Nuprl Lemma : glb-com

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  ∀[glb:T ⟶ T ⟶ T]
    ∀[a,b:T].  ((glb b) (glb a) ∈ T) supposing ∀[a,b:T].  greatest-lower-bound(T;x,y.R[x;y];a;b;glb b) 
  supposing Order(T;x,y.R[x;y])


Proof




Definitions occuring in Statement :  greatest-lower-bound: greatest-lower-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 :  greatest-lower-bound-com uall_wf greatest-lower-bound_wf order_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality because_Cache lemma_by_obid sqequalRule lambdaEquality applyEquality independent_isectElimination isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}[glb:T  {}\mrightarrow{}  T  {}\mrightarrow{}  T]
        \mforall{}[a,b:T].    ((glb  a  b)  =  (glb  b  a)) 
        supposing  \mforall{}[a,b:T].    greatest-lower-bound(T;x,y.R[x;y];a;b;glb  a  b) 
    supposing  Order(T;x,y.R[x;y])



Date html generated: 2016_05_13-PM-04_18_32
Last ObjectModification: 2015_12_26-AM-11_27_41

Theory : rel_1


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