Nuprl Lemma : fpf-join-list-ap
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]
∀L:a:A fp-> B[a] List. ∀x:A. (∃f∈L. (↑x ∈ dom(f)) ∧ (⊕(L)(x) = f(x) ∈ B[x])) supposing ↑x ∈ dom(⊕(L))
Proof
Definitions occuring in Statement :
fpf-join-list: ⊕(L),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
l_exists: (∃x∈L. P[x]),
list: T List,
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
fpf-join-list: ⊕(L),
fpf-empty: ⊗,
fpf-dom: x ∈ dom(f),
pi1: fst(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
squash: ↓T,
true: True,
guard: {T},
not: ¬A,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
list_induction,
fpf_wf,
all_wf,
isect_wf,
assert_wf,
fpf-dom_wf,
fpf-join-list_wf,
top_wf,
subtype_rel_list,
subtype-fpf2,
l_exists_wf,
l_member_wf,
equal_wf,
fpf-ap_wf,
list_wf,
reduce_nil_lemma,
deq_member_nil_lemma,
false_wf,
reduce_cons_lemma,
assert_witness,
fpf-join_wf,
l_exists_cons,
fpf-join-dom,
decidable__assert,
deq_wf,
squash_wf,
true_wf,
fpf-join-ap-left,
iff_weakening_equal,
select_wf,
int_seg_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,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
fpf-join-ap,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
productEquality,
setEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
unionElimination,
functionEquality,
universeEquality,
inlFormation,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inrFormation,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
equalityElimination,
promote_hyp,
instantiate
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}L:a:A fp-> B[a] List. \mforall{}x:A.
(\mexists{}f\mmember{}L. (\muparrow{}x \mmember{} dom(f)) \mwedge{} (\moplus{}(L)(x) = f(x))) supposing \muparrow{}x \mmember{} dom(\moplus{}(L))
Date html generated:
2018_05_21-PM-09_22_55
Last ObjectModification:
2018_02_09-AM-10_18_59
Theory : finite!partial!functions
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