Nuprl Lemma : causal_order_sigma
∀[T,A:Type].
∀L:T List
∀[R:ℕ||L|| ⟶ ℕ||L|| ⟶ ℙ]. ∀[P,Q:A ⟶ ℕ||L|| ⟶ ℙ].
(Trans(ℕ||L||)(R _1 _2)
⇒ (∀x:A. causal_order(L;R;λi.P[x;i];λi.Q[x;i]))
⇒ causal_order(L;R;λi.∃x:A. P[x;i];λi.∃x:A. Q[x;i]))
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,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
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,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
and: P ∧ Q,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
cand: A c∧ B
Lemmas referenced :
exists_wf,
int_seg_wf,
length_wf,
all_wf,
le_wf,
subtype_rel_self,
trans_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
natural_numberEquality,
functionEquality,
productEquality,
setElimination,
rename,
instantiate,
universeEquality,
because_Cache,
inhabitedIsType,
functionIsType,
universeIsType,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}[T,A:Type].
\mforall{}L:T List
\mforall{}[R:\mBbbN{}||L|| {}\mrightarrow{} \mBbbN{}||L|| {}\mrightarrow{} \mBbbP{}]. \mforall{}[P,Q:A {}\mrightarrow{} \mBbbN{}||L|| {}\mrightarrow{} \mBbbP{}].
(Trans(\mBbbN{}||L||)(R $_{1}$ $_{2}$)
{}\mRightarrow{} (\mforall{}x:A. causal\_order(L;R;\mlambda{}i.P[x;i];\mlambda{}i.Q[x;i]))
{}\mRightarrow{} causal\_order(L;R;\mlambda{}i.\mexists{}x:A. P[x;i];\mlambda{}i.\mexists{}x:A. Q[x;i]))
Date html generated:
2019_10_15-AM-10_57_41
Last ObjectModification:
2018_09_27-AM-09_52_36
Theory : list!
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