Nuprl Lemma : int-int-retraction-reals
∃r:(ℤ ⟶ ℤ) ⟶ ℝ. ∀x:ℝ. (x = (r (λi.if i <z 1 then x 1 else x i fi )))
Proof
Definitions occuring in Statement :
req: x = y,
real: ℝ,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
int_upper: {i...},
le: A ≤ B,
false: False,
not: ¬A,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real: ℝ,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
top: Top
Lemmas referenced :
real-regular,
less_than_wf,
real_wf,
int-int-retraction-reals-1,
false_wf,
le_wf,
all_wf,
squash_wf,
true_wf,
req_wf,
iff_weakening_equal,
req-iff-bdd-diff,
accelerate_wf,
regular-int-seq_wf,
nat_plus_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
trivial-bdd-diff,
bdd-diff_functionality,
bdd-diff_weakening,
accelerate-bdd-diff
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
hypothesis,
dependent_functionElimination,
productElimination,
dependent_pairFormation,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
setElimination,
rename,
functionExtensionality,
intEquality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
voidElimination,
isect_memberEquality,
voidEquality
Latex:
\mexists{}r:(\mBbbZ{} {}\mrightarrow{} \mBbbZ{}) {}\mrightarrow{} \mBbbR{}. \mforall{}x:\mBbbR{}. (x = (r (\mlambda{}i.if i <z 1 then x 1 else x i fi )))
Date html generated:
2017_10_03-AM-10_07_18
Last ObjectModification:
2017_07_28-AM-08_53_12
Theory : reals
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