Nuprl Lemma : map_equal
∀[T,T':Type]. ∀[a:T List]. ∀[f,g:T ⟶ T'].
map(f;a) = map(g;a) ∈ (T' List) supposing ∀i:ℕ. (i < ||a|| ⇒ ((f a[i]) = (g a[i]) ∈ T'))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
and: P ∧ Q,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
select_wf,
equal_wf,
length_wf,
less_than_wf,
nat_wf,
all_wf,
map_wf,
list_extensionality,
map_length,
istype-less_than,
istype-nat,
full-omega-unsat,
istype-int,
istype-void,
istype-le,
iff_weakening_equal,
squash_wf,
true_wf,
subtype_rel_self,
istype-universe,
map_select
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
isect_memberFormation,
universeEquality,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
natural_numberEquality,
dependent_functionElimination,
independent_isectElimination,
functionExtensionality,
applyEquality,
hypothesisEquality,
cumulativity,
because_Cache,
rename,
setElimination,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :lambdaFormation_alt,
Error :dependent_set_memberEquality_alt,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :productIsType,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
Error :inhabitedIsType,
instantiate
Latex:
\mforall{}[T,T':Type]. \mforall{}[a:T List]. \mforall{}[f,g:T {}\mrightarrow{} T'].
map(f;a) = map(g;a) supposing \mforall{}i:\mBbbN{}. (i < ||a|| {}\mRightarrow{} ((f a[i]) = (g a[i])))
Date html generated:
2019_06_20-PM-01_45_16
Last ObjectModification:
2019_01_10-PM-08_46_14
Theory : list_1
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