Nuprl Lemma : fpf-sub-join-right2
∀[A:Type]. ∀[B,C:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> B[a]]. ∀[g:a:A fp-> C[a]].
  g ⊆ f ⊕ g supposing ∀x:A. (((↑x ∈ dom(f)) ∧ (↑x ∈ dom(g))) 
⇒ ((B[x] ⊆r C[x]) c∧ (f(x) = g(x) ∈ C[x])))
Proof
Definitions occuring in Statement : 
fpf-join: f ⊕ g
, 
fpf-sub: f ⊆ g
, 
fpf-ap: f(x)
, 
fpf-dom: x ∈ dom(f)
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
assert: ↑b
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
cand: A c∧ B
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
fpf-sub: f ⊆ g
, 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
cand: A c∧ B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
and: P ∧ Q
, 
respects-equality: respects-equality(S;T)
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
or: P ∨ Q
, 
fpf-join: f ⊕ g
, 
fpf-ap: f(x)
, 
pi2: snd(t)
, 
fpf-cap: f(x)?z
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
Lemmas referenced : 
assert_witness, 
fpf-dom_wf, 
fpf-join_wf, 
top_wf, 
subtype-fpf2, 
istype-assert, 
subtype_rel_wf, 
fpf-ap_wf, 
subtype-respects-equality, 
fpf_wf, 
deq_wf, 
istype-universe, 
fpf-join-dom2, 
equal-wf-T-base, 
bool_wf, 
assert_wf, 
bnot_wf, 
not_wf, 
eqtt_to_assert, 
uiff_transitivity, 
eqff_to_assert, 
assert_of_bnot, 
trivial-equal
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation_alt, 
introduction, 
cut, 
thin, 
lambdaFormation_alt, 
independent_pairFormation, 
hypothesis, 
because_Cache, 
sqequalHypSubstitution, 
dependent_functionElimination, 
hypothesisEquality, 
independent_functionElimination, 
lambdaEquality_alt, 
productElimination, 
independent_pairEquality, 
extract_by_obid, 
isectElimination, 
inhabitedIsType, 
applyEquality, 
independent_isectElimination, 
Error :memTop, 
universeIsType, 
axiomEquality, 
functionIsTypeImplies, 
functionIsType, 
productIsType, 
equalityIstype, 
instantiate, 
universeEquality, 
inrFormation_alt, 
equalityTransitivity, 
equalitySymmetry, 
baseClosed, 
unionElimination, 
equalityElimination
Latex:
\mforall{}[A:Type].  \mforall{}[B,C:A  {}\mrightarrow{}  Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[f:a:A  fp->  B[a]].  \mforall{}[g:a:A  fp->  C[a]].
    g  \msubseteq{}  f  \moplus{}  g  supposing  \mforall{}x:A.  (((\muparrow{}x  \mmember{}  dom(f))  \mwedge{}  (\muparrow{}x  \mmember{}  dom(g)))  {}\mRightarrow{}  ((B[x]  \msubseteq{}r  C[x])  c\mwedge{}  (f(x)  =  g(x))))
Date html generated:
2020_05_20-AM-09_02_43
Last ObjectModification:
2020_01_28-PM-03_39_43
Theory : finite!partial!functions
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