Nuprl Lemma : fpf-join-compatible-right
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B,C,D,E,F,G:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[g:a:A fp-> C[a]]. ∀[h:a:A fp-> D[a]].
({(h || f ⊕ g) supposing (h || g and h || f)}) supposing
((∀a:A. (E[a] ⊆r G[a])) and
(∀a:A. (F[a] ⊆r G[a])) and
(∀a:A. (D[a] ⊆r F[a])) and
(∀a:A. (D[a] ⊆r E[a])) and
(∀a:A. (C[a] ⊆r F[a])) and
(∀a:A. (B[a] ⊆r E[a])))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-compatible: f || g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-compatible: f || g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
fpf-join: f ⊕ g,
fpf-ap: f(x),
pi2: snd(t),
fpf-cap: f(x)?z,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
cand: A c∧ B,
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
fpf-join-dom2,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf-join_wf,
fpf-compatible_wf,
all_wf,
subtype_rel_wf,
fpf_wf,
deq_wf,
bool_wf,
equal_functionality_wrt_subtype_rel2,
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,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
productElimination,
thin,
extract_by_obid,
isectElimination,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
unionElimination,
productEquality,
applyEquality,
lambdaEquality,
functionExtensionality,
cumulativity,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
independent_pairFormation,
baseClosed,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B,C,D,E,F,G:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[g:a:A fp-> C[a]].
\mforall{}[h:a:A fp-> D[a]].
(\{(h || f \moplus{} g) supposing (h || g and h || f)\}) supposing
((\mforall{}a:A. (E[a] \msubseteq{}r G[a])) and
(\mforall{}a:A. (F[a] \msubseteq{}r G[a])) and
(\mforall{}a:A. (D[a] \msubseteq{}r F[a])) and
(\mforall{}a:A. (D[a] \msubseteq{}r E[a])) and
(\mforall{}a:A. (C[a] \msubseteq{}r F[a])) and
(\mforall{}a:A. (B[a] \msubseteq{}r E[a])))
Date html generated:
2018_05_21-PM-09_28_16
Last ObjectModification:
2018_02_09-AM-10_23_39
Theory : finite!partial!functions
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