Nuprl Lemma : from-upto-split
∀[n,m,k:ℤ]. ([n, m) ~ [n, k) @ [k, m)) supposing ((k ≤ m) and (n ≤ k))
Proof
Definitions occuring in Statement :
from-upto: [n, m),
append: as @ bs,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
from-upto: [n, m),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
has-value: (a)↓
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
from-upto-is-nil,
list_ind_nil_lemma,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_ind_cons_lemma,
value-type-has-value,
int-value-type,
itermAdd_wf,
int_term_value_add_lemma,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
comment,
unionElimination,
because_Cache,
productElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
addEquality,
callbyvalueReduce,
isect_memberFormation,
dependent_set_memberEquality
Latex:
\mforall{}[n,m,k:\mBbbZ{}]. ([n, m) \msim{} [n, k) @ [k, m)) supposing ((k \mleq{} m) and (n \mleq{} k))
Date html generated:
2017_04_17-AM-07_53_55
Last ObjectModification:
2017_02_27-PM-04_27_18
Theory : list_1
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