Nuprl Lemma : RankEx2-defop-extract
∀[T,S,P:Type]. ∀[R:P ⟶ RankEx2(S;T) ⟶ ℙ].
((∀t:T. (∃x:{P| (R x RankEx2_LeafT(t))}))
⇒ (∀s:S. (∃x:{P| (R x RankEx2_LeafS(s))}))
⇒ (∀d:RankEx2(S;T). ∀s:S. ∀t:T. ((∃x:{P| (R x d)}) ⇒ (∃x:{P| (R x RankEx2_Prod(<<d, s>, t>))})))
⇒ (∀z:S × RankEx2(S;T) + RankEx2(S;T)
(case z of inl(p) => ∃x:{P| (R x (snd(p)))} | inr(d) => ∃x:{P| (R x d)} ⇒ (∃x:{P| (R x RankEx2_Union(z))})))
⇒ (∀L:(S × RankEx2(S;T)) List. ((∀p∈L.∃x:{P| (R x (snd(p)))}) ⇒ (∃x:{P| (R x RankEx2_ListProd(L))})))
⇒ (∀z:T + (RankEx2(S;T) List)
(case z of inl(p) => True | inr(L) => (∀p∈L.∃x:{P| (R x p)}) ⇒ (∃x:{P| (R x RankEx2_UnionList(z))})))
⇒ {∀t:RankEx2(S;T). (∃x:{P| (R x t)})})
Proof
Definitions occuring in Statement :
RankEx2_UnionList: RankEx2_UnionList(unionlist),
RankEx2_ListProd: RankEx2_ListProd(listprod),
RankEx2_Union: RankEx2_Union(union),
RankEx2_Prod: RankEx2_Prod(prod),
RankEx2_LeafS: RankEx2_LeafS(leafs),
RankEx2_LeafT: RankEx2_LeafT(leaft),
RankEx2: RankEx2(S;T),
l_all: (∀x∈L.P[x]),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
pi2: snd(t),
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
RankEx2-defop,
RankEx2-definition,
RankEx2-induction,
eq_atom: x =a y,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
any: any x,
btrue: tt,
assert-bnot,
eqff_to_assert,
bfalse: ff,
bool_cases_sqequal,
RankEx2-ext,
uniform-comp-nat-induction,
it: ⋅,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
genrec-ap: genrec-ap
Lemmas referenced :
RankEx2-defop,
RankEx2-definition,
RankEx2-induction,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
assert-bnot,
eqff_to_assert,
bool_cases_sqequal,
RankEx2-ext,
uniform-comp-nat-induction,
lifting-strict-spread,
base_wf,
lifting-strict-atom_eq,
is-exception_wf,
has-value_wf_base,
top_wf
Rules used in proof :
introduction,
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
instantiate,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
thin,
lemma_by_obid,
lambdaFormation,
because_Cache,
sqequalSqle,
divergentSqle,
callbyvalueDecide,
unionEquality,
unionElimination,
sqleReflexivity,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
decideExceptionCases,
axiomSqleEquality,
exceptionSqequal,
baseApply,
closedConclusion,
baseClosed,
isectElimination,
independent_isectElimination,
independent_pairFormation,
inrFormation,
imageMemberEquality,
imageElimination,
inlFormation,
callbyvalueApply,
applyExceptionCases,
callbyvalueSpread,
productEquality,
productElimination,
spreadExceptionCases
Latex:
\mforall{}[T,S,P:Type]. \mforall{}[R:P {}\mrightarrow{} RankEx2(S;T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}t:T. (\mexists{}x:\{P| (R x RankEx2\_LeafT(t))\}))
{}\mRightarrow{} (\mforall{}s:S. (\mexists{}x:\{P| (R x RankEx2\_LeafS(s))\}))
{}\mRightarrow{} (\mforall{}d:RankEx2(S;T). \mforall{}s:S. \mforall{}t:T. ((\mexists{}x:\{P| (R x d)\}) {}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_Prod(<<d, s>, t>))\}))\000C)
{}\mRightarrow{} (\mforall{}z:S \mtimes{} RankEx2(S;T) + RankEx2(S;T)
(case z of inl(p) => \mexists{}x:\{P| (R x (snd(p)))\} | inr(d) => \mexists{}x:\{P| (R x d)\}
{}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_Union(z))\})))
{}\mRightarrow{} (\mforall{}L:(S \mtimes{} RankEx2(S;T)) List
((\mforall{}p\mmember{}L.\mexists{}x:\{P| (R x (snd(p)))\}) {}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_ListProd(L))\})))
{}\mRightarrow{} (\mforall{}z:T + (RankEx2(S;T) List)
(case z of inl(p) => True | inr(L) => (\mforall{}p\mmember{}L.\mexists{}x:\{P| (R x p)\})
{}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_UnionList(z))\})))
{}\mRightarrow{} \{\mforall{}t:RankEx2(S;T). (\mexists{}x:\{P| (R x t)\})\})
Date html generated:
2016_05_16-AM-09_03_09
Last ObjectModification:
2016_01_17-AM-09_45_02
Theory : C-semantics
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