Nuprl Lemma : apply-alist-inr
∀[A,T:Type].
∀eq:EqDecider(T). ∀x:T. ∀u:Unit. ∀L:(T × A) List.
((apply-alist(eq;L;x) = (inr u ) ∈ (A?)) ⇒ (¬(∃z:A. (<x, z> ∈ L))))
Proof
Definitions occuring in Statement :
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
unit: Unit,
pair: <a, b>,
product: x:A × B[x],
inr: inr x ,
union: left + right,
universe: Type,
equal: s = t ∈ 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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
subtype_rel: A ⊆r B,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
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),
isl: isl(x),
iff: P ⇐⇒ Q,
pi1: fst(t),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
list-cases,
l_member_wf,
nil_wf,
apply-alist_wf,
unit_subtype_base,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
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,
le_wf,
apply_alist_cons_lemma,
cons_wf,
ifthenelse_wf,
eqof_wf,
pi1_wf,
unit_wf2,
pi2_wf,
istype-nat,
list_wf,
deq_wf,
istype-universe,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
bfalse_wf,
assert_wf,
bnot_wf,
not_wf,
equal_wf,
istype-assert,
cons_member,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
productEquality,
unionElimination,
productIsType,
independent_pairEquality,
equalityIstype,
unionIsType,
because_Cache,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
productElimination,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
applyLambdaEquality,
imageElimination,
intEquality,
unionEquality,
inlEquality_alt,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
functionIsType,
cumulativity
Latex:
\mforall{}[A,T:Type].
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}u:Unit. \mforall{}L:(T \mtimes{} A) List.
((apply-alist(eq;L;x) = (inr u )) {}\mRightarrow{} (\mneg{}(\mexists{}z:A. (<x, z> \mmember{} L))))
Date html generated:
2020_05_19-PM-09_41_53
Last ObjectModification:
2020_01_29-AM-11_32_06
Theory : list_1
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