Nuprl Lemma : comparison-linear
∀[T:Type]. ∀cmp:comparison(T). Linorder(cmp-type(T;cmp);x,y.cmp-le(cmp;x;y))
Proof
Definitions occuring in Statement :
cmp-le: cmp-le(cmp;x;y),
cmp-type: cmp-type(T;cmp),
comparison: comparison(T),
linorder: Linorder(T;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
linorder: Linorder(T;x,y.R[x; y]),
and: P ∧ Q,
order: Order(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
member: t ∈ T,
refl: Refl(T;x,y.E[x; y]),
cmp-le: cmp-le(cmp;x;y),
uimplies: b supposing a,
cmp-type: cmp-type(T;cmp),
quotient: x,y:A//B[x; y],
implies: P ⇒ Q,
comparison: comparison(T),
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
anti_sym: AntiSym(T;x,y.R[x; y]),
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
true: True,
or: P ∨ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
squash: ↓T,
rev_implies: P ⇐ Q,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
comparison_wf,
istype-universe,
cmp-type_wf,
subtype_base_sq,
int_subtype_base,
comparison-reflexive,
istype-int,
istype-false,
cmp-le_wf,
decidable__cmp-le,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
equal_wf,
comparison-equiv,
equal-wf-base,
quotient-member-eq,
le_wf,
int_formula_prop_wf,
int_term_value_minus_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermMinus_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
isectElimination,
universeEquality,
cumulativity,
intEquality,
independent_isectElimination,
pointwiseFunctionalityForEquality,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :inhabitedIsType,
Error :equalityIsType1,
independent_functionElimination,
Error :productIsType,
Error :equalityIstype,
sqequalBase,
because_Cache,
applyEquality,
setElimination,
rename,
baseClosed,
natural_numberEquality,
voidElimination,
unionElimination,
pointwiseFunctionality,
imageMemberEquality,
imageElimination,
Error :lambdaEquality_alt,
Error :equalityIsType4,
promote_hyp,
hyp_replacement,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
Error :inrFormation_alt,
Error :inlFormation_alt,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}cmp:comparison(T). Linorder(cmp-type(T;cmp);x,y.cmp-le(cmp;x;y))
Date html generated:
2019_06_20-PM-01_42_08
Last ObjectModification:
2018_11_22-PM-10_09_55
Theory : list_1
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