Nuprl Lemma : pairwise-map
∀[T,T2:Type].
∀L:T List. ∀[P:T2 ⟶ T2 ⟶ ℙ']. ∀[f:{x:T| (x ∈ L)} ⟶ T2]. ((∀x,y∈map(f;L). P[x;y]) ⇐⇒ (∀x,y∈L. P[f x;f y]))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
pairwise: (∀x,y∈L. P[x; y]),
top: Top,
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
le: A ≤ B
Lemmas referenced :
list-subtype,
list_wf,
l_member_wf,
length-map,
int_seg_wf,
length_wf,
pairwise_wf2,
map_wf,
equal_wf,
set_wf,
subtype_rel_list,
top_wf,
int_seg_properties,
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-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
setEquality,
independent_pairFormation,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
because_Cache,
setElimination,
rename,
natural_numberEquality,
instantiate,
functionExtensionality,
applyEquality,
lambdaEquality,
dependent_set_memberEquality,
independent_functionElimination,
functionEquality,
universeEquality,
independent_isectElimination,
productElimination,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[T,T2:Type].
\mforall{}L:T List
\mforall{}[P:T2 {}\mrightarrow{} T2 {}\mrightarrow{} \mBbbP{}']. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} T2].
((\mforall{}x,y\mmember{}map(f;L). P[x;y]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x,y\mmember{}L. P[f x;f y]))
Date html generated:
2017_04_17-AM-07_44_23
Last ObjectModification:
2017_02_27-PM-04_16_07
Theory : list_1
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