Nuprl Lemma : greatest-lower-bound-unique

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  ∀[a,b,c,d:T].
    (c d ∈ T) supposing (greatest-lower-bound(T;x,y.R[x;y];a;b;c) and greatest-lower-bound(T;x,y.R[x;y];a;b;d)) 
  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] 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 prop: so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] guard: {T} greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c) and: P ∧ Q all: x:A. B[x] implies:  Q order: Order(T;x,y.R[x; y]) anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :  greatest-lower-bound_wf order_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality productElimination dependent_functionElimination independent_functionElimination

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



Date html generated: 2016_05_13-PM-04_18_27
Last ObjectModification: 2015_12_26-AM-11_27_27

Theory : rel_1


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