Nuprl Lemma : from-upto-singleton
∀[n,m,k:ℤ]. uiff([n, m) = [k] ∈ (ℤ List);(m = (n + 1) ∈ ℤ) ∧ (k = n ∈ ℤ))
Proof
Definitions occuring in Statement :
from-upto: [n, m),
cons: [a / b],
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
from-upto: [n, m),
has-value: (a)↓,
all: ∀x:A. B[x],
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
squash: ↓T,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
equal-wf-base,
list_wf,
int_subtype_base,
lt_int_wf,
value-type-has-value,
int-value-type,
null_nil_lemma,
btrue_wf,
reduce_tl_cons_lemma,
nil_wf,
and_wf,
equal_wf,
tl_wf,
cons_wf,
from-upto_wf,
subtype_rel_list,
le_wf,
less_than_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
assert_wf,
bnot_wf,
not_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
ge_wf,
length_wf,
length_cons_ge_one,
top_wf,
from-upto-nil,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
bool_cases_sqequal,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
extract_by_obid,
isectElimination,
intEquality,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
because_Cache,
productEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
independent_isectElimination,
addEquality,
natural_numberEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
setEquality,
lambdaEquality,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
instantiate,
cumulativity,
lambdaFormation,
impliesFunctionality,
imageElimination,
universeEquality,
imageMemberEquality,
equalityElimination,
promote_hyp
Latex:
\mforall{}[n,m,k:\mBbbZ{}]. uiff([n, m) = [k];(m = (n + 1)) \mwedge{} (k = n))
Date html generated:
2017_04_17-AM-07_56_04
Last ObjectModification:
2017_02_27-PM-04_28_26
Theory : list_1
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