Nuprl Lemma : rev-map-append-sq
∀[A,B:Type].
∀[f:A ⟶ B]. ∀[as:A List]. ∀[bs:B List]. (rev-map-append(f;as;bs) ~ rev(map(f;as)) @ bs) supposing value-type(B)
Proof
Definitions occuring in Statement :
rev-map-append: rev-map-append(f;as;bs),
reverse: rev(as),
map: map(f;as),
append: as @ bs,
list: T List,
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
rev-map-append: rev-map-append(f;as;bs),
nil: [],
it: ⋅,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
has-value: (a)↓
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,
less_than_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
map_nil_lemma,
reverse_nil_lemma,
list_ind_nil_lemma,
list_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
map_cons_lemma,
reverse-cons,
value-type-has-value,
cons_wf,
append_assoc,
list_ind_cons_lemma,
nat_wf,
istype-universe,
value-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomSqEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
callbyvalueReduce,
sqleReflexivity,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIsType1,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
imageElimination,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
Error :functionIsType,
universeEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[as:A List]. \mforall{}[bs:B List]. (rev-map-append(f;as;bs) \msim{} rev(map(f;as)) @ bs)
supposing value-type(B)
Date html generated:
2019_06_20-PM-01_49_01
Last ObjectModification:
2018_10_07-AM-00_07_24
Theory : list_1
Home
Index