Nuprl Lemma : fpf-union-compatible-subtype
∀[A,C:Type]. ∀[B1,B2:A ⟶ Type].
(∀eq:EqDecider(A). ∀f,g:x:A fp-> B1[x] List. ∀R:(C List) ⟶ C ⟶ 𝔹.
(fpf-union-compatible(A;C;x.B1[x];eq;R;f;g) ⇒ fpf-union-compatible(A;C;x.B2[x];eq;R;f;g))) supposing
((∀x:A. (B1[x] ⊆r B2[x])) and
(∀a:A. (B2[a] ⊆r C)))
Proof
Definitions occuring in Statement :
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
fpf: a:A fp-> B[a],
list: T List,
deq: EqDecider(T),
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,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
so_apply: x[s],
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
prop: ℙ,
top: Top
Lemmas referenced :
l_member_subtype,
fpf-ap_wf,
list_wf,
equal_wf,
l_member_wf,
subtype_rel_list,
or_wf,
not_wf,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf-union-compatible_wf,
bool_wf,
fpf_wf,
deq_wf,
all_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
rename,
lambdaFormation,
independent_functionElimination,
productElimination,
dependent_pairFormation,
applyEquality,
independent_pairFormation,
extract_by_obid,
isectElimination,
functionExtensionality,
cumulativity,
independent_isectElimination,
because_Cache,
productEquality,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,C:Type]. \mforall{}[B1,B2:A {}\mrightarrow{} Type].
(\mforall{}eq:EqDecider(A). \mforall{}f,g:x:A fp-> B1[x] List. \mforall{}R:(C List) {}\mrightarrow{} C {}\mrightarrow{} \mBbbB{}.
(fpf-union-compatible(A;C;x.B1[x];eq;R;f;g)
{}\mRightarrow{} fpf-union-compatible(A;C;x.B2[x];eq;R;f;g))) supposing
((\mforall{}x:A. (B1[x] \msubseteq{}r B2[x])) and
(\mforall{}a:A. (B2[a] \msubseteq{}r C)))
Date html generated:
2018_05_21-PM-09_18_21
Last ObjectModification:
2018_02_09-AM-10_17_06
Theory : finite!partial!functions
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