Nuprl Lemma : do-apply-p-first
∀[A,B:Type]. ∀[L:(A ⟶ (B + Top)) List]. ∀[x:A].
do-apply(p-first(L);x) = do-apply(hd(filter(λf.can-apply(f;x);L));x) ∈ B supposing ↑can-apply(p-first(L);x)
Proof
Definitions occuring in Statement :
p-first: p-first(L),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
hd: hd(l),
filter: filter(P;l),
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
lambda: λx.A[x],
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,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
bool: 𝔹,
unit: Unit,
btrue: tt,
do-apply: do-apply(f;x),
p-first: p-first(L),
can-apply: can-apply(f;x),
isl: isl(x),
outl: outl(x),
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
p-conditional: [f?g]
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
assert_wf,
can-apply_wf,
p-first_wf,
top_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,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtype_rel_list,
subtype_rel_dep_function,
subtype_rel_union,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
list_wf,
p_first_nil_lemma,
filter_nil_lemma,
false_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
assert_functionality_wrt_uiff,
cons_wf,
squash_wf,
true_wf,
p-first-append,
nil_wf,
p-conditional_wf,
p-conditional-to-p-first,
p-conditional-domain,
decidable__assert,
append_wf,
filter_cons_lemma,
bool_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
reduce_hd_cons_lemma,
list_accum_cons_lemma,
list_induction,
all_wf,
list_accum_wf,
list_accum_nil_lemma,
p-first-singleton,
iff_weakening_equal,
do-apply_wf,
outl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
because_Cache,
applyEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
unionEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
universeEquality,
functionExtensionality,
imageMemberEquality,
equalityElimination,
inlEquality,
hyp_replacement,
inrFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}[L:(A {}\mrightarrow{} (B + Top)) List]. \mforall{}[x:A].
do-apply(p-first(L);x) = do-apply(hd(filter(\mlambda{}f.can-apply(f;x);L));x)
supposing \muparrow{}can-apply(p-first(L);x)
Date html generated:
2018_05_21-PM-06_44_41
Last ObjectModification:
2017_07_26-PM-04_55_12
Theory : general
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