Nuprl Lemma : canonicalizable-function
∀[T:Type]
∀[B:T ⟶ Type]. retractible(T) ⇒ canonicalizable(x:T ⟶ B[x]) supposing ∀x:T. (B[x] ⊆r Base)
supposing (T ⊆r Base) ∧ value-type(T)
Proof
Definitions occuring in Statement :
retractible: retractible(T),
canonicalizable: canonicalizable(T),
value-type: value-type(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
value-type: value-type(T),
has-value: (a)↓,
all: ∀x:A. B[x],
implies: P ⇒ Q,
retractible: retractible(T),
exists: ∃x:A. B[x],
canonicalizable: canonicalizable(T),
prop: ℙ,
so_apply: x[s]
Lemmas referenced :
base_wf,
retractible_wf,
subtype_rel_wf,
value-type_wf,
equal-wf-base,
subtype_rel-equal,
has-value_wf_base,
is-exception_wf,
sqle_wf_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
Error :isect_memberEquality_alt,
isectElimination,
hypothesisEquality,
axiomSqleEquality,
Error :equalityIsType4,
Error :universeIsType,
because_Cache,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
rename,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :lambdaFormation_alt,
Error :dependent_pairFormation_alt,
independent_isectElimination,
Error :functionIsType,
applyEquality,
Error :inhabitedIsType,
Error :productIsType,
universeEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
sqequalSqle,
divergentSqle,
callbyvalueCallbyvalue,
callbyvalueReduce,
independent_functionElimination,
functionExtensionality,
cumulativity,
Error :equalityIsType1,
sqleReflexivity,
hyp_replacement,
applyLambdaEquality,
callbyvalueExceptionCases,
exceptionSqequal,
Error :equalityIsType2
Latex:
\mforall{}[T:Type]
\mforall{}[B:T {}\mrightarrow{} Type]. retractible(T) {}\mRightarrow{} canonicalizable(x:T {}\mrightarrow{} B[x]) supposing \mforall{}x:T. (B[x] \msubseteq{}r Base)
supposing (T \msubseteq{}r Base) \mwedge{} value-type(T)
Date html generated:
2019_06_20-AM-11_28_26
Last ObjectModification:
2018_09_28-PM-02_49_59
Theory : call!by!value_2
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