Nuprl Lemma : RankEx2-induction
∀[S,T:Type]. ∀[P:RankEx2(S;T) ⟶ ℙ].
((∀leaft:T. P[RankEx2_LeafT(leaft)])
⇒ (∀leafs:S. P[RankEx2_LeafS(leafs)])
⇒ (∀prod:RankEx2(S;T) × S × T. (let u,u1 = prod in let u1,u2 = u in P[u1] ⇒ P[RankEx2_Prod(prod)]))
⇒ (∀union:S × RankEx2(S;T) + RankEx2(S;T)
(case union of inl(u) => let u1,u2 = u in P[u2] | inr(u1) => P[u1] ⇒ P[RankEx2_Union(union)]))
⇒ (∀listprod:(S × RankEx2(S;T)) List. ((∀u∈listprod.let u1,u2 = u in P[u2]) ⇒ P[RankEx2_ListProd(listprod)]))
⇒ (∀unionlist:T + (RankEx2(S;T) List)
(case unionlist of inl(u) => True | inr(u1) => (∀u∈u1.P[u]) ⇒ P[RankEx2_UnionList(unionlist)]))
⇒ {∀v:RankEx2(S;T). P[v]})
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},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
true: True,
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],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
RankEx2_LeafT: RankEx2_LeafT(leaft),
RankEx2_size: RankEx2_size(p),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
RankEx2_LeafS: RankEx2_LeafS(leafs),
RankEx2_Prod: RankEx2_Prod(prod),
pi1: fst(t),
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
RankEx2_Union: RankEx2_Union(union),
pi2: snd(t),
RankEx2_ListProd: RankEx2_ListProd(listprod),
l_all: (∀x∈L.P[x]),
RankEx2_UnionList: RankEx2_UnionList(unionlist),
true: True
Lemmas referenced :
RankEx2_LeafT_wf,
RankEx2_LeafS_wf,
RankEx2_Prod_wf,
RankEx2_Union_wf,
RankEx2_ListProd_wf,
RankEx2_UnionList_wf,
l_member_wf,
l_all_wf2,
true_wf,
uall_wf,
list_wf,
sum-nat-less,
int_seg_wf,
pi2_wf,
length_wf,
int_seg_properties,
select_wf,
length_wf_nat,
sum-nat,
lelt_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__le,
subtract_wf,
int_formula_prop_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
neg_assert_of_eq_atom,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
atom_subtype_base,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
bool_wf,
eq_atom_wf,
RankEx2-ext,
less_than'_wf,
nat_wf,
RankEx2_size_wf,
le_wf,
isect_wf,
RankEx2_wf,
all_wf,
uniform-comp-nat-induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
setElimination,
rename,
independent_functionElimination,
introduction,
productElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
tokenEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
instantiate,
cumulativity,
atomEquality,
dependent_pairFormation,
setEquality,
intEquality,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
imageElimination,
inlEquality,
inrEquality,
productEquality,
equalityEquality,
unionEquality,
functionEquality,
decideEquality,
spreadEquality,
universeEquality
Latex:
\mforall{}[S,T:Type]. \mforall{}[P:RankEx2(S;T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}leaft:T. P[RankEx2\_LeafT(leaft)])
{}\mRightarrow{} (\mforall{}leafs:S. P[RankEx2\_LeafS(leafs)])
{}\mRightarrow{} (\mforall{}prod:RankEx2(S;T) \mtimes{} S \mtimes{} T
(let u,u1 = prod in let u1,u2 = u in P[u1] {}\mRightarrow{} P[RankEx2\_Prod(prod)]))
{}\mRightarrow{} (\mforall{}union:S \mtimes{} RankEx2(S;T) + RankEx2(S;T)
(case union of inl(u) => let u1,u2 = u in P[u2] | inr(u1) => P[u1]
{}\mRightarrow{} P[RankEx2\_Union(union)]))
{}\mRightarrow{} (\mforall{}listprod:(S \mtimes{} RankEx2(S;T)) List
((\mforall{}u\mmember{}listprod.let u1,u2 = u in P[u2]) {}\mRightarrow{} P[RankEx2\_ListProd(listprod)]))
{}\mRightarrow{} (\mforall{}unionlist:T + (RankEx2(S;T) List)
(case unionlist of inl(u) => True | inr(u1) => (\mforall{}u\mmember{}u1.P[u])
{}\mRightarrow{} P[RankEx2\_UnionList(unionlist)]))
{}\mRightarrow{} \{\mforall{}v:RankEx2(S;T). P[v]\})
Date html generated:
2016_05_16-AM-09_02_27
Last ObjectModification:
2016_01_17-AM-09_42_12
Theory : C-semantics
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