Nuprl Lemma : rat_term-induction
∀[P:rat_term() ⟶ ℙ]
((∀const:ℤ. P["const"])
⇒ (∀var:ℤ. P[rtermVar(var)])
⇒ (∀left,right:rat_term(). (P[left] ⇒ P[right] ⇒ P[left "+" right]))
⇒ (∀left,right:rat_term(). (P[left] ⇒ P[right] ⇒ P[left "-" right]))
⇒ (∀left,right:rat_term(). (P[left] ⇒ P[right] ⇒ P[left "*" right]))
⇒ (∀num,denom:rat_term(). (P[num] ⇒ P[denom] ⇒ P[num "/" denom]))
⇒ (∀num:rat_term(). (P[num] ⇒ P[rtermMinus(num)]))
⇒ {∀v:rat_term(). P[v]})
Proof
Definitions occuring in Statement :
rtermMinus: rtermMinus(num),
rtermDivide: num "/" denom,
rtermMultiply: left "*" right,
rtermSubtract: left "-" right,
rtermAdd: left "+" right,
rtermVar: rtermVar(var),
rtermConstant: "const",
rat_term: rat_term(),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
rtermConstant: "const",
rat_term_size: rat_term_size(p),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
rtermVar: rtermVar(var),
rtermAdd: left "+" right,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
rtermSubtract: left "-" right,
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermMinus: rtermMinus(num)
Lemmas referenced :
uniform-comp-nat-induction,
rat_term_wf,
le_wf,
rat_term_size_wf,
istype-nat,
le_witness_for_triv,
rat_term-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,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
itermAdd_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
istype-less_than,
int_seg_wf,
rtermMinus_wf,
rtermDivide_wf,
rtermMultiply_wf,
rtermSubtract_wf,
rtermAdd_wf,
rtermVar_wf,
rtermConstant_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesis,
isectEquality,
hypothesisEquality,
applyEquality,
because_Cache,
setElimination,
rename,
independent_functionElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
promote_hyp,
hypothesis_subsumption,
tokenEquality,
inhabitedIsType,
unionElimination,
equalityElimination,
instantiate,
cumulativity,
atomEquality,
dependent_functionElimination,
dependent_pairFormation_alt,
equalityIstype,
voidElimination,
independent_pairFormation,
applyLambdaEquality,
natural_numberEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
universeIsType,
dependent_set_memberEquality_alt,
productIsType,
imageElimination,
isectIsType,
functionIsType,
universeEquality
Latex:
\mforall{}[P:rat\_term() {}\mrightarrow{} \mBbbP{}]
((\mforall{}const:\mBbbZ{}. P["const"])
{}\mRightarrow{} (\mforall{}var:\mBbbZ{}. P[rtermVar(var)])
{}\mRightarrow{} (\mforall{}left,right:rat\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left "+" right]))
{}\mRightarrow{} (\mforall{}left,right:rat\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left "-" right]))
{}\mRightarrow{} (\mforall{}left,right:rat\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left "*" right]))
{}\mRightarrow{} (\mforall{}num,denom:rat\_term(). (P[num] {}\mRightarrow{} P[denom] {}\mRightarrow{} P[num "/" denom]))
{}\mRightarrow{} (\mforall{}num:rat\_term(). (P[num] {}\mRightarrow{} P[rtermMinus(num)]))
{}\mRightarrow{} \{\mforall{}v:rat\_term(). P[v]\})
Date html generated:
2019_10_29-AM-09_30_46
Last ObjectModification:
2019_03_31-PM-05_21_32
Theory : reals
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