Nuprl Lemma : causal_order_filter_iseg
∀[T,T':Type].
∀L:T List. ∀P,Q:ℕ||L|| ⟶ 𝔹. ∀f,g:T ⟶ T'.
((∀L':T List. (L' ≤ L ⇒ map(f;filter2(P;L')) ≤ map(g;filter2(Q;L'))))
⇒ causal_order(L;λi,j. ((g L[i]) = (f L[j]) ∈ T');λi.(↑Q[i]);λi.(↑P[i])))
Proof
Definitions occuring in Statement :
causal_order: causal_order(L;R;P;Q),
filter2: filter2(P;L),
iseg: l1 ≤ l2,
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
causal_order: causal_order(L;R;P;Q),
nat: ℕ,
guard: {T},
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
int_iseg: {i...j},
cand: A c∧ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
list_wf,
iseg_wf,
map_wf,
filter2_wf,
iseg_length,
decidable__lt,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
int_seg_wf,
bool_wf,
assert_wf,
firstn_wf,
firstn-iseg,
int_seg_properties,
decidable__le,
itermConstant_wf,
itermAdd_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
le_wf,
iseg_member,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
squash_wf,
true_wf,
length_firstn_eq,
iff_weakening_equal,
subtype_rel_self,
select_wf,
member_filter2,
less_than_wf,
select_firstn,
equal_wf,
length_firstn,
member_map,
l_member_wf,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
functionExtensionality,
applyEquality,
independent_isectElimination,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
imageElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
universeEquality,
addEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T,T':Type].
\mforall{}L:T List. \mforall{}P,Q:\mBbbN{}||L|| {}\mrightarrow{} \mBbbB{}. \mforall{}f,g:T {}\mrightarrow{} T'.
((\mforall{}L':T List. (L' \mleq{} L {}\mRightarrow{} map(f;filter2(P;L')) \mleq{} map(g;filter2(Q;L'))))
{}\mRightarrow{} causal\_order(L;\mlambda{}i,j. ((g L[i]) = (f L[j]));\mlambda{}i.(\muparrow{}Q[i]);\mlambda{}i.(\muparrow{}P[i])))
Date html generated:
2017_10_01-AM-08_37_56
Last ObjectModification:
2017_07_26-PM-04_26_42
Theory : list!
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