Nuprl Lemma : fpf-contains-union-join-left2
∀[A:Type]. ∀[B:A ⟶ Type].
∀eq:EqDecider(A). ∀f,h,g:a:A fp-> B[a] List. ∀R:⋂a:A. ((B[a] List) ⟶ B[a] ⟶ 𝔹).
(h ⊆⊆ f ⇒ h ⊆⊆ fpf-union-join(eq;R;f;g))
Proof
Definitions occuring in Statement :
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf-contains: f ⊆⊆ g,
fpf: a:A fp-> B[a],
list: T List,
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
fpf-contains: f ⊆⊆ g,
member: t ∈ T,
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
or: P ∨ Q,
true: True,
l_contains: A ⊆ B,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
fpf-cap: f(x)?z,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff
Lemmas referenced :
assert_wf,
fpf-dom_wf,
subtype-fpf2,
list_wf,
top_wf,
fpf-contains_wf,
bool_wf,
fpf_wf,
deq_wf,
fpf-union-join-dom,
assert_elim,
subtype_base_sq,
bool_subtype_base,
fpf-union-join-ap,
fpf-union-contains,
l_all_iff,
fpf-cap_wf,
nil_wf,
l_member_wf,
fpf-union_wf,
select_wf,
fpf-ap_wf,
int_seg_properties,
length_wf,
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,
int_seg_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
isectEquality,
functionEquality,
universeEquality,
productElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
inlFormation,
natural_numberEquality,
setElimination,
rename,
setEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
baseClosed,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eq:EqDecider(A). \mforall{}f,h,g:a:A fp-> B[a] List. \mforall{}R:\mcap{}a:A. ((B[a] List) {}\mrightarrow{} B[a] {}\mrightarrow{} \mBbbB{}).
(h \msubseteq{}\msubseteq{} f {}\mRightarrow{} h \msubseteq{}\msubseteq{} fpf-union-join(eq;R;f;g))
Date html generated:
2018_05_21-PM-09_23_49
Last ObjectModification:
2018_02_09-AM-10_19_25
Theory : finite!partial!functions
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