Nuprl Lemma : eval-mklist-sq
∀[T:Type]
∀[n,offset:ℕ]. ∀[f:{offset..n + offset-} ⟶ T]. (eval-mklist(n;f;offset) ~ mklist(n;λi.(f (i + offset))))
supposing value-type(T)
Proof
Definitions occuring in Statement :
eval-mklist: eval-mklist(n;f;offset),
mklist: mklist(n;f),
int_seg: {i..j-},
nat: ℕ,
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
mklist: mklist(n;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
eval-mklist: eval-mklist(n;f;offset),
nil: [],
it: ⋅,
primrec: primrec(n;b;c),
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
has-value: (a)↓,
decidable: Dec(P),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
int_seg_wf,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
subtract-1-ge-0,
istype-nat,
value-type_wf,
istype-universe,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
value-type-has-value,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
istype-le,
int-value-type,
subtract_wf,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
zero-add,
subtype_rel_self,
list_wf,
list-value-type,
primrec_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
nil_wf,
append_wf,
cons_wf,
add-member-int_seg2,
decidable__equal_int,
primrec0_lemma,
list_ind_nil_lemma,
le_reflexive,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomSqEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
Error :functionIsType,
addEquality,
because_Cache,
productElimination,
instantiate,
universeEquality,
Error :equalityIstype,
applyEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
unionElimination,
equalityElimination,
equalityTransitivity,
promote_hyp,
cumulativity,
int_eqReduceFalseSq,
callbyvalueReduce,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
intEquality,
minusEquality,
multiplyEquality,
closedConclusion
Latex:
\mforall{}[T:Type]
\mforall{}[n,offset:\mBbbN{}]. \mforall{}[f:\{offset..n + offset\msupminus{}\} {}\mrightarrow{} T].
(eval-mklist(n;f;offset) \msim{} mklist(n;\mlambda{}i.(f (i + offset))))
supposing value-type(T)
Date html generated:
2019_06_20-PM-01_31_17
Last ObjectModification:
2019_01_21-AM-11_12_10
Theory : list_1
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