Nuprl Lemma : unshuffle-odd-length
∀[f:ℕ ⟶ Top]. ∀[m:ℕ]. ∀[L:ℕ List]. ∀[x:ℕ].
unshuffle(map(f;L @ [x])) ~ unshuffle(map(f;L)) supposing ||L|| = (2 * m) ∈ ℤ
Proof
Definitions occuring in Statement :
unshuffle: unshuffle(L),
length: ||as||,
map: map(f;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t,
equal: s = t ∈ 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,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
unshuffle: unshuffle(L),
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
le: A ≤ B,
assert: ↑b,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
le_wf,
subtype_rel_self,
nat_wf,
list-cases,
list_ind_nil_lemma,
map_nil_lemma,
map_cons_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
reduce_tl_nil_lemma,
product_subtype_list,
list_ind_cons_lemma,
list_subtype_base,
list_wf,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-top,
int_term_value_mul_lemma,
itermMultiply_wf,
satisfiable-full-omega-tt,
subtract_nat_wf,
non_neg_length,
length_wf,
length_wf_nat,
top_wf,
map_wf,
append_wf,
cons_wf,
nil_wf,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
lt_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
assert_wf,
false_wf,
iff_weakening_uiff,
assert_of_lt_int,
iff_weakening_equal
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,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
promote_hyp,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
addEquality,
Error :functionIsType,
computeAll,
lambdaEquality,
dependent_pairFormation,
voidEquality,
isect_memberEquality,
Error :equalityIsType1,
multiplyEquality,
cumulativity
Latex:
\mforall{}[f:\mBbbN{} {}\mrightarrow{} Top]. \mforall{}[m:\mBbbN{}]. \mforall{}[L:\mBbbN{} List]. \mforall{}[x:\mBbbN{}].
unshuffle(map(f;L @ [x])) \msim{} unshuffle(map(f;L)) supposing ||L|| = (2 * m)
Date html generated:
2019_06_20-PM-01_47_47
Last ObjectModification:
2018_10_07-PM-00_28_20
Theory : list_1
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