Nuprl Lemma : cycle-transitive2
∀n:ℕ. ∀L:ℕn List. (∀x∈L.(∀y∈L.∃m:ℕ||L||. ((cycle(L)^m x) = y ∈ ℕn))) supposing no_repeats(ℕn;L)
Proof
Definitions occuring in Statement :
cycle: cycle(L),
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
int_seg: {i..j-},
sq_type: SQType(T),
guard: {T},
lelt: i ≤ j < k
Lemmas referenced :
no_repeats_witness,
int_seg_wf,
l_all_iff,
l_member_wf,
l_all_wf,
exists_wf,
length_wf,
equal_wf,
fun_exp_wf,
int_seg_subtype_nat,
false_wf,
cycle_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
cycle-transitive,
no_repeats_wf,
list_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
setEquality,
productElimination,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}L:\mBbbN{}n List. (\mforall{}x\mmember{}L.(\mforall{}y\mmember{}L.\mexists{}m:\mBbbN{}||L||. ((cycle(L)\^{}m x) = y))) supposing no\_repeats(\mBbbN{}n;L)
Date html generated:
2016_05_14-PM-02_27_25
Last ObjectModification:
2015_12_26-PM-04_23_56
Theory : list_1
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