Nuprl Lemma : cycle-injection
∀[n:ℕ]. ∀[L:ℕn List]. Inj(ℕn;ℕn;cycle(L)) supposing no_repeats(ℕn;L)
Proof
Definitions occuring in Statement :
cycle: cycle(L),
no_repeats: no_repeats(T;l),
list: T List,
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
inject: Inj(A;B;f),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
guard: {T},
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
bfalse: ff,
rev_implies: P ⇐ Q,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
squash: ↓T,
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
ge: i ≥ j ,
true: True,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
less_than: a < b,
less_than': less_than'(a;b),
no_repeats: no_repeats(T;l)
Lemmas referenced :
decidable__equal_int_seg,
decidable__l_member,
nat_wf,
list_wf,
no_repeats_wf,
cycle_wf,
int_seg_wf,
equal_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
subtype_base_sq,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
bool_cases,
iff_weakening_equal,
apply-cycle-member,
true_wf,
squash_wf,
not_wf,
bnot_wf,
assert_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
select_wf,
nat_properties,
int_seg_properties,
subtract_wf,
eq_int_wf,
length_wf,
equal-wf-base-T,
int_term_value_subtract_lemma,
itermSubtract_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
decidable__le,
le_wf,
false_wf,
decidable__equal_int,
less_than_wf,
cycle-closed,
istype-int,
full-omega-unsat,
istype-void,
istype-universe,
apply-cycle-non-member,
subtype_rel_self,
istype-le,
istype-less_than,
l_member_wf
Rules used in proof :
unionElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
hypothesisEquality,
applyEquality,
because_Cache,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
intEquality,
independent_isectElimination,
cumulativity,
instantiate,
impliesFunctionality,
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination,
computeAll,
voidEquality,
voidElimination,
int_eqEquality,
dependent_pairFormation,
applyLambdaEquality,
independent_pairFormation,
dependent_set_memberEquality,
levelHypothesis,
equalityUniverse,
addEquality,
productEquality,
Error :lambdaEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :dependent_set_memberEquality_alt,
Error :inhabitedIsType,
Error :productIsType,
Error :lambdaFormation_alt,
hyp_replacement,
Error :equalityIstype
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[L:\mBbbN{}n List]. Inj(\mBbbN{}n;\mBbbN{}n;cycle(L)) supposing no\_repeats(\mBbbN{}n;L)
Date html generated:
2019_06_20-PM-01_40_24
Last ObjectModification:
2019_01_13-PM-02_03_53
Theory : list_1
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