Nuprl Lemma : subtype_rel_tuple-type
∀[As,Bs:Type List]. tuple-type(As) ⊆r tuple-type(Bs) supposing (||As|| = ||Bs|| ∈ ℤ) ∧ (∀i:ℕ||As||. (As[i] ⊆r Bs[i]))
Proof
Definitions occuring in Statement :
tuple-type: tuple-type(L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
so_apply: x[s],
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
ge: i ≥ j ,
le: A ≤ B,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
less_than': less_than'(a;b),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
subtract: n - m,
iff: P ⇐⇒ Q
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
equal_wf,
length_wf,
all_wf,
int_seg_wf,
subtype_rel_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
tuple-type_wf,
equal-wf-base-T,
nil_wf,
length_of_nil_lemma,
subtype_rel_self,
equal-wf-base,
tupletype_nil_lemma,
tupletype_cons_lemma,
subtype_rel-equal,
unit_wf2,
ifthenelse_wf,
null_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
length_of_cons_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
cons_wf,
add-is-int-iff,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
lelt_wf,
list-cases,
stuck-spread,
base_wf,
null_nil_lemma,
product_subtype_list,
subtype_rel_transitivity,
null_cons_lemma,
btrue_wf,
and_wf,
bfalse_wf,
btrue_neq_bfalse,
subtype_rel_product,
decidable__equal_int,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel,
int_subtype_base,
squash_wf,
true_wf,
select_cons_tl,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
sqequalRule,
lambdaEquality,
hypothesis,
isectEquality,
productEquality,
intEquality,
because_Cache,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
cumulativity,
independent_functionElimination,
axiomEquality,
lambdaFormation,
equalityElimination,
promote_hyp,
baseClosed,
pointwiseFunctionality,
baseApply,
closedConclusion,
dependent_set_memberEquality,
imageMemberEquality,
applyLambdaEquality,
addEquality,
hypothesis_subsumption,
hyp_replacement,
imageElimination
Latex:
\mforall{}[As,Bs:Type List].
tuple-type(As) \msubseteq{}r tuple-type(Bs) supposing (||As|| = ||Bs||) \mwedge{} (\mforall{}i:\mBbbN{}||As||. (As[i] \msubseteq{}r Bs[i]))
Date html generated:
2017_04_17-AM-09_29_07
Last ObjectModification:
2017_02_27-PM-05_31_29
Theory : tuples
Home
Index