Nuprl Lemma : fpf-join-list-ap-disjoint
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[L:a:A fp-> B[a] List]. ∀[x:A].
(∀[f:a:A fp-> B[a]]. (⊕(L)(x) = f(x) ∈ B[x]) supposing ((↑x ∈ dom(f)) and (f ∈ L))) supposing
((∀f,g∈L. ∀x:A. (¬((↑x ∈ dom(f)) ∧ (↑x ∈ dom(g))))) and
(↑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],
pairwise: (∀x,y∈L. P[x; y]),
l_member: (x ∈ l),
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],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
l_member: (x ∈ l),
cand: A c∧ B,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
pairwise: (∀x,y∈L. P[x; y]),
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
not: ¬A,
implies: P ⇒ Q,
squash: ↓T,
true: True,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
fpf-join-list-ap,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
l_member_wf,
fpf_wf,
pairwise_wf2,
all_wf,
not_wf,
fpf-join-list_wf,
subtype_rel_list,
list_wf,
equal_wf,
fpf-ap_wf,
decidable__lt,
lelt_wf,
length_wf,
assert_functionality_wrt_uiff,
squash_wf,
true_wf,
deq_wf,
nat_properties,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
select_wf,
le_wf,
less_than_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
cumulativity,
applyEquality,
sqequalRule,
lambdaEquality,
functionExtensionality,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
productEquality,
functionEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
unionElimination,
dependent_set_memberEquality,
independent_pairFormation,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[L:a:A fp-> B[a] List]. \mforall{}[x:A].
(\mforall{}[f:a:A fp-> B[a]]. (\moplus{}(L)(x) = f(x)) supposing ((\muparrow{}x \mmember{} dom(f)) and (f \mmember{} L))) supposing
((\mforall{}f,g\mmember{}L. \mforall{}x:A. (\mneg{}((\muparrow{}x \mmember{} dom(f)) \mwedge{} (\muparrow{}x \mmember{} dom(g))))) and
(\muparrow{}x \mmember{} dom(\moplus{}(L))))
Date html generated:
2018_05_21-PM-09_23_04
Last ObjectModification:
2018_02_09-AM-10_19_03
Theory : finite!partial!functions
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