Nuprl Lemma : RankEx4-induction
∀[P:RankEx4() ⟶ ℙ]
((∀foo:ℤ + RankEx4(). (case foo of inl(u) => True | inr(u1) => P[u1] ⇒ P[RankEx4_Foo(foo)])) ⇒ {∀v:RankEx4(). P[v]})
Proof
Definitions occuring in Statement :
RankEx4_Foo: RankEx4_Foo(foo),
RankEx4: RankEx4(),
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],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
int: ℤ
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 ,
RankEx4_Foo: RankEx4_Foo(foo),
RankEx4_size: RankEx4_size(p),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
true: True,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
RankEx4_Foo_wf,
true_wf,
int_seg_wf,
uall_wf,
neg_assert_of_eq_atom,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
subtract_wf,
atom_subtype_base,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
bool_wf,
eq_atom_wf,
RankEx4-ext,
less_than'_wf,
nat_wf,
RankEx4_size_wf,
le_wf,
isect_wf,
RankEx4_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,
hypothesis,
hypothesisEquality,
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,
inlEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
setEquality,
inrEquality,
imageElimination,
equalityEquality,
unionEquality,
functionEquality,
decideEquality,
universeEquality
Latex:
\mforall{}[P:RankEx4() {}\mrightarrow{} \mBbbP{}]
((\mforall{}foo:\mBbbZ{} + RankEx4(). (case foo of inl(u) => True | inr(u1) => P[u1] {}\mRightarrow{} P[RankEx4\_Foo(foo)]))
{}\mRightarrow{} \{\mforall{}v:RankEx4(). P[v]\})
Date html generated:
2016_05_16-AM-09_04_36
Last ObjectModification:
2016_01_17-AM-09_41_45
Theory : C-semantics
Home
Index