Nuprl Lemma : flip-conjugate-rotate
∀[n:ℕ]. ∀[i:ℕn - 1]. ((i, i + 1) = (rot(n)^i o ((0, 1) o rot(n)^n - i)) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
flip: (i, j),
rotate: rot(n),
fun_exp: f^n,
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
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,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
guard: {T},
subtract: n - m,
compose: f o g,
le: A ≤ B,
less_than': less_than'(a;b),
sq_type: SQType(T),
flip: (i, j),
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
add-member-int_seg2,
int_seg_wf,
subtract_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
iterated-rotate,
int_seg_subtype_nat,
istype-false,
itermAdd_wf,
int_term_value_add_lemma,
subtype_base_sq,
int_subtype_base,
eq_int_wf,
equal-wf-base,
bool_wf,
set_subtype_base,
lelt_wf,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal-wf-T-base,
lt_int_wf,
ifthenelse_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
equal_wf,
false_wf,
less_than_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
assert_of_lt_int,
le_wf,
le_int_wf,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
neg_assert_of_eq_int,
istype-nat,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
subtract-add-cancel,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
Error :dependent_set_memberEquality_alt,
hypothesisEquality,
productElimination,
independent_pairFormation,
hypothesis,
extract_by_obid,
isectElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
Error :universeIsType,
Error :productIsType,
because_Cache,
closedConclusion,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :functionExtensionality_alt,
Error :lambdaFormation_alt,
Error :inhabitedIsType,
axiomEquality,
Error :isectIsTypeImplies,
instantiate,
cumulativity,
intEquality,
baseApply,
baseClosed,
Error :equalityIstype,
sqequalBase,
Error :functionIsType,
equalityElimination,
Error :equalityIsType1,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
lambdaFormation,
impliesFunctionality,
functionExtensionality,
hyp_replacement,
promote_hyp,
dependent_set_memberEquality,
addEquality,
Error :equalityIsType4,
multiplyEquality,
minusEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[i:\mBbbN{}n - 1]. ((i, i + 1) = (rot(n)\^{}i o ((0, 1) o rot(n)\^{}n - i)))
Date html generated:
2019_06_20-PM-01_35_56
Last ObjectModification:
2018_11_22-AM-10_00_45
Theory : list_1
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