Nuprl Lemma : map_equal2
∀[T,T':Type]. ∀[a:T List]. ∀[f,g:T ⟶ T'].
map(f;a) = map(g;a) ∈ (T' List) supposing ∀x:T. ((x ∈ a) ⇒ ((f x) = (g x) ∈ T'))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
map: map(f;as),
list: T List,
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 :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
l_member_wf,
equal_wf,
list_wf,
map_equal,
squash_wf,
true_wf,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
select_member,
lelt_wf,
length_wf,
subtype_rel_self,
iff_weakening_equal,
less_than_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
applyEquality,
because_Cache,
universeEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaFormation,
imageElimination,
dependent_functionElimination,
cumulativity,
setElimination,
rename,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
instantiate,
productElimination
Latex:
\mforall{}[T,T':Type]. \mforall{}[a:T List]. \mforall{}[f,g:T {}\mrightarrow{} T'].
map(f;a) = map(g;a) supposing \mforall{}x:T. ((x \mmember{} a) {}\mRightarrow{} ((f x) = (g x)))
Date html generated:
2018_05_21-PM-06_20_19
Last ObjectModification:
2018_05_19-PM-05_32_25
Theory : list!
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