Nuprl Lemma : cyclic-map-equipollent
∀n:ℕ+. Combination(n - 1;ℕn - 1) ~ cyclic-map(ℕn)
Proof
Definitions occuring in Statement :
cyclic-map: cyclic-map(T),
combination: Combination(n;T),
equipollent: A ~ B,
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
combination: Combination(n;T),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
cand: A c∧ B,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
l_member: (x ∈ l),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True,
nat: ℕ,
ge: i ≥ j ,
guard: {T},
equipollent: A ~ B,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
select: L[n],
cons: [a / b],
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
cyclic-map: cyclic-map(T),
injection: A →⟶ B,
sq_stable: SqStable(P),
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
orbit: orbit(T;f;L),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
bfalse: ff,
compose: f o g,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
cardinality-le: |T| ≤ n,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
no_repeats: no_repeats(T;l)
Lemmas referenced :
nat_plus_wf,
cons_wf,
int_seg_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
decidable__lt,
lelt_wf,
no_repeats_cons,
no_repeats-strong-subtype,
strong-subtype-set1,
subtract_wf,
length_of_cons_lemma,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
no_repeats_wf,
equal_wf,
length_wf,
combination_wf,
l_member_wf,
subtype_rel_list,
int_seg_subtype,
false_wf,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
select_wf,
nat_properties,
int_seg_properties,
cycle_wf2,
nat_plus_subtype_nat,
cyclic-map_wf,
biject_wf,
list_extensionality,
less_than_wf,
nat_wf,
cycle-transitive1,
le_wf,
squash_wf,
true_wf,
select_cons_tl,
subtype_rel_self,
iff_weakening_equal,
le_weakening2,
non_neg_length,
add-subtract-cancel,
add_nat_plus,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
fun_exp_wf,
zero-add,
orbit-decomp2,
decidable__equal-int_seg,
finite-type-int_seg,
sq_stable__inject,
list_wf,
orbit_wf,
cyclic-map-transitive,
orbit-closed,
l_all_iff,
all_wf,
int_seg_subtype_nat,
exists_wf,
inject_wf,
apply-cycle-member,
orbit-iterates,
fun_exp_unroll,
fun_exp0_lemma,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
rem_rec_case,
less_than_transitivity2,
rem_base_case,
l_member_decomp,
list-cardinality-le,
cardinality-le-no_repeats-length,
surject_wf,
cardinality-le-int_seg,
length_wf_nat,
list_subtype_base,
length-append,
length_of_nil_lemma,
no_repeats-append,
append_wf,
nil_wf,
l_disjoint_append,
l_disjoint_singleton,
list_ind_cons_lemma,
list_ind_nil_lemma,
list-set-type2,
add-is-int-iff,
no_repeats-permute,
no_repeats-sublist,
append_assoc,
sublist_append2,
cycle-append,
cycle_wf,
cycle-injection,
cycle-transitive2,
l_all_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
productElimination,
isectElimination,
natural_numberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
productEquality,
applyEquality,
addEquality,
minusEquality,
multiplyEquality,
applyLambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
hyp_replacement,
functionExtensionality,
setEquality,
equalityElimination,
impliesFunctionality,
promote_hyp,
pointwiseFunctionality,
baseApply,
closedConclusion,
isect_memberFormation,
functionEquality,
allFunctionality
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. Combination(n - 1;\mBbbN{}n - 1) \msim{} cyclic-map(\mBbbN{}n)
Date html generated:
2018_05_21-PM-08_26_31
Last ObjectModification:
2018_05_19-PM-05_02_43
Theory : general
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