Nuprl Lemma : fpf-union-compatible-self
∀[A,C:Type]. ∀[B:A ⟶ Type].
  ∀eq:EqDecider(A). ∀f:x:A fp-> B[x] List. ∀R:(C List) ⟶ C ⟶ 𝔹.  fpf-union-compatible(A;C;x.B[x];eq;R;f;f) 
  supposing ∀a:A. (B[a] ⊆r C)
Proof
Definitions occuring in Statement : 
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]
, 
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-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g)
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
l_member: (x ∈ l)
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
nat: ℕ
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
false: False
, 
not: ¬A
, 
top: Top
, 
prop: ℙ
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
le: A ≤ B
Lemmas referenced : 
select_wf, 
fpf-ap_wf, 
nat_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, 
select_member, 
list_wf, 
lelt_wf, 
length_wf, 
equal_wf, 
l_member_wf, 
or_wf, 
not_wf, 
assert_wf, 
subtype_rel_list, 
fpf-dom_wf, 
subtype-fpf2, 
top_wf, 
bool_wf, 
fpf_wf, 
deq_wf, 
all_wf, 
subtype_rel_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
cut, 
introduction, 
sqequalRule, 
sqequalHypSubstitution, 
lambdaEquality, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
axiomEquality, 
hypothesis, 
rename, 
lambdaFormation, 
unionElimination, 
productElimination, 
dependent_pairFormation, 
extract_by_obid, 
isectElimination, 
applyEquality, 
functionExtensionality, 
because_Cache, 
independent_isectElimination, 
setElimination, 
natural_numberEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
equalitySymmetry, 
cumulativity, 
dependent_set_memberEquality, 
productEquality, 
functionEquality, 
universeEquality
Latex:
\mforall{}[A,C:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].
    \mforall{}eq:EqDecider(A).  \mforall{}f:x:A  fp->  B[x]  List.  \mforall{}R:(C  List)  {}\mrightarrow{}  C  {}\mrightarrow{}  \mBbbB{}.
        fpf-union-compatible(A;C;x.B[x];eq;R;f;f) 
    supposing  \mforall{}a:A.  (B[a]  \msubseteq{}r  C)
Date html generated:
2018_05_21-PM-09_18_24
Last ObjectModification:
2018_02_09-AM-10_17_07
Theory : finite!partial!functions
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