### Nuprl Lemma : causal_order_transitivity

`∀[T:Type]`
`  ∀L:T List`
`    ∀[R:ℕ||L|| ⟶ ℕ||L|| ⟶ ℙ]. ∀[P1,P2,P3:ℕ||L|| ⟶ ℙ].`
`      (Trans(ℕ||L||)(R _1 _2) `` causal_order(L;R;P1;P2) `` causal_order(L;R;P2;P3) `` causal_order(L;R;P1;P3))`

Proof

Definitions occuring in Statement :  causal_order: `causal_order(L;R;P;Q)` length: `||as||` list: `T List` trans: `Trans(T;x,y.E[x; y])` 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 :  causal_order: `causal_order(L;R;P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` exists: `∃x:A. B[x]` and: `P ∧ Q` cand: `A c∧ B` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` trans: `Trans(T;x,y.E[x; y])`
Lemmas referenced :  int_seg_properties length_wf decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf le_wf int_seg_wf all_wf exists_wf subtype_rel_self trans_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination productElimination dependent_pairFormation introduction extract_by_obid isectElimination natural_numberEquality setElimination rename because_Cache unionElimination independent_isectElimination approximateComputation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation productEquality applyEquality functionEquality instantiate universeEquality cumulativity

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List
\mforall{}[R:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[P1,P2,P3:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].
(Trans(\mBbbN{}||L||)(R  \$_{1}\$  \$_{2}\$)
{}\mRightarrow{}  causal\_order(L;R;P1;P2)
{}\mRightarrow{}  causal\_order(L;R;P2;P3)
{}\mRightarrow{}  causal\_order(L;R;P1;P3))

Date html generated: 2018_05_21-PM-06_20_38
Last ObjectModification: 2018_05_19-PM-05_32_49

Theory : list!

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