Nuprl Lemma : iterated-rotate
∀[n,i:ℕ]. rot(n)^i = (λx.if x + i <z n then x + i else (x + i) - n fi ) ∈ (ℕn ⟶ ℕn) supposing i ≤ n
Proof
Definitions occuring in Statement :
rotate: rot(n),
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
lambda: λx.A[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,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
fun_exp: f^n,
subtype_rel: A ⊆r B,
compose: f o g,
rotate: rot(n),
nequal: a ≠ b ∈ T ,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
fun_exp0_lemma,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_properties,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
decidable__lt,
lelt_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
subtract_wf,
int_seg_wf,
subtract-1-ge-0,
nat_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
primrec-unroll,
equal-wf-base,
int_subtype_base,
ifthenelse_wf,
le_int_wf,
bnot_wf,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
equal-wf-T-base,
equal_wf,
add-member-int_seg2,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :functionExtensionality_alt,
addEquality,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
Error :dependent_set_memberEquality_alt,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
hyp_replacement,
applyLambdaEquality,
functionExtensionality
Latex:
\mforall{}[n,i:\mBbbN{}]. rot(n)\^{}i = (\mlambda{}x.if x + i <z n then x + i else (x + i) - n fi ) supposing i \mleq{} n
Date html generated:
2019_06_20-PM-01_35_33
Last ObjectModification:
2018_10_03-PM-11_00_52
Theory : list_1
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