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],
deq: EqDecider(T),
l_contains: A ⊆ B,
append: as @ bs,
list: T List,
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
Lemmas :
assert_witness,
assert_wf,
l_contains_wf,
list_wf,
append_wf,
fpf-union-join-dom,
subtype-fpf2,
top_wf,
subtype_top,
or_wf,
l_member_wf,
fpf-ap_wf,
fpf-union-join_wf,
subtype_rel_dep_function,
bool_wf,
subtype_rel_list,
subtype_rel_self,
not_wf,
fpf-dom_wf,
fpf-union-compatible_wf,
fpf_wf,
all_wf,
isect_wf,
subtype_rel_wf,
deq_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
fpf-union-join-ap,
member_append,
filter_wf5,
set_wf,
member_filter,
subtype_rel_transitivity,
and_wf,
assert_elim,
decidable__assert,
l_all_iff
\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:
2015_07_17-AM-11_07_40
Last ObjectModification:
2015_01_28-AM-08_00_22
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