;y),
tuple-type: tuple-type(L),
length: ||as||,
list: T List,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
select-tuple: x.n,
update-tuple: update-tuple(len;x;n;y),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi1: fst(t),
decidable: Dec(P),
subtype_rel: A ⊆r B,
pi2: snd(t),
cons: [a / b],
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
top_wf,
tuple-type_wf,
length_wf,
list_wf,
nat_wf,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformeq_wf,
intformnot_wf,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
equal-wf-base,
int_subtype_base,
list-cases,
tupletype_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
tupletype_cons_lemma,
length_of_cons_lemma,
add-subtract-cancel,
le_wf,
decidable__lt,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
null_wf,
bool_cases,
assert_of_null,
pi2_wf,
length_wf_nat,
non_neg_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
baseClosed,
productElimination,
because_Cache,
promote_hyp,
cumulativity,
impliesFunctionality,
baseApply,
closedConclusion,
applyEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
pointwiseFunctionality,
imageElimination,
independent_pairEquality,
hyp_replacement,
applyLambdaEquality,
addEquality
Latex:
\mforall{}[m,n:\mBbbN{}]. \mforall{}[L:Type List].
(\mforall{}[x:tuple-type(L)]. \mforall{}[y:Top].
(update-tuple(||L||;x;n;y).m \msim{} if (n =\msubz{} m) then y else x.m fi )) supposing
(m < ||L|| and
n < ||L||)
Date html generated:
2017_04_17-AM-09_30_25
Last ObjectModification:
2017_02_27-PM-05_31_27
Theory : tuples
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