Nuprl Lemma : RankEx2-ext
∀[S,T:Type].
RankEx2(S;T) ≡ lbl:Atom × if lbl =a "LeafT" then T
if lbl =a "LeafS" then S
if lbl =a "Prod" then RankEx2(S;T) × S × T
if lbl =a "Union" then S × RankEx2(S;T) + RankEx2(S;T)
if lbl =a "ListProd" then (S × RankEx2(S;T)) List
if lbl =a "UnionList" then T + (RankEx2(S;T) List)
else Void
fi
Proof
Definitions occuring in Statement :
RankEx2: RankEx2(S;T),
list: T List,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
product: x:A × B[x],
union: left + right,
token: "$token",
atom: Atom,
void: Void,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
RankEx2: RankEx2(S;T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
eq_atom: x =a y,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
RankEx2co_size: RankEx2co_size(p),
pi1: fst(t),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
pi2: snd(t),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
nequal: a ≠ b ∈ T ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
RankEx2_size: RankEx2_size(p),
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
sum-nat,
nat_properties,
top_wf,
pi1_wf_top,
RankEx2_size_wf,
false_wf,
add-nat,
subtype_rel_list,
subtype_rel_union,
subtype_rel_product,
list-set-type2,
list_wf,
sum-partial-nat,
int_seg_wf,
pi2_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf,
length_wf_nat,
sum-partial-has-value,
RankEx2co_wf,
list-prod-set-type,
RankEx2_wf,
set-value-type,
has-value_wf-partial,
int-value-type,
value-type-has-value,
base_wf,
le_wf,
set_subtype_base,
nat_wf,
subtype_partial_sqtype_base,
RankEx2co_size_wf,
int_subtype_base,
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,
RankEx2co-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
lemma_by_obid,
hypothesis,
isectElimination,
hypothesisEquality,
promote_hyp,
productElimination,
hypothesis_subsumption,
applyEquality,
sqequalRule,
dependent_pairEquality,
tokenEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
instantiate,
cumulativity,
atomEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
voidElimination,
independent_pairEquality,
dependent_set_memberEquality,
natural_numberEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
equalityEquality,
callbyvalueAdd,
inlEquality,
inrEquality,
productEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination,
unionEquality,
sqleReflexivity,
universeEquality
Latex:
\mforall{}[S,T:Type].
RankEx2(S;T) \mequiv{} lbl:Atom \mtimes{} if lbl =a "LeafT" then T
if lbl =a "LeafS" then S
if lbl =a "Prod" then RankEx2(S;T) \mtimes{} S \mtimes{} T
if lbl =a "Union" then S \mtimes{} RankEx2(S;T) + RankEx2(S;T)
if lbl =a "ListProd" then (S \mtimes{} RankEx2(S;T)) List
if lbl =a "UnionList" then T + (RankEx2(S;T) List)
else Void
fi
Date html generated:
2016_05_16-AM-08_59_39
Last ObjectModification:
2016_01_17-AM-09_42_19
Theory : C-semantics
Home
Index