Nuprl Lemma : ring_term_polynomial
∀r:CRng. ∀t:int_term(). ipolynomial-term(int_term_to_ipoly(t)) ≡ t
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
crng: CRng,
int_term_to_ipoly: int_term_to_ipoly(t),
ipolynomial-term: ipolynomial-term(p),
int_term: int_term(),
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
iPolynomial: iPolynomial(),
crng: CRng,
so_apply: x[s],
implies: P ⇒ Q,
int_term_to_ipoly: int_term_to_ipoly(t),
itermConstant: "const",
int_term_ind: int_term_ind,
itermVar: vvar,
itermAdd: left (+) right,
prop: ℙ,
itermSubtract: left (-) right,
itermMultiply: left (*) right,
itermMinus: "-"num,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
false: False,
not: ¬A,
ringeq_int_terms: t1 ≡ t2,
ring_term_value: ring_term_value(f;t),
ipolynomial-term: ipolynomial-term(p),
ifthenelse: if b then t else f fi ,
null: null(as),
nil: [],
it: ⋅,
btrue: tt,
int-to-ring: int-to-ring(r;n),
lt_int: i <z j,
bfalse: ff,
rng_nat_op: n ⋅r e,
mon_nat_op: n ⋅ e,
nat_op: n x(op;id) e,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
grp_id: e,
pi1: fst(t),
pi2: snd(t),
add_grp_of_rng: r↓+gp,
rng_zero: 0,
rng: Rng,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
imonomial-term: imonomial-term(m),
true: True,
and: P ∧ Q,
squash: ↓T,
infix_ap: x f y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_term-induction,
ringeq_int_terms_wf,
ipolynomial-term_wf,
int_term_to_ipoly_wf,
iPolynomial_wf,
int_term_wf,
crng_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
ring_term_value_wf,
itermConstant_wf,
rng_car_wf,
null_cons_lemma,
spread_cons_lemma,
list_accum_nil_lemma,
list_accum_cons_lemma,
ring_term_value_mul_lemma,
ring_term_value_const_lemma,
ring_term_value_var_lemma,
rng_times_wf,
rng_times_one,
equal_wf,
squash_wf,
true_wf,
int-to-ring-one,
subtype_rel_self,
iff_weakening_equal,
add-ipoly-ringeq,
add_ipoly_wf,
itermAdd_wf,
add-ipoly_wf1,
uiff_transitivity,
add_ipoly-sq,
ringeq_int_terms_functionality,
ringeq_int_terms_weakening,
itermAdd_functionality_wrt_ringeq,
minus-poly-ringeq,
minus-poly_wf,
itermSubtract_wf,
itermMinus_wf,
ring_term_value_add_lemma,
ring_term_value_minus_lemma,
ring_term_value_sub_lemma,
rng_plus_wf,
rng_minus_wf,
ringeq_int_terms_transitivity,
itermMinus_functionality_wrt_ringeq,
mul-ipoly-ringeq,
mul_ipoly_wf,
itermMultiply_wf,
mul-ipoly_wf,
mul_poly-sq,
itermMultiply_functionality_wrt_ringeq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesisEquality,
hypothesis,
applyEquality,
setElimination,
rename,
independent_functionElimination,
intEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
int_eqReduceFalseSq,
functionEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}r:CRng. \mforall{}t:int\_term(). ipolynomial-term(int\_term\_to\_ipoly(t)) \mequiv{} t
Date html generated:
2018_05_21-PM-03_17_24
Last ObjectModification:
2018_05_19-AM-08_08_35
Theory : rings_1
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