Nuprl Lemma : vdf-eq-append1
∀A:Type. ∀f:Top. ∀L:(a:Top × b:Top × Top) List. ∀a,b,c:Top.
(vdf-eq(A;f;L @ [<a, b, c>]) ~ x:vdf-eq(A;f;L) ⋂ a = (f L b) ∈ A)
Proof
Definitions occuring in Statement :
vdf-eq: vdf-eq(A;f;L),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
dep-isect: x:A ⋂ B[x],
top: Top,
all: ∀x:A. B[x],
apply: f a,
pair: <a, b>,
product: x:A × B[x],
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
select: L[n],
cons: [a / b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
less_than: a < b,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
let: let,
pi1: fst(t),
pi2: snd(t),
nat: ℕ,
less_than': less_than'(a;b)
Lemmas referenced :
vdf-eq-firstn,
append_wf,
top_wf,
cons_wf,
nil_wf,
non_neg_length,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
length-append,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__lt,
length_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
list_wf,
istype-top,
istype-universe,
select-append,
subtype_rel_list,
length_wf_nat,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
firstn-append,
add_nat_wf,
istype-void,
nat_properties,
add-is-int-iff,
false_wf,
le_int_wf,
assert_of_le_int,
le_wf,
firstn_all,
subtract_wf,
length-singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
productEquality,
hypothesis,
dependent_pairEquality_alt,
inhabitedIsType,
productIsType,
dependent_set_memberEquality_alt,
because_Cache,
independent_pairFormation,
unionElimination,
productElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
universeIsType,
voidElimination,
addEquality,
independent_pairEquality,
instantiate,
universeEquality,
applyEquality,
closedConclusion,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
imageElimination,
equalityIstype,
promote_hyp,
applyLambdaEquality,
setElimination,
rename,
pointwiseFunctionality,
baseApply,
baseClosed
Latex:
\mforall{}A:Type. \mforall{}f:Top. \mforall{}L:(a:Top \mtimes{} b:Top \mtimes{} Top) List. \mforall{}a,b,c:Top.
(vdf-eq(A;f;L @ [<a, b, c>]) \msim{} x:vdf-eq(A;f;L) \mcap{} a = (f L b))
Date html generated:
2020_05_19-PM-09_40_41
Last ObjectModification:
2020_03_09-PM-01_39_15
Theory : co-recursion-2
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