Nuprl Lemma : from-upto-decomp-last
∀[n,m:ℤ]. [n, m) = ([n, m - 1) @ [m - 1]) ∈ (ℤ List) supposing n < m
Proof
Definitions occuring in Statement :
from-upto: [n, m),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
ge: i ≥ j ,
less_than: a < b,
less_than': less_than'(a;b),
from-upto: [n, m),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
subtract: n - m,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
decidable__le,
subtract_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
equal-wf-base-T,
int_subtype_base,
subtype_base_sq,
less_than_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
nat_properties,
decidable__lt,
ge_wf,
lt_int_wf,
bool_wf,
value-type-has-value,
int-value-type,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
cons_wf,
list_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
nil_wf,
add-subtract-cancel,
list_ind_nil_lemma,
list_ind_cons_lemma,
list_subtype_base,
set_subtype_base,
from-upto_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_pairFormation,
dependent_set_memberEquality,
because_Cache,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
natural_numberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
addEquality,
setElimination,
rename,
productElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
intWeakElimination,
lambdaFormation,
axiomEquality,
callbyvalueReduce,
equalityElimination,
promote_hyp,
minusEquality,
setEquality,
productEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. [n, m) = ([n, m - 1) @ [m - 1]) supposing n < m
Date html generated:
2018_05_21-PM-00_40_26
Last ObjectModification:
2018_05_19-AM-06_46_05
Theory : list_1
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