Nuprl Lemma : canonical-function_wf
∀X:{X:Type| X ⊆r Base} . ∀f:ℕ ⟶ X. (canonical-function(f) ∈ {g:Base| g = f ∈ (ℕ ⟶ X)} )
Proof
Definitions occuring in Statement :
canonical-function: canonical-function(f),
nat: ℕ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
canonical-function: canonical-function(f),
prop: ℙ,
has-value: (a)↓,
and: P ∧ Q,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
gt: i > j
Lemmas referenced :
nat_wf,
subtype_rel_wf,
base_wf,
equal-wf-base,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
bottom-sqle,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
not-gt-2,
le_wf,
has-value_wf_base,
is-exception_wf,
exception-not-value,
value-type-has-value,
int-value-type,
less_than_transitivity1,
less_than_irreflexivity,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
hypothesis,
Error :functionIsType,
Error :universeIsType,
introduction,
extract_by_obid,
hypothesisEquality,
Error :setIsType,
universeEquality,
isectElimination,
pointwiseFunctionality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
because_Cache,
sqequalSqle,
divergentSqle,
callbyvalueLess,
productElimination,
Error :inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isect_memberEquality_alt,
independent_pairFormation,
voidElimination,
natural_numberEquality,
imageMemberEquality,
imageElimination,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType2,
promote_hyp,
dependent_functionElimination,
instantiate,
applyEquality,
Error :dependent_set_memberEquality_alt,
Error :equalityIsType1,
cumulativity,
sqleReflexivity,
lessExceptionCases,
axiomSqleEquality,
intEquality,
exceptionSqequal,
functionExtensionality
Latex:
\mforall{}X:\{X:Type| X \msubseteq{}r Base\} . \mforall{}f:\mBbbN{} {}\mrightarrow{} X. (canonical-function(f) \mmember{} \{g:Base| g = f\} )
Date html generated:
2019_06_20-AM-11_28_10
Last ObjectModification:
2018_09_29-PM-11_20_57
Theory : call!by!value_2
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