Nuprl Lemma : lub-assoc

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  ∀[lub:T ⟶ T ⟶ T]
    ∀[a,b,c:T].  ((lub (lub c)) (lub (lub b) c) ∈ T) 
    supposing ∀[a,b:T].  least-upper-bound(T;x,y.R[x;y];a;b;lub b) 
  supposing Order(T;x,y.R[x;y])


Proof




Definitions occuring in Statement :  least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c) order: Order(T;x,y.R[x; y]) uimplies: supposing a uall: [x:A]. B[x] prop: so_apply: x[s1;s2] apply: a function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] prop: so_lambda: λ2x.t[x] so_apply: x[s]
Lemmas referenced :  least-upper-bound-assoc uall_wf least-upper-bound_wf order_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality because_Cache applyEquality lemma_by_obid sqequalRule lambdaEquality independent_isectElimination equalitySymmetry isect_memberEquality axiomEquality equalityTransitivity functionEquality cumulativity universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}[lub:T  {}\mrightarrow{}  T  {}\mrightarrow{}  T]
        \mforall{}[a,b,c:T].    ((lub  a  (lub  b  c))  =  (lub  (lub  a  b)  c)) 
        supposing  \mforall{}[a,b:T].    least-upper-bound(T;x,y.R[x;y];a;b;lub  a  b) 
    supposing  Order(T;x,y.R[x;y])



Date html generated: 2016_05_13-PM-04_18_22
Last ObjectModification: 2015_12_26-AM-11_27_25

Theory : rel_1


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