Nuprl Lemma : urefl_shift

[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ].
  ∀f:A ⟶ B
    ((∀[x,y:A].  R[x;y] supposing R[x;y])
     RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)
     UniformlyRefl(B;x,y.S[x;y])
     UniformlyRefl(A;x,y.R[x;y]))


Proof




Definitions occuring in Statement :  rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f) urefl: UniformlyRefl(T;x,y.E[x; y]) uimplies: supposing a uall: [x:A]. B[x] prop: so_apply: x[s1;s2] all: x:A. B[x] implies:  Q function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] all: x:A. B[x] implies:  Q urefl: UniformlyRefl(T;x,y.E[x; y]) rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f) member: t ∈ T prop: so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] so_lambda: λ2x.t[x] uimplies: supposing a subtype_rel: A ⊆B so_apply: x[s] guard: {T} iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q
Lemmas referenced :  urefl_wf rels_iso_wf uall_wf isect_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution hypothesisEquality cut lemma_by_obid isectElimination thin sqequalRule lambdaEquality applyEquality hypothesis universeEquality because_Cache functionEquality cumulativity independent_isectElimination dependent_functionElimination productElimination independent_functionElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[S:B  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}f:A  {}\mrightarrow{}  B
        ((\mforall{}[x,y:A].    R[x;y]  supposing  R[x;y])
        {}\mRightarrow{}  RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)
        {}\mRightarrow{}  UniformlyRefl(B;x,y.S[x;y])
        {}\mRightarrow{}  UniformlyRefl(A;x,y.R[x;y]))



Date html generated: 2016_05_15-PM-00_03_44
Last ObjectModification: 2015_12_26-PM-11_24_51

Theory : gen_algebra_1


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