Nuprl Lemma : do-apply-p-first-disjoint
∀[A,B:Type]. ∀[L:(A ⟶ (B + Top)) List]. ∀[x:A].
∀[f:A ⟶ (B + Top)]. (do-apply(p-first(L);x) = do-apply(f;x) ∈ B) supposing ((↑can-apply(f;x)) and (f ∈ L))
supposing (∀f,g∈L. p-disjoint(A;f;g))
Proof
Definitions occuring in Statement :
p-disjoint: p-disjoint(A;f;g),
p-first: p-first(L),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
pairwise: (∀x,y∈L. P[x; y]),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
l_member: (x ∈ l),
exists: ∃x:A. B[x],
l_exists: (∃x∈L. P[x]),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
cand: A c∧ B,
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
less_than: a < b,
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
less_than': less_than'(a;b),
pairwise: (∀x,y∈L. P[x; y]),
select: L[n],
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
bfalse: ff,
p-disjoint: p-disjoint(A;f;g)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
do-apply-p-first,
do-apply_wf,
iff_weakening_equal,
assert_wf,
can-apply_wf,
subtype_rel_union,
top_wf,
l_member_wf,
pairwise_wf2,
subtype_rel_list,
subtype_rel_dep_function,
p-disjoint_wf,
list_wf,
can-apply-p-first,
lelt_wf,
length_wf,
and_wf,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
less_than_transitivity1,
less_than_irreflexivity,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
length_of_nil_lemma,
stuck-spread,
base_wf,
filter_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
all_wf,
int_seg_wf,
cons_wf,
filter_cons_lemma,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
reduce_hd_cons_lemma,
false_wf,
length_of_cons_lemma,
pairwise-cons,
cons_member,
not_assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
independent_isectElimination,
cumulativity,
functionExtensionality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
functionEquality,
unionEquality,
instantiate,
lambdaFormation,
dependent_functionElimination,
dependent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
unionElimination,
int_eqEquality,
intEquality,
computeAll,
intWeakElimination,
promote_hyp,
hypothesis_subsumption,
addEquality,
equalityElimination,
hyp_replacement,
addLevel,
levelHypothesis
Latex:
\mforall{}[A,B:Type]. \mforall{}[L:(A {}\mrightarrow{} (B + Top)) List]. \mforall{}[x:A].
\mforall{}[f:A {}\mrightarrow{} (B + Top)]
(do-apply(p-first(L);x) = do-apply(f;x)) supposing ((\muparrow{}can-apply(f;x)) and (f \mmember{} L))
supposing (\mforall{}f,g\mmember{}L. p-disjoint(A;f;g))
Date html generated:
2018_05_21-PM-06_50_10
Last ObjectModification:
2017_07_26-PM-04_57_18
Theory : general
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