Nuprl Lemma : find_transitions_wf
∀[A:Type]. ∀[f:A ⟶ A ⟶ 𝔹]. ∀[L:A List]. find_transitions(f;L) ∈ Unit + ℤ × (Unit + ℤ) supposing 1 < ||L||
Proof
Definitions occuring in Statement :
find_transitions: find_transitions(f;L),
length: ||as||,
list: T List,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
unit: Unit,
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
union: left + right,
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
cons: [a / b],
top: Top,
find_transitions: find_transitions(f;L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
has-value: (a)↓,
bool: 𝔹,
ge: i ≥ j ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
spreadn: spread4,
isr: isr(x),
ifthenelse: if b then t else f fi ,
isl: isl(x),
btrue: tt,
bfalse: ff,
band: p ∧b q,
unit: Unit,
it: ⋅,
bnot: ¬bb,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
assert: ↑b
Lemmas referenced :
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
spread_cons_lemma,
reduce_tl_cons_lemma,
value-type-has-value,
bool_wf,
union-value-type,
unit_wf2,
hd_wf,
decidable__le,
length_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
add-swap,
add-commutes,
minus-one-mul,
minus-one-mul-top,
zero-add,
add_functionality_wrt_le,
le-add-cancel2,
it_wf,
equal_wf,
less_than_wf,
list_wf,
pi2_wf,
list_accum'_wf,
null_nil_lemma,
eqtt_to_assert,
int-value-type,
bnot_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
eqff_to_assert,
assert-bnot,
bfalse_wf,
btrue_wf,
null_cons_lemma,
reduce_hd_cons_lemma,
and_wf,
isl_wf,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
callbyvalueReduce,
independent_isectElimination,
because_Cache,
applyEquality,
functionExtensionality,
cumulativity,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
independent_functionElimination,
addEquality,
lambdaEquality,
intEquality,
minusEquality,
inlEquality,
unionEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
universeEquality,
productEquality,
setElimination,
rename,
sqleReflexivity,
equalityElimination,
dependent_pairFormation,
instantiate,
dependent_pairEquality,
independent_pairEquality,
inrEquality,
setEquality,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List].
find\_transitions(f;L) \mmember{} Unit + \mBbbZ{} \mtimes{} (Unit + \mBbbZ{}) supposing 1 < ||L||
Date html generated:
2017_09_29-PM-05_49_47
Last ObjectModification:
2017_07_26-PM-01_38_42
Theory : list_0
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