Nuprl Lemma : null-mklist
∀[n:ℤ]. ∀[f:Top]. (null(mklist(n;f)) ~ n ≤z 0)
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
null: null(as),
le_int: i ≤z j,
uall: ∀[x:A]. B[x],
top: Top,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
not: ¬A,
null: null(as),
mklist: mklist(n;f),
primrec: primrec(n;b;c),
exposed-btrue: exposed-btrue,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
nil: [],
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nat: ℕ,
decidable: Dec(P)
Lemmas referenced :
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
top_wf,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
trivial-equal,
it_wf,
subtype_rel_union,
unit_wf2,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract-add-cancel,
mklist-add1-cons,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
null_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
sqequalRule,
voidElimination,
sqequalAxiom,
isect_memberEquality,
intEquality,
lessCases,
independent_pairFormation,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
inlEquality,
applyEquality,
lambdaEquality,
approximateComputation,
int_eqEquality,
dependent_set_memberEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[f:Top]. (null(mklist(n;f)) \msim{} n \mleq{}z 0)
Date html generated:
2018_05_21-PM-00_37_17
Last ObjectModification:
2018_05_19-AM-06_43_52
Theory : list_1
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