Nuprl Lemma : fpf-sub-functionality2

  ∀[B:A ⟶ Type]. ∀[C:A' ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[eq':EqDecider(A')]. ∀[f,g:a:A fp-> B[a]].
    (f ⊆ g) supposing (f ⊆ and (∀a:A. (B[a] ⊆C[a]))) 
  supposing strong-subtype(A;A')


Definitions occuring in Statement :  fpf-sub: f ⊆ g fpf: a:A fp-> B[a] deq: EqDecider(T) strong-subtype: strong-subtype(A;B) uimplies: supposing a subtype_rel: A ⊆B uall: [x:A]. B[x] so_apply: x[s] all: x:A. B[x] function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a fpf-sub: f ⊆ g all: x:A. B[x] implies:  Q subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] top: Top prop: strong-subtype: strong-subtype(A;B) cand: c∧ B guard: {T} fpf-ap: f(x)
Lemmas referenced :  fpf-dom-type subtype-fpf2 top_wf assert_wf fpf-dom_wf subtype-fpf3 fpf-sub_witness fpf-sub_wf all_wf subtype_rel_wf fpf_wf deq_wf strong-subtype_wf fpf-dom_functionality2 strong-subtype-deq-subtype fpf-ap_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution lambdaFormation extract_by_obid isectElimination thin hypothesisEquality because_Cache applyEquality sqequalRule lambdaEquality functionExtensionality cumulativity hypothesis independent_isectElimination isect_memberEquality voidElimination voidEquality dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination productElimination functionEquality universeEquality independent_pairFormation hyp_replacement applyLambdaEquality

    \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[C:A'  {}\mrightarrow{}  Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[eq':EqDecider(A')].  \mforall{}[f,g:a:A  fp->  B[a]].
        (f  \msubseteq{}  g)  supposing  (f  \msubseteq{}  g  and  (\mforall{}a:A.  (B[a]  \msubseteq{}r  C[a]))) 
    supposing  strong-subtype(A;A')

Date html generated: 2018_05_21-PM-09_19_00
Last ObjectModification: 2018_02_09-AM-10_17_21

Theory : finite!partial!functions

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