Nuprl Lemma : from-upto-is-nil
∀[n,m:ℤ]. uiff([n, m) ~ [];m ≤ n)
Proof
Definitions occuring in Statement :
from-upto: [n, m),
nil: [],
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
from-upto: [n, m),
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True
Lemmas referenced :
equal-wf-base,
list_wf,
int_subtype_base,
le_wf,
length_of_nil_lemma,
list_subtype_base,
length-from-upto,
lt_int_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
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,
equal_wf,
bool_cases_sqequal,
assert-bnot,
nil_wf,
le_witness_for_triv,
from-upto_wf,
set_wf,
list-cases,
product_subtype_list,
istype-sqequal,
full-omega-unsat,
istype-int,
istype-void,
list_ind_nil_lemma,
list_ind_cons_lemma,
less_than'_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesis,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
because_Cache,
natural_numberEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
independent_isectElimination,
equalityTransitivity,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
unionElimination,
instantiate,
cumulativity,
independent_functionElimination,
productElimination,
impliesFunctionality,
equalityElimination,
promote_hyp,
axiomSqEquality,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
isect_memberFormation_alt,
lambdaFormation_alt,
equalityIstype,
lambdaEquality_alt,
productEquality,
universeIsType,
hypothesis_subsumption,
rename,
sqequalBase,
setElimination,
approximateComputation,
dependent_pairFormation_alt,
isect_memberFormation,
axiomEquality,
sqequalIntensionalEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. uiff([n, m) \msim{} [];m \mleq{} n)
Date html generated:
2019_10_15-AM-10_22_39
Last ObjectModification:
2019_08_05-PM-01_55_33
Theory : list_1
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