Nuprl Lemma : fpf-sub-val
∀[A:Type]. ∀[B:A ⟶ Type].
  ∀eq:EqDecider(A). ∀f,g:a:A fp-> B[a]. ∀x:A.
    ∀[P:a:A ⟶ B[a] ⟶ ℙ]. z != f(x) ==> P[x;z] 
⇒ z != g(x) ==> P[x;z] supposing g ⊆ f
Proof
Definitions occuring in Statement : 
fpf-sub: f ⊆ g
, 
fpf-val: z != f(x) ==> P[a; z]
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s1;s2]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
fpf-val: z != f(x) ==> P[a; z]
, 
fpf-sub: f ⊆ g
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
cand: A c∧ B
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
top: Top
, 
prop: ℙ
, 
so_apply: x[s1;s2]
Lemmas referenced : 
assert_witness, 
fpf-dom_wf, 
subtype-fpf2, 
top_wf, 
assert_wf, 
fpf-ap_wf, 
all_wf, 
equal_wf, 
fpf_wf, 
deq_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
lambdaFormation, 
cut, 
introduction, 
sqequalHypSubstitution, 
lambdaEquality, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
productElimination, 
independent_pairEquality, 
extract_by_obid, 
isectElimination, 
cumulativity, 
applyEquality, 
functionExtensionality, 
hypothesis, 
independent_isectElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
because_Cache, 
independent_functionElimination, 
axiomEquality, 
rename, 
functionEquality, 
productEquality, 
universeEquality, 
hyp_replacement, 
equalitySymmetry, 
applyLambdaEquality
Latex:
\mforall{}[A:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].
    \mforall{}eq:EqDecider(A).  \mforall{}f,g:a:A  fp->  B[a].  \mforall{}x:A.
        \mforall{}[P:a:A  {}\mrightarrow{}  B[a]  {}\mrightarrow{}  \mBbbP{}].  z  !=  f(x)  ==>  P[x;z]  {}\mRightarrow{}  z  !=  g(x)  ==>  P[x;z]  supposing  g  \msubseteq{}  f
Date html generated:
2018_05_21-PM-09_23_57
Last ObjectModification:
2018_02_09-AM-10_19_29
Theory : finite!partial!functions
Home
Index