Nuprl Lemma : comparison-equiv
∀[T:Type]. ∀cmp:comparison(T). EquivRel(T;x,y.(cmp x y) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
comparison: comparison(T),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
comparison: comparison(T),
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
trans: Trans(T;x,y.E[x; y]),
guard: {T},
prop: ℙ,
uimplies: b supposing a,
sq_type: SQType(T),
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
uiff: uiff(P;Q)
Lemmas referenced :
istype-int,
int_subtype_base,
comparison_wf,
subtype_base_sq,
equal_wf,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
minus-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
hypothesis,
Error :universeIsType,
hypothesisEquality,
Error :equalityIsType4,
extract_by_obid,
applyEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
Error :lambdaEquality_alt,
independent_pairEquality,
axiomEquality,
Error :functionIsTypeImplies,
universeEquality,
independent_functionElimination,
hyp_replacement,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}cmp:comparison(T). EquivRel(T;x,y.(cmp x y) = 0)
Date html generated:
2019_06_20-PM-01_41_44
Last ObjectModification:
2018_10_04-PM-04_24_27
Theory : list_1
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