Nuprl Lemma : fpf-contains-union-join-right2
∀[A,V:Type]. ∀[B:A ⟶ Type].
∀eq:EqDecider(A). ∀f,h,g:a:A fp-> B[a] List. ∀R:(V List) ⟶ V ⟶ 𝔹.
fpf-union-compatible(A;V;x.B[x];eq;R;f;g) ⇒ h ⊆⊆ g ⇒ h ⊆⊆ fpf-union-join(eq;R;f;g)
supposing fpf-single-valued(A;eq;x.B[x];V;g)
supposing ∀a:A. (B[a] ⊆r V)
Proof
Definitions occuring in Statement :
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf-contains: f ⊆⊆ g,
fpf-single-valued: fpf-single-valued(A;eq;x.B[x];V;g),
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
fpf: a:A fp-> B[a],
list: T List,
deq: EqDecider(T),
bool: 𝔹,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
fpf-single-valued: fpf-single-valued(A;eq;x.B[x];V;g),
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
fpf-contains: f ⊆⊆ g,
cand: A c∧ B,
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 :
equal_wf,
l_member_wf,
fpf-ap_wf,
list_wf,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf-contains_wf,
fpf-union-compatible_wf,
fpf-single-valued_wf,
bool_wf,
fpf_wf,
deq_wf,
all_wf,
subtype_rel_wf,
fpf-union-join-dom,
assert_elim,
subtype_base_sq,
bool_subtype_base,
fpf-union-contains2,
fpf-union-join-ap,
l_all_iff,
fpf-cap_wf,
nil_wf,
fpf-union_wf,
subtype_rel_dep_function,
subtype_rel_list,
subtype_rel_self,
select_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
rename,
lambdaFormation,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
independent_functionElimination,
independent_pairFormation,
functionEquality,
universeEquality,
productElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
inrFormation,
natural_numberEquality,
setElimination,
setEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
baseClosed,
equalityElimination
Latex:
\mforall{}[A,V:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eq:EqDecider(A). \mforall{}f,h,g:a:A fp-> B[a] List. \mforall{}R:(V List) {}\mrightarrow{} V {}\mrightarrow{} \mBbbB{}.
fpf-union-compatible(A;V;x.B[x];eq;R;f;g) {}\mRightarrow{} h \msubseteq{}\msubseteq{} g {}\mRightarrow{} h \msubseteq{}\msubseteq{} fpf-union-join(eq;R;f;g)
supposing fpf-single-valued(A;eq;x.B[x];V;g)
supposing \mforall{}a:A. (B[a] \msubseteq{}r V)
Date html generated:
2018_05_21-PM-09_23_54
Last ObjectModification:
2018_02_09-AM-10_19_27
Theory : finite!partial!functions
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