Nuprl Lemma : nat-int-retraction-reals-1
∀k:{2...}. ∃r:(ℕ ⟶ ℤ) ⟶ ℝ. ∀x:ℝ. (accelerate(k;x) = (r (λn.(x (n + 1)))) ∈ ℝ)
Proof
Definitions occuring in Statement :
accelerate: accelerate(k;f),
real: ℝ,
int_upper: {i...},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
int_upper: {i...},
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
nat: ℕ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x.t[x],
real: ℝ,
subtract: n - m,
so_apply: x[s],
squash: ↓T,
less_than: a < b
Lemmas referenced :
int-int-retraction-reals-1,
real-regular,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
real_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_wf,
le_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
subtract_wf,
int_upper_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,
all_wf,
accelerate_wf,
regular-int-seq_wf,
nat_plus_wf,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
int_upper_wf,
squash_wf,
true_wf,
minus-minus,
add-swap,
iff_weakening_equal,
subtract-add-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
dependent_set_memberEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
dependent_pairFormation,
functionExtensionality,
functionEquality,
because_Cache,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
approximateComputation,
int_eqEquality,
addEquality,
minusEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}k:\{2...\}. \mexists{}r:(\mBbbN{} {}\mrightarrow{} \mBbbZ{}) {}\mrightarrow{} \mBbbR{}. \mforall{}x:\mBbbR{}. (accelerate(k;x) = (r (\mlambda{}n.(x (n + 1)))))
Date html generated:
2017_10_03-AM-10_07_06
Last ObjectModification:
2017_07_05-PM-03_52_56
Theory : reals
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