Nuprl Lemma : trans_shift

[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ].
  ∀f:A ⟶ B. (RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)  Trans(B;x,y.S[x;y])  Trans(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) trans: Trans(T;x,y.E[x; y]) 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 trans: Trans(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_apply: x[s1;s2] so_lambda: λ2y.t[x; y] iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q
Lemmas referenced :  trans_wf rels_iso_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution applyEquality hypothesisEquality cut lemma_by_obid isectElimination thin sqequalRule lambdaEquality hypothesis functionEquality cumulativity universeEquality 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.  (RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)  {}\mRightarrow{}  Trans(B;x,y.S[x;y])  {}\mRightarrow{}  Trans(A;x,y.R[x;y]))



Date html generated: 2016_05_15-PM-00_03_28
Last ObjectModification: 2015_12_26-PM-11_24_59

Theory : gen_algebra_1


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