Nuprl Lemma : select-shuffle2
∀[T:Type]. ∀[ps:(T × T) List].
∀i:ℕ||ps||. ((shuffle(ps)[2 * i] ~ fst(ps[i])) ∧ (shuffle(ps)[(2 * i) + 1] ~ snd(ps[i])))
Proof
Definitions occuring in Statement :
shuffle: shuffle(ps),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
and: P ∧ Q,
product: x:A × B[x],
multiply: n * m,
add: n + m,
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],
and: P ∧ Q,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
int_nzero: ℤ-o,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
select-shuffle,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
length-shuffle,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
shuffle_wf,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf,
list_wf,
rem_invariant,
false_wf,
le_wf,
int_seg_subtype_nat,
less_than_wf,
eq_int_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
div-cancel2,
nequal_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
mul-commutes,
zero-add,
div-cancel3,
add-commutes,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
dependent_set_memberEquality,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
independent_pairFormation,
productEquality,
cumulativity,
productElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
imageElimination,
addEquality,
independent_pairEquality,
sqequalAxiom,
universeEquality,
applyEquality,
imageMemberEquality,
baseClosed,
remainderEquality,
addLevel,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
equalityElimination,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[ps:(T \mtimes{} T) List].
\mforall{}i:\mBbbN{}||ps||. ((shuffle(ps)[2 * i] \msim{} fst(ps[i])) \mwedge{} (shuffle(ps)[(2 * i) + 1] \msim{} snd(ps[i])))
Date html generated:
2017_04_17-AM-08_56_31
Last ObjectModification:
2017_02_27-PM-05_13_02
Theory : list_1
Home
Index