Nuprl Lemma : unshuffle-map
∀[f:ℕ ⟶ Top]. ∀[m:ℕ]. (unshuffle(map(f;upto(2 * m))) ~ map(λi.<f (2 * i), f ((2 * i) + 1)>;upto(m)))
Proof
Definitions occuring in Statement :
unshuffle: unshuffle(L),
upto: upto(n),
map: map(f;as),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
multiply: n * m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
upto: upto(n),
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,
int_seg: {i..j-},
from-upto: [n, m),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
lelt: i ≤ j < k,
shuffle: shuffle(ps),
concat: concat(ll),
decidable: Dec(P),
pi1: fst(t),
pi2: snd(t),
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
has-value: (a)↓,
subtract: n - m,
cand: A c∧ B
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
le_wf,
subtract_wf,
int_seg_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
map_cons_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_seg_properties,
itermMultiply_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
map_nil_lemma,
reduce_nil_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
false_wf,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
list_wf,
list_subtype_base,
set_subtype_base,
int_subtype_base,
from-upto_wf,
squash_wf,
true_wf,
mul-commutes,
unshuffle-shuffle,
top_wf,
map_wf,
upto_wf,
nat_wf,
reduce_cons_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
value-type-has-value,
int-value-type,
add-member-int_seg2,
add-subtract-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_list_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
sqequalAxiom,
addEquality,
because_Cache,
multiplyEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
instantiate,
cumulativity,
dependent_set_memberEquality,
setEquality,
productEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_pairEquality,
functionEquality,
callbyvalueReduce
Latex:
\mforall{}[f:\mBbbN{} {}\mrightarrow{} Top]. \mforall{}[m:\mBbbN{}].
(unshuffle(map(f;upto(2 * m))) \msim{} map(\mlambda{}i.<f (2 * i), f ((2 * i) + 1)>upto(m)))
Date html generated:
2018_05_21-PM-00_44_48
Last ObjectModification:
2018_05_19-AM-06_49_28
Theory : list_1
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