Nuprl Lemma : rotate-by-id
∀[n,i:ℕ]. uiff(rotate-by(n;i) = (λx.x) ∈ (ℕn ⟶ ℕn);(i rem n) = 0 ∈ ℤ supposing 0 < n)
Proof
Definitions occuring in Statement :
rotate-by: rotate-by(n;i),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
remainder: n rem m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
rotate-by: rotate-by(n;i),
guard: {T},
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
sq_type: SQType(T),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
less_than_wf,
equal-wf-T-base,
int_seg_wf,
rotate-by_wf,
isect_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
nat_wf,
decidable__equal_int,
intformnot_wf,
int_formula_prop_not_lemma,
false_wf,
lelt_wf,
int_seg_properties,
zero-add,
decidable__lt,
intformle_wf,
int_formula_prop_le_lemma,
rem_addition,
int_seg_subtype_nat,
less_than_transitivity2,
subtype_base_sq,
int_subtype_base,
decidable__le,
equal-wf-base,
add_nat_wf,
remainder_wf,
le_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
equal_wf,
add-zero,
rem_bounds_1,
squash_wf,
true_wf,
rem_base_case,
add_functionality_wrt_eq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
baseClosed,
lambdaEquality,
remainderEquality,
lambdaFormation,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
computeAll,
productElimination,
independent_pairEquality,
applyLambdaEquality,
applyEquality,
functionExtensionality,
unionElimination,
dependent_set_memberEquality,
instantiate,
cumulativity,
independent_functionElimination,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[n,i:\mBbbN{}]. uiff(rotate-by(n;i) = (\mlambda{}x.x);(i rem n) = 0 supposing 0 < n)
Date html generated:
2018_05_21-PM-08_17_44
Last ObjectModification:
2017_07_26-PM-05_51_11
Theory : general
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