Nuprl Lemma : ap-tuple_wf
∀[n:ℕ]. ∀[A,B:Type List].
∀[f:tuple-type(map(λp.((fst(p)) ⟶ (snd(p)));zip(A;B)))]. ∀[t:tuple-type(A)]. (ap-tuple(n;f;t) ∈ tuple-type(B))
supposing (||A|| = n ∈ ℤ) ∧ (||B|| = n ∈ ℤ)
Proof
Definitions occuring in Statement :
ap-tuple: ap-tuple(len;f;t),
tuple-type: tuple-type(L),
zip: zip(as;bs),
length: ||as||,
map: map(f;as),
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
or: P ∨ Q,
ap-tuple: ap-tuple(len;f;t),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
decidable: Dec(P)
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,
less_than_wf,
list-cases,
tupletype_nil_lemma,
zip_nil_lemma,
map_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
non_neg_length,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
tupletype_cons_lemma,
null_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
list_wf,
tuple-type_wf,
map_wf,
zip_wf,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
subtract-1-ge-0,
zip_cons_cons_lemma,
map_cons_lemma,
eq_int_wf,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
neg_assert_of_eq_int,
null_nil_lemma,
null_cons_lemma,
zip_cons_nil_lemma,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
nil_wf,
intformnot_wf,
int_formula_prop_not_lemma,
nat_wf,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
assert_of_bnot,
decidable__equal_int,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
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,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
unionElimination,
promote_hyp,
hypothesis_subsumption,
Error :inhabitedIsType,
equalityElimination,
because_Cache,
Error :equalityIsType1,
cumulativity,
baseClosed,
Error :equalityIsType3,
productEquality,
functionEquality,
Error :productIsType,
Error :equalityIsType4,
applyEquality,
intEquality,
Error :equalityIsType2,
baseApply,
closedConclusion,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
independent_pairEquality,
pointwiseFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[A,B:Type List].
\mforall{}[f:tuple-type(map(\mlambda{}p.((fst(p)) {}\mrightarrow{} (snd(p)));zip(A;B)))]. \mforall{}[t:tuple-type(A)].
(ap-tuple(n;f;t) \mmember{} tuple-type(B))
supposing (||A|| = n) \mwedge{} (||B|| = n)
Date html generated:
2019_06_20-PM-02_03_12
Last ObjectModification:
2018_10_06-AM-08_30_10
Theory : tuples
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