Nuprl Lemma : pairwise-mapl
∀[T,T':Type].
∀L:T List. ∀f:{x:T| (x ∈ L)} ⟶ T'.
∀[P:T' ⟶ T' ⟶ ℙ']. ((∀x,y:T. ((x ∈ L) ⇒ (y ∈ L) ⇒ P[f x;f y])) ⇒ (∀x,y∈mapl(f;L). P[x;y]))
Proof
Definitions occuring in Statement :
mapl: mapl(f;l),
pairwise: (∀x,y∈L. P[x; y]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
ge: i ≥ j ,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
nat_plus: ℕ+,
cons: [a / b],
select: L[n],
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l),
false: False,
not: ¬A,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
mapl: mapl(f;l),
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
true: True,
or: P ∨ Q,
uimplies: b supposing a,
guard: {T},
cand: A c∧ B
Lemmas referenced :
subtype_rel_sets_simple,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
nat_properties,
select_wf,
length_wf,
false_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
intformand_wf,
add-is-int-iff,
nat_plus_properties,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
length_wf_nat,
add_nat_plus,
length_of_cons_lemma,
istype-le,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
istype-void,
l_all_iff,
subtype_rel_self,
list_induction,
all_wf,
l_member_wf,
uall_wf,
pairwise_wf2,
mapl_wf,
list_wf,
map_nil_lemma,
pairwise-nil,
nil_wf,
map_cons_lemma,
pairwise-cons,
cons_wf,
cons_member,
subtype_rel_dep_function,
subtype_rel_sets,
equal_wf,
set_wf
Rules used in proof :
hyp_replacement,
inrFormation_alt,
int_eqEquality,
baseClosed,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
applyLambdaEquality,
approximateComputation,
unionElimination,
dependent_pairFormation_alt,
productIsType,
isect_memberEquality_alt,
productEquality,
equalityIstype,
setIsType,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
universeIsType,
inhabitedIsType,
functionIsType,
lambdaFormation_alt,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
setEquality,
because_Cache,
hypothesis,
applyEquality,
universeEquality,
setElimination,
rename,
functionExtensionality,
dependent_set_memberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
natural_numberEquality,
inlFormation,
independent_isectElimination,
inrFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation
Latex:
\mforall{}[T,T':Type].
\mforall{}L:T List. \mforall{}f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} T'.
\mforall{}[P:T' {}\mrightarrow{} T' {}\mrightarrow{} \mBbbP{}']
((\mforall{}x,y:T. ((x \mmember{} L) {}\mRightarrow{} (y \mmember{} L) {}\mRightarrow{} P[f x;f y])) {}\mRightarrow{} (\mforall{}x,y\mmember{}mapl(f;L). P[x;y]))
Date html generated:
2019_10_15-AM-10_23_20
Last ObjectModification:
2019_09_24-PM-06_05_38
Theory : list_1
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