Nuprl Lemma : pa-add_functionality
∀[p:{2...}]. ∀[x1,y1,x2,y2:basic-padic(p)].
(pa-add(p;x1;y1) = pa-add(p;x2;y2) ∈ padic(p)) supposing (bpa-equiv(p;x1;x2) and bpa-equiv(p;y1;y2))
Proof
Definitions occuring in Statement :
pa-add: pa-add(p;x;y),
padic: padic(p),
bpa-equiv: bpa-equiv(p;x;y),
basic-padic: basic-padic(p),
int_upper: {i...},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
int_upper: {i...},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
pa-add: pa-add(p;x;y),
all: ∀x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
iff: P ⇐⇒ Q,
basic-padic: basic-padic(p),
bpa-equiv: bpa-equiv(p;x;y),
bpa-add: bpa-add(p;x;y),
cand: A c∧ B,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
has-value: (a)↓,
le: A ≤ B,
guard: {T},
true: True,
squash: ↓T,
subtype_rel: A ⊆r B
Lemmas referenced :
equal-padics,
pa-add_wf,
bpa-equiv-iff-norm,
bpa-add_wf,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
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_formula_prop_wf,
istype-less_than,
imax_ub,
nat_properties,
decidable__le,
istype-le,
value-type-has-value,
int-value-type,
imax_wf,
fastexp_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
bpa-equiv_wf,
basic-padic_wf,
istype-int_upper,
imax_nat,
intformeq_wf,
int_formula_prop_eq_lemma,
p-adics_wf,
p-int_wf,
exp_wf2,
p-mul_wf,
p-add_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
p-distrib,
nat_plus_wf,
p-mul-assoc,
exp-fastexp,
p-mul-int,
subtype_rel_self,
exp_add,
iff_weakening_equal,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
productElimination,
independent_isectElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
inlFormation_alt,
inrFormation_alt,
callbyvalueReduce,
intEquality,
inhabitedIsType,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
equalityIstype,
productIsType,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
addEquality,
hyp_replacement
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[x1,y1,x2,y2:basic-padic(p)].
(pa-add(p;x1;y1) = pa-add(p;x2;y2)) supposing (bpa-equiv(p;x1;x2) and bpa-equiv(p;y1;y2))
Date html generated:
2020_05_19-PM-10_08_57
Last ObjectModification:
2020_01_08-PM-06_54_20
Theory : rings_1
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