Nuprl Lemma : RankEx1_ind_wf_simple
∀[T,A:Type]. ∀[v:RankEx1(T)]. ∀[Leaf:leaf:T ⟶ A]. ∀[Prod:prod:(RankEx1(T) × RankEx1(T))
⟶ let u,u1 = prod
in A ∧ A
⟶ A]. ∀[ProdL:prodl:(T × RankEx1(T))
⟶ let u,u1 = prodl
in A
⟶ A]. ∀[ProdR:prodr:(RankEx1(T) × T)
⟶ let u,u1 = prodr
in A
⟶ A].
∀[List:list:(RankEx1(T) List) ⟶ (∀u∈list.A) ⟶ A].
(RankEx1_ind(v;
RankEx1_Leaf(leaf)⇒ Leaf[leaf];
RankEx1_Prod(prod)⇒ rec1.Prod[prod;rec1];
RankEx1_ProdL(prodl)⇒ rec2.ProdL[prodl;rec2];
RankEx1_ProdR(prodr)⇒ rec3.ProdR[prodr;rec3];
RankEx1_List(list)⇒ rec4.List[list;rec4]) ∈ A)
Proof
Definitions occuring in Statement :
RankEx1_ind: RankEx1_ind,
RankEx1: RankEx1(T),
l_all: (∀x∈L.P[x]),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
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,
and: P ∧ Q,
implies: P ⇒ Q,
guard: {T}
Lemmas referenced :
RankEx1_ind_wf,
true_wf,
RankEx1_wf,
subtype_rel_dep_function,
and_wf,
set_wf,
subtype_rel_product,
list_wf,
l_all_wf2,
l_member_wf,
subtype-l_all
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,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[v:RankEx1(T)]. \mforall{}[Leaf:leaf:T {}\mrightarrow{} A]. \mforall{}[Prod:prod:(RankEx1(T) \mtimes{} RankEx1(T))
{}\mrightarrow{} let u,u1 = prod
in A \mwedge{} A
{}\mrightarrow{} A]. \mforall{}[ProdL:prodl:(T \mtimes{} RankEx1(T))
{}\mrightarrow{} let u,u1 = prodl
in A
{}\mrightarrow{} A].
\mforall{}[ProdR:prodr:(RankEx1(T) \mtimes{} T) {}\mrightarrow{} let u,u1 = prodr in A {}\mrightarrow{} A]. \mforall{}[List:list:(RankEx1(T) List)
{}\mrightarrow{} (\mforall{}u\mmember{}list.A)
{}\mrightarrow{} A].
(RankEx1\_ind(v;
RankEx1\_Leaf(leaf){}\mRightarrow{} Leaf[leaf];
RankEx1\_Prod(prod){}\mRightarrow{} rec1.Prod[prod;rec1];
RankEx1\_ProdL(prodl){}\mRightarrow{} rec2.ProdL[prodl;rec2];
RankEx1\_ProdR(prodr){}\mRightarrow{} rec3.ProdR[prodr;rec3];
RankEx1\_List(list){}\mRightarrow{} rec4.List[list;rec4]) \mmember{} A)
Date html generated:
2016_05_16-AM-08_58_49
Last ObjectModification:
2015_12_28-PM-06_51_51
Theory : C-semantics
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