Nuprl Lemma : prior-cases
∀[T:Type]
∀f:ℕ ⟶ (T + Top). ∀n:ℕ.
case prior(n;f)
of inl(p) =>
let m,x = p
in ((f m) = (inl x) ∈ (T + Top)) ∧ (∀k:{m + 1..n-}. (¬↑isl(f k)))
| inr(q) =>
∀k:ℕn. (¬↑isl(f k))
Proof
Definitions occuring in Statement :
prior: prior(n;f),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inl: inl x,
union: left + right,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
prior: prior(n;f),
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
top: Top,
so_apply: x[s1;s2],
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
has-value: (a)↓,
cand: A c∧ B,
isl: isl(x),
less_than: a < b
Lemmas referenced :
natrec-unroll,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
assert_wf,
isl_wf,
top_wf,
nat_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
prior_wf,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
unit_wf2,
equal_wf,
all_wf,
not_wf,
itermAdd_wf,
int_term_value_add_lemma,
set_wf,
less_than_wf,
primrec-wf2,
nat_properties,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
value-type-has-value,
int-value-type,
subtract-add-cancel,
decidable__lt,
lelt_wf,
int_subtype_base,
decidable__equal_int,
assert_elim,
and_wf,
bfalse_wf,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
natural_numberEquality,
because_Cache,
hypothesisEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
independent_pairFormation,
computeAll,
independent_functionElimination,
cumulativity,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
unionElimination,
unionEquality,
productEquality,
inlEquality,
addEquality,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
equalityElimination,
promote_hyp,
instantiate,
callbyvalueReduce,
addLevel,
levelHypothesis,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:\mBbbN{} {}\mrightarrow{} (T + Top). \mforall{}n:\mBbbN{}.
case prior(n;f)
of inl(p) =>
let m,x = p
in ((f m) = (inl x)) \mwedge{} (\mforall{}k:\{m + 1..n\msupminus{}\}. (\mneg{}\muparrow{}isl(f k)))
| inr(q) =>
\mforall{}k:\mBbbN{}n. (\mneg{}\muparrow{}isl(f k))
Date html generated:
2017_10_01-AM-09_12_06
Last ObjectModification:
2017_07_26-PM-04_47_53
Theory : general
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