Nuprl Lemma : rat_term-ext
rat_term() ≡ lbl:Atom × if lbl =a "Constant" then ℤ
if lbl =a "Var" then ℤ
if lbl =a "Add" then left:rat_term() × rat_term()
if lbl =a "Subtract" then left:rat_term() × rat_term()
if lbl =a "Multiply" then left:rat_term() × rat_term()
if lbl =a "Divide" then num:rat_term() × rat_term()
if lbl =a "Minus" then rat_term()
else Void
fi
Proof
Definitions occuring in Statement :
rat_term: rat_term(),
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
ext-eq: A ≡ B,
product: x:A × B[x],
int: ℤ,
token: "$token",
atom: Atom,
void: Void
Definitions unfolded in proof :
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
rat_term: rat_term(),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
eq_atom: x =a y,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
rat_termco_size: rat_termco_size(p),
pi1: fst(t),
pi2: snd(t),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
prop: ℙ,
rat_term_size: rat_term_size(p),
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nequal: a ≠ b ∈ T
Lemmas referenced :
rat_termco-ext,
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
ifthenelse_wf,
rat_term_wf,
int_subtype_base,
rat_termco_size_wf,
subtype_partial_sqtype_base,
nat_wf,
set_subtype_base,
le_wf,
istype-int,
value-type-has-value,
int-value-type,
istype-universe,
has-value_wf-partial,
set-value-type,
rat_termco_wf,
istype-atom,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
nat_properties,
add-nat,
rat_term_size_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
hypothesis,
promote_hyp,
productElimination,
hypothesis_subsumption,
hypothesisEquality,
applyEquality,
sqequalRule,
dependent_pairEquality_alt,
isectElimination,
tokenEquality,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
instantiate,
cumulativity,
atomEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIstype,
voidElimination,
universeIsType,
universeEquality,
intEquality,
productEquality,
voidEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
baseApply,
closedConclusion,
baseClosed,
callbyvalueAdd,
productIsType,
approximateComputation,
isect_memberEquality_alt,
independent_pairEquality
Latex:
rat\_term() \mequiv{} lbl:Atom \mtimes{} if lbl =a "Constant" then \mBbbZ{}
if lbl =a "Var" then \mBbbZ{}
if lbl =a "Add" then left:rat\_term() \mtimes{} rat\_term()
if lbl =a "Subtract" then left:rat\_term() \mtimes{} rat\_term()
if lbl =a "Multiply" then left:rat\_term() \mtimes{} rat\_term()
if lbl =a "Divide" then num:rat\_term() \mtimes{} rat\_term()
if lbl =a "Minus" then rat\_term()
else Void
fi
Date html generated:
2019_10_29-AM-09_25_35
Last ObjectModification:
2019_03_31-PM-05_16_40
Theory : reals
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