Nuprl Lemma : pairwise-sublist
∀[T:Type]. ∀L1,L2:T List. ∀[P:T ⟶ T ⟶ ℙ']. (L1 ⊆ L2 ⇒ (∀x,y∈L2. P[x;y]) ⇒ (∀x,y∈L1. P[x;y]))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
sublist: L1 ⊆ L2,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
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 y.t[x; y],
so_apply: x[s1;s2],
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
pairwise: (∀x,y∈L. P[x; y]),
and: P ∧ Q,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
nat: ℕ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
pairwise_wf2,
sublist_wf,
list_wf,
equal_wf,
squash_wf,
true_wf,
int_seg_properties,
length_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
select_wf,
int_seg_wf,
non_neg_length,
decidable__le,
length_wf_nat,
nat_properties,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
iff_weakening_equal,
increasing_implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
applyLambdaEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. \mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}']. (L1 \msubseteq{} L2 {}\mRightarrow{} (\mforall{}x,y\mmember{}L2. P[x;y]) {}\mRightarrow{} (\mforall{}x,y\mmember{}L1. P[x;y]))
Date html generated:
2017_04_17-AM-07_44_32
Last ObjectModification:
2017_02_27-PM-04_16_58
Theory : list_1
Home
Index