Nuprl Lemma : split-tuple-append-tuple
∀[L1,L2:Type List].
∀[x:tuple-type(L1)]. ∀[y:tuple-type(L2)]. (split-tuple(append-tuple(||L1||;||L2||;x;y);||L1||) ~ <x, y>) supposing 0 \000C< ||L2||
Proof
Definitions occuring in Statement :
append-tuple: append-tuple(n;m;x;y),
split-tuple: split-tuple(x;n),
tuple-type: tuple-type(L),
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pair: <a, b>,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
split-tuple: split-tuple(x;n),
eq_int: (i =z j),
append-tuple: append-tuple(n;m;x;y),
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,
sq_type: SQType(T),
guard: {T},
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
nil: [],
it: ⋅,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
decidable: Dec(P),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
le: A ≤ B,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
pi1: fst(t),
pi2: snd(t),
length: ||as||,
list_ind: list_ind
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,
tuple-type_wf,
length_wf,
nat_wf,
colength_wf_list,
int_subtype_base,
list_wf,
list-cases,
tupletype_nil_lemma,
length_of_nil_lemma,
subtype_base_sq,
unit_wf2,
unit_subtype_base,
equal-unit,
it_wf,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
set_subtype_base,
le_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__le,
tupletype_cons_lemma,
length_of_cons_lemma,
ifthenelse_wf,
null_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
le_int_wf,
assert_of_le_int,
eqff_to_assert,
non_neg_length,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
subtype_rel-equal,
assert_of_null,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
nequal-le-implies,
add-is-int-iff,
false_wf,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
assert_of_bnot,
length_wf_nat,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
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,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
Error :inhabitedIsType,
Error :equalityIsType3,
applyEquality,
unionElimination,
cumulativity,
because_Cache,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
Error :equalityIsType1,
imageElimination,
Error :dependent_set_memberEquality_alt,
productEquality,
addEquality,
equalityElimination,
pointwiseFunctionality,
hyp_replacement
Latex:
\mforall{}[L1,L2:Type List].
\mforall{}[x:tuple-type(L1)]. \mforall{}[y:tuple-type(L2)].
(split-tuple(append-tuple(||L1||;||L2||;x;y);||L1||) \msim{} <x, y>)
supposing 0 < ||L2||
Date html generated:
2019_06_20-PM-02_03_40
Last ObjectModification:
2018_09_30-PM-02_47_08
Theory : tuples
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