Nuprl Lemma : RankEx2_ind_wf_simple
∀[S,T,A:Type]. ∀[v:RankEx2(S;T)]. ∀[LeafT:leaft:T ⟶ A]. ∀[LeafS:leafs:S ⟶ A]. ∀[Prod:prod:(RankEx2(S;T) × S × T)
⟶ let u,u1 = prod
in let u1,u2 = u
in A
⟶ A].
∀[Union:union:(S × RankEx2(S;T) + RankEx2(S;T)) ⟶ case union of inl(u) => let u1,u2 = u in A | inr(u1) => A ⟶ A].
∀[ListProd:listprod:((S × RankEx2(S;T)) List) ⟶ (∀u∈listprod.let u1,u2 = u in A) ⟶ A].
∀[UnionList:unionlist:(T + (RankEx2(S;T) List)) ⟶ case unionlist of inl(u) => True | inr(u1) => (∀u∈u1.A) ⟶ A].
(RankEx2_ind(v;
RankEx2_LeafT(leaft)⇒ LeafT[leaft];
RankEx2_LeafS(leafs)⇒ LeafS[leafs];
RankEx2_Prod(prod)⇒ rec1.Prod[prod;rec1];
RankEx2_Union(union)⇒ rec2.Union[union;rec2];
RankEx2_ListProd(listprod)⇒ rec3.ListProd[listprod;rec3];
RankEx2_UnionList(unionlist)⇒ rec4.UnionList[unionlist;rec4]) ∈ A)
Proof
Definitions occuring in Statement :
RankEx2_ind: RankEx2_ind,
RankEx2: RankEx2(S;T),
l_all: (∀x∈L.P[x]),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
true: True,
member: t ∈ T,
function: x:A ⟶ B[x],
spread: spread def,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
true: True,
implies: P ⇒ Q,
guard: {T}
Lemmas referenced :
RankEx2_ind_wf,
true_wf,
RankEx2_wf,
subtype_rel_dep_function,
set_wf,
list_wf,
l_all_wf2,
l_member_wf,
subtype-l_all,
subtype_rel_self
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
setEquality,
independent_isectElimination,
lambdaFormation,
dependent_set_memberEquality,
natural_numberEquality,
productEquality,
functionEquality,
spreadEquality,
productElimination,
setElimination,
rename,
unionEquality,
decideEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[S,T,A:Type]. \mforall{}[v:RankEx2(S;T)]. \mforall{}[LeafT:leaft:T {}\mrightarrow{} A]. \mforall{}[LeafS:leafs:S {}\mrightarrow{} A].
\mforall{}[Prod:prod:(RankEx2(S;T) \mtimes{} S \mtimes{} T) {}\mrightarrow{} let u,u1 = prod in let u1,u2 = u in A {}\mrightarrow{} A].
\mforall{}[Union:union:(S \mtimes{} RankEx2(S;T) + RankEx2(S;T))
{}\mrightarrow{} case union of inl(u) => let u1,u2 = u in A | inr(u1) => A
{}\mrightarrow{} A]. \mforall{}[ListProd:listprod:((S \mtimes{} RankEx2(S;T)) List)
{}\mrightarrow{} (\mforall{}u\mmember{}listprod.let u1,u2 = u
in A)
{}\mrightarrow{} A]. \mforall{}[UnionList:unionlist:(T + (RankEx2(S;T) List))
{}\mrightarrow{} case unionlist
of inl(u) =>
True
| inr(u1) =>
(\mforall{}u\mmember{}u1.A)
{}\mrightarrow{} A].
(RankEx2\_ind(v;
RankEx2\_LeafT(leaft){}\mRightarrow{} LeafT[leaft];
RankEx2\_LeafS(leafs){}\mRightarrow{} LeafS[leafs];
RankEx2\_Prod(prod){}\mRightarrow{} rec1.Prod[prod;rec1];
RankEx2\_Union(union){}\mRightarrow{} rec2.Union[union;rec2];
RankEx2\_ListProd(listprod){}\mRightarrow{} rec3.ListProd[listprod;rec3];
RankEx2\_UnionList(unionlist){}\mRightarrow{} rec4.UnionList[unionlist;rec4]) \mmember{} A)
Date html generated:
2016_05_16-AM-09_02_47
Last ObjectModification:
2015_12_28-PM-06_50_22
Theory : C-semantics
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