Nuprl Lemma : apply-cycle-non-member
∀[n:ℕ]. ∀[L:ℕn List]. ∀[x:ℕn]. (cycle(L) x) = x ∈ ℕn supposing ¬(x ∈ L)
Proof
Definitions occuring in Statement :
cycle: cycle(L),
l_member: (x ∈ l),
list: T List,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cycle: cycle(L),
let: let,
prop: ℙ,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
or: P ∨ Q,
false: False,
cons: [a / b],
top: Top,
guard: {T},
int_seg: {i..j-},
le: A ≤ B,
decidable: Dec(P),
lelt: i ≤ j < k,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
listp: A List+,
so_lambda: λ2x.t[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_apply: x[s],
squash: ↓T,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
not_wf,
l_member_wf,
int_seg_wf,
list_wf,
nat_wf,
null_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_null,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
hd_wf,
listp_properties,
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
length_wf,
list_induction,
list_ind_wf,
eq_int_wf,
assert_of_eq_int,
null_nil_lemma,
null_cons_lemma,
cons_wf,
length_cons_ge_one,
subtype_rel_list,
top_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
cons_member,
or_wf,
squash_wf,
true_wf,
iff_weakening_equal,
int_seg_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseClosed,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
impliesFunctionality,
dependent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
voidEquality,
addEquality,
applyEquality,
lambdaEquality,
intEquality,
minusEquality,
dependent_set_memberEquality,
functionEquality,
addLevel,
levelHypothesis,
impliesLevelFunctionality,
imageElimination,
universeEquality,
inrFormation,
imageMemberEquality,
inlFormation,
dependent_pairFormation,
int_eqEquality,
computeAll
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[L:\mBbbN{}n List]. \mforall{}[x:\mBbbN{}n]. (cycle(L) x) = x supposing \mneg{}(x \mmember{} L)
Date html generated:
2017_04_17-AM-08_17_59
Last ObjectModification:
2017_02_27-PM-04_42_04
Theory : list_1
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