Nuprl Lemma : fpf-all-join-decl
∀[A:Type]
∀eq:EqDecider(A)
∀[P:x:A ⟶ Type ⟶ ℙ]
∀f,g:x:A fp-> Type.
(∀y∈dom(f). w=f(y) ⇒ P[y;w] ⇒ ∀y∈dom(g). w=g(y) ⇒ P[y;w] ⇒ ∀y∈dom(f ⊕ g). w=f ⊕ g(y) ⇒ P[y;w])
Proof
Definitions occuring in Statement :
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
not: ¬A,
false: False
Lemmas referenced :
fpf-join-dom2,
fpf-join-ap-sq,
assert_wf,
fpf-dom_wf,
fpf-join_wf,
top_wf,
subtype-fpf2,
fpf-all_wf,
fpf_wf,
deq_wf,
bool_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,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
cumulativity,
lambdaEquality,
applyEquality,
instantiate,
universeEquality,
independent_isectElimination,
functionExtensionality,
setElimination,
rename,
setEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
equalityElimination
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[P:x:A {}\mrightarrow{} Type {}\mrightarrow{} \mBbbP{}]
\mforall{}f,g:x:A fp-> Type.
(\mforall{}y\mmember{}dom(f). w=f(y) {}\mRightarrow{} P[y;w]
{}\mRightarrow{} \mforall{}y\mmember{}dom(g). w=g(y) {}\mRightarrow{} P[y;w]
{}\mRightarrow{} \mforall{}y\mmember{}dom(f \moplus{} g). w=f \moplus{} g(y) {}\mRightarrow{} P[y;w])
Date html generated:
2018_05_21-PM-09_30_13
Last ObjectModification:
2018_02_09-AM-10_24_47
Theory : finite!partial!functions
Home
Index