Nuprl Lemma : rec-value-evalall
∀[x:Base]. uiff((evalall(x))↓;x ∈ rec-value())
Proof
Definitions occuring in Statement :
rec-value: rec-value(),
has-value: (a)↓,
evalall: evalall(t),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
member: t ∈ T,
base: Base
Definitions unfolded in proof :
has-value: (a)↓,
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
evalall: evalall(t),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
guard: {T},
pi2: snd(t),
ext-eq: A ≡ B,
outl: outl(x),
outr: outr(x),
atomic-values: atomic-values(),
Value: Value(),
is-atomic: is-atomic(x),
assert: ↑b,
true: True,
le: A ≤ B,
co-value-height: co-value-height(t),
rec-value-height: rec-value-height(v),
unit: Unit,
bool: 𝔹
Lemmas referenced :
istype-base,
rec-value_wf,
has-value_wf_base,
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,
int_subtype_base,
base_wf,
fun_exp0_lemma,
strictness-apply,
bottom_diverge,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
fun_exp_unroll_1,
has-value-implies-dec-ispair-2,
bfalse_wf,
ifthenelse_wf,
atomic-values_wf,
b-union_wf,
rec-value-ext,
has-value-implies-dec-isinl-2,
has-value-implies-dec-isinr-2,
btrue_wf,
istype-assert,
is-atomic_wf,
istype-nat,
false_wf,
add-is-int-iff,
is-exception_wf,
subtract-1-ge-0,
istype-le,
int_term_value_add_lemma,
itermAdd_wf,
rec-value-height_wf,
istype-less_than,
istype-void,
istype-int,
full-omega-unsat,
atomic-values_subtype_base,
atomic-values-evalall
Rules used in proof :
Error :inhabitedIsType,
Error :isectIsTypeImplies,
Error :isect_memberEquality_alt,
independent_pairEquality,
productElimination,
sqequalBase,
Error :equalityIstype,
axiomSqleEquality,
hypothesisEquality,
baseClosed,
closedConclusion,
baseApply,
thin,
isectElimination,
extract_by_obid,
Error :universeIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
sqequalHypSubstitution,
independent_pairFormation,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
compactness,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
applyEquality,
unionElimination,
because_Cache,
dependent_set_memberEquality,
callbyvalueCallbyvalue,
callbyvalueReduce,
imageMemberEquality,
Error :dependent_pairEquality_alt,
instantiate,
universeEquality,
productEquality,
unionEquality,
Error :inlEquality_alt,
Error :inrEquality_alt,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
promote_hyp,
pointwiseFunctionality,
sqleReflexivity,
divergentSqle,
applyLambdaEquality,
equalityElimination,
imageElimination,
hypothesis_subsumption,
Error :functionIsTypeImplies,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
approximateComputation
Latex:
\mforall{}[x:Base]. uiff((evalall(x))\mdownarrow{};x \mmember{} rec-value())
Date html generated:
2019_06_20-PM-01_54_29
Last ObjectModification:
2019_01_11-PM-03_12_31
Theory : rec_values
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