### 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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