Nuprl Lemma : causal_order_monotonic2

[T:Type]
  ∀L:T List
    ∀[P1,P2,Q:ℕ||L|| ⟶ ℙ]. ∀[R:ℕ||L|| ⟶ ℕ||L|| ⟶ ℙ].
      ((∀i:ℕ||L||. ((P1 i)  (P2 i)))  causal_order(L;R;P1;Q)  causal_order(L;R;P2;Q))


Proof




Definitions occuring in Statement :  causal_order: causal_order(L;R;P;Q) length: ||as|| list: List int_seg: {i..j-} uall: [x:A]. B[x] prop: all: x:A. B[x] implies:  Q apply: a function: x:A ⟶ B[x] natural_number: $n universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] all: x:A. B[x] implies:  Q causal_order: causal_order(L;R;P;Q) member: t ∈ T exists: x:A. B[x] and: P ∧ Q cand: c∧ B prop: int_seg: {i..j-} subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] guard: {T}
Lemmas referenced :  le_wf subtype_rel_self causal_order_wf all_wf int_seg_wf length_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation sqequalHypSubstitution lambdaFormation_alt cut hypothesis dependent_functionElimination thin hypothesisEquality independent_functionElimination productElimination dependent_pairFormation independent_pairFormation productEquality introduction extract_by_obid isectElimination setElimination rename applyEquality because_Cache sqequalRule instantiate universeEquality universeIsType natural_numberEquality lambdaEquality functionEquality functionIsType inhabitedIsType

Latex:
\mforall{}[T:Type]
    \mforall{}L:T  List
        \mforall{}[P1,P2,Q:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[R:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].
            ((\mforall{}i:\mBbbN{}||L||.  ((P1  i)  {}\mRightarrow{}  (P2  i)))  {}\mRightarrow{}  causal\_order(L;R;P1;Q)  {}\mRightarrow{}  causal\_order(L;R;P2;Q))



Date html generated: 2019_10_15-AM-10_57_46
Last ObjectModification: 2018_09_27-AM-09_50_20

Theory : list!


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