### Nuprl Lemma : causal_order_monotonic3

`∀[T:Type]`
`  ∀L:T List`
`    ∀[P1,P2,Q1,Q2:ℕ||L|| ⟶ ℙ]. ∀[R1,R2:ℕ||L|| ⟶ ℕ||L|| ⟶ ℙ].`
`      ((∀i:ℕ||L||. ((P1 i) `` (P2 i)))`
`      `` (∀i:ℕ||L||. ((Q2 i) `` (Q1 i)))`
`      `` (∀i,j:ℕ||L||.  ((R1 i j) `` (R2 i j)))`
`      `` causal_order(L;R1;P1;Q1)`
`      `` causal_order(L;R2;P2;Q2))`

Proof

Definitions occuring in Statement :  causal_order: `causal_order(L;R;P;Q)` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f 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: `P `` Q` causal_order: `causal_order(L;R;P;Q)` member: `t ∈ T` exists: `∃x:A. B[x]` and: `P ∧ Q` cand: `A c∧ B` prop: `ℙ` int_seg: `{i..j-}` subtype_rel: `A ⊆r 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 inhabitedIsType functionIsType

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List
\mforall{}[P1,P2,Q1,Q2:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[R1,R2:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}i:\mBbbN{}||L||.  ((P1  i)  {}\mRightarrow{}  (P2  i)))
{}\mRightarrow{}  (\mforall{}i:\mBbbN{}||L||.  ((Q2  i)  {}\mRightarrow{}  (Q1  i)))
{}\mRightarrow{}  (\mforall{}i,j:\mBbbN{}||L||.    ((R1  i  j)  {}\mRightarrow{}  (R2  i  j)))
{}\mRightarrow{}  causal\_order(L;R1;P1;Q1)
{}\mRightarrow{}  causal\_order(L;R2;P2;Q2))

Date html generated: 2019_10_15-AM-10_57_49
Last ObjectModification: 2018_09_27-AM-09_50_18

Theory : list!

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