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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