Nuprl Lemma : retraction-nat-nsub
∀k:ℕ+. ∃r:ℕ ⟶ ℕk. ∀x:ℕk. ((r x) = x ∈ ℕk)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
prop: ℙ,
lelt: i ≤ j < k,
int_seg: {i..j-},
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
nat_plus_wf,
int_seg_subtype_nat,
all_wf,
int_seg_wf,
nat_wf,
false_wf,
assert_of_bnot,
iff_weakening_uiff,
not_wf,
bnot_wf,
assert_wf,
iff_transitivity,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
less_than_wf,
le_wf,
and_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf
Rules used in proof :
functionExtensionality,
applyEquality,
impliesFunctionality,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
natural_numberEquality,
independent_pairFormation,
hypothesisEquality,
dependent_set_memberEquality,
independent_isectElimination,
productElimination,
sqequalRule,
equalityElimination,
unionElimination,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaEquality,
dependent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}\msupplus{}. \mexists{}r:\mBbbN{} {}\mrightarrow{} \mBbbN{}k. \mforall{}x:\mBbbN{}k. ((r x) = x)
Date html generated:
2017_09_29-PM-05_47_03
Last ObjectModification:
2017_09_04-PM-00_14_44
Theory : arithmetic
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