Nuprl Lemma : cycle-transitive1
∀[n:ℕ]. ∀[L:ℕn List].
∀[a,b:ℕ]. ((cycle(L)^b - a L[a]) = L[b] ∈ ℕn) supposing ((a ≤ b) and b < ||L||) supposing no_repeats(ℕn;L)
Proof
Definitions occuring in Statement :
cycle: cycle(L),
no_repeats: no_repeats(T;l),
select: L[n],
length: ||as||,
list: T List,
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
compose: f o g,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtype_base_sq,
nat_wf,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__lt,
length_wf,
int_seg_wf,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
less_than_wf,
no_repeats_wf,
list_wf,
ge_wf,
fun_exp_unroll,
false_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
select_wf,
squash_wf,
add-zero,
subtract-is-int-iff,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
cycle_wf,
fun_exp_wf,
int_seg_properties,
lelt_wf,
equal-wf-base,
assert_wf,
bnot_wf,
not_wf,
true_wf,
apply-cycle-member,
iff_weakening_equal,
add-associates,
add-swap,
add-commutes,
zero-add,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
lambdaFormation,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
addEquality,
independent_functionElimination,
imageElimination,
productElimination,
axiomEquality,
intWeakElimination,
equalityElimination,
applyEquality,
productEquality,
universeEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
imageMemberEquality,
minusEquality,
impliesFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[L:\mBbbN{}n List].
\mforall{}[a,b:\mBbbN{}]. ((cycle(L)\^{}b - a L[a]) = L[b]) supposing ((a \mleq{} b) and b < ||L||)
supposing no\_repeats(\mBbbN{}n;L)
Date html generated:
2017_04_17-AM-08_18_48
Last ObjectModification:
2017_02_27-PM-04_44_29
Theory : list_1
Home
Index