Nuprl Lemma : comparison-seq-zero
∀[T:Type]. ∀[c1:comparison(T)]. ∀[c2:⋂a:T. comparison({b:T| (c1 a b) = 0 ∈ ℤ} )]. ∀[x,y:T].
uiff((comparison-seq(c1; c2) x y) = 0 ∈ ℤ;((c1 x y) = 0 ∈ ℤ) ∧ ((c2 x y) = 0 ∈ ℤ))
Proof
Definitions occuring in Statement :
comparison-seq: comparison-seq(c1; c2),
comparison: comparison(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
isect: ⋂x:A. B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
comparison-seq: comparison-seq(c1; c2),
has-value: (a)↓,
member: t ∈ T,
uimplies: b supposing a,
comparison: comparison(T),
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
false: False,
not: ¬A,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
value-type-has-value,
int-value-type,
isect_wf,
comparison_wf,
equal-wf-T-base,
comparison-reflexive,
equal_wf,
squash_wf,
sq_stable__uiff,
sq_stable__equal,
sq_stable__and,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalRule,
callbyvalueReduce,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
applyEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
lambdaEquality,
dependent_functionElimination,
setEquality,
cumulativity,
baseClosed,
universeEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_set_memberEquality,
independent_functionElimination,
int_eqEquality,
productEquality,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
imageMemberEquality,
imageElimination,
unionElimination,
equalityElimination,
productElimination,
int_eqReduceTrueSq,
independent_pairFormation,
independent_pairEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
int_eqReduceFalseSq,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[c1:comparison(T)]. \mforall{}[c2:\mcap{}a:T. comparison(\{b:T| (c1 a b) = 0\} )]. \mforall{}[x,y:T].
uiff((comparison-seq(c1; c2) x y) = 0;((c1 x y) = 0) \mwedge{} ((c2 x y) = 0))
Date html generated:
2017_04_17-AM-08_28_36
Last ObjectModification:
2017_02_27-PM-04_50_07
Theory : list_1
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