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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