Nuprl Lemma : apply-Id-alist-function
∀[x:Id]. ∀[F:Top]. ∀[L:Id List]. apply-alist(IdDeq;map(λx.<x, F[x]>;L);x) ~ inl F[x] supposing (x ∈ L)
Proof
Definitions occuring in Statement :
id-deq: IdDeq,
Id: Id,
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
apply-alist: apply-alist(eq;L;x),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pi1: fst(t),
pi2: snd(t),
eq_id: a = b,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
Id: Id,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
l_member_wf,
Id_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
nil_member,
nil_wf,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
cons_member,
map_cons_lemma,
list_ind_cons_lemma,
eq_id_wf,
bool_wf,
eqtt_to_assert,
assert-eq-id,
atom2_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
cons_wf,
list_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
unionElimination,
productElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
cumulativity,
imageElimination,
equalityElimination
Latex:
\mforall{}[x:Id]. \mforall{}[F:Top]. \mforall{}[L:Id List].
apply-alist(IdDeq;map(\mlambda{}x.<x, F[x]>L);x) \msim{} inl F[x] supposing (x \mmember{} L)
Date html generated:
2017_04_17-AM-09_18_39
Last ObjectModification:
2017_02_27-PM-05_22_13
Theory : decidable!equality
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