Nuprl Lemma : fpf-union-compatible-property
∀[T,V:Type].
∀eq:EqDecider(T)
∀[X:T ⟶ Type]
∀R:(V List) ⟶ V ⟶ 𝔹
(∀A,B,C:t:T fp-> X[t] List.
(fpf-union-compatible(T;V;t.X[t];eq;R;A;B)
⇒ fpf-union-compatible(T;V;t.X[t];eq;R;C;A)
⇒ fpf-union-compatible(T;V;t.X[t];eq;R;C;B)
⇒ fpf-union-compatible(T;V;t.X[t];eq;R;C;fpf-union-join(eq;R;A;B)))) supposing
((∀L1,L2:V List. ∀x:V. (↑(R (L1 @ L2) x)) supposing ((↑(R L2 x)) and (↑(R L1 x)))) and
(∀L1,L2:V List. ∀x:V. (L1 ⊆ L2 ⇒ ↑(R L1 x) supposing ↑(R L2 x))))
supposing ∀t:T. (X[t] ⊆r V)
Proof
Definitions occuring in Statement :
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
fpf: a:A fp-> B[a],
l_contains: A ⊆ B,
append: as @ bs,
list: T List,
deq: EqDecider(T),
assert: ↑b,
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],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
istype: istype(T),
prop: ℙ,
not: ¬A,
false: False,
fpf-union: fpf-union(f;g;eq;R;x),
fpf-cap: f(x)?z,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
respects-equality: respects-equality(S;T),
rev_implies: P ⇐ Q,
true: True,
decidable: Dec(P),
l_contains: A ⊆ B
Lemmas referenced :
assert_witness,
append_wf,
fpf-union-join-dom,
subtype-fpf2,
list_wf,
top_wf,
istype-void,
l_member_wf,
fpf-ap_wf,
fpf-union-join_wf,
subtype_rel_dep_function,
bool_wf,
subtype_rel_list,
istype-assert,
fpf-dom_wf,
fpf-union-compatible_wf,
fpf_wf,
l_contains_wf,
subtype_rel_wf,
deq_wf,
istype-universe,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
fpf-union-join-ap,
member_append,
filter_wf5,
member_filter,
subtype-respects-equality,
subtype_rel_transitivity,
assert_elim,
decidable__assert,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
rename,
isect_memberEquality_alt,
isectElimination,
extract_by_obid,
applyEquality,
independent_functionElimination,
isectIsTypeImplies,
independent_isectElimination,
voidElimination,
universeIsType,
because_Cache,
productElimination,
unionIsType,
productIsType,
functionEquality,
functionIsType,
isectIsType,
instantiate,
universeEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
cumulativity,
setEquality,
setIsType,
setElimination,
inlFormation_alt,
inrFormation_alt,
independent_pairFormation,
dependent_set_memberEquality_alt,
applyLambdaEquality,
natural_numberEquality
Latex:
\mforall{}[T,V:Type].
\mforall{}eq:EqDecider(T)
\mforall{}[X:T {}\mrightarrow{} Type]
\mforall{}R:(V List) {}\mrightarrow{} V {}\mrightarrow{} \mBbbB{}
(\mforall{}A,B,C:t:T fp-> X[t] List.
(fpf-union-compatible(T;V;t.X[t];eq;R;A;B)
{}\mRightarrow{} fpf-union-compatible(T;V;t.X[t];eq;R;C;A)
{}\mRightarrow{} fpf-union-compatible(T;V;t.X[t];eq;R;C;B)
{}\mRightarrow{} fpf-union-compatible(T;V;t.X[t];eq;R;C;fpf-union-join(eq;R;A;B)))) supposing
((\mforall{}L1,L2:V List. \mforall{}x:V. (\muparrow{}(R (L1 @ L2) x)) supposing ((\muparrow{}(R L2 x)) and (\muparrow{}(R L1 x)))) and
(\mforall{}L1,L2:V List. \mforall{}x:V. (L1 \msubseteq{} L2 {}\mRightarrow{} \muparrow{}(R L1 x) supposing \muparrow{}(R L2 x))))
supposing \mforall{}t:T. (X[t] \msubseteq{}r V)
Date html generated:
2019_10_16-AM-11_25_47
Last ObjectModification:
2019_06_25-PM-01_23_02
Theory : finite!partial!functions
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