Nuprl Lemma : comparison-seq_wf
∀[T:Type]. ∀[c1:comparison(T)]. ∀[c2:⋂a:T. comparison({b:T| (c1 a b) = 0 ∈ ℤ} )]. (comparison-seq(c1; c2) ∈ comparison(\000CT))
Proof
Definitions occuring in Statement :
comparison-seq: comparison-seq(c1; c2),
comparison: comparison(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
member: t ∈ T,
comparison: comparison(T),
comparison-seq: comparison-seq(c1; c2),
and: P ∧ Q,
has-value: (a)↓,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
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,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
prop: ℙ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
le: A ≤ B
Lemmas referenced :
value-type-has-value,
eqtt_to_assert,
assert_of_eq_int,
int_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
istype-int,
istype-le,
comparison_wf,
equal-wf-base,
istype-universe,
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,
int-value-type,
comparison-reflexive,
minus-is-int-iff,
false_wf,
eq_int_wf,
bool_wf,
sq_stable__le,
nequal-le-implies,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
intformle_wf,
int_formula_prop_le_lemma,
decidable__le,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
Error :dependent_set_memberEquality_alt,
productElimination,
Error :lambdaEquality_alt,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
int_eqReduceTrueSq,
equalityTransitivity,
equalitySymmetry,
applyEquality,
hypothesisEquality,
Error :equalityIstype,
baseClosed,
sqequalBase,
dependent_functionElimination,
independent_functionElimination,
Error :dependent_pairFormation_alt,
promote_hyp,
instantiate,
voidElimination,
int_eqReduceFalseSq,
independent_pairFormation,
Error :universeIsType,
Error :productIsType,
Error :functionIsType,
minusEquality,
natural_numberEquality,
axiomEquality,
Error :isectIsType,
setEquality,
intEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
universeEquality,
approximateComputation,
int_eqEquality,
cumulativity,
pointwiseFunctionality,
closedConclusion,
imageMemberEquality,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[c1:comparison(T)]. \mforall{}[c2:\mcap{}a:T. comparison(\{b:T| (c1 a b) = 0\} )].
(comparison-seq(c1; c2) \mmember{} comparison(T))
Date html generated:
2019_06_20-PM-01_42_24
Last ObjectModification:
2019_05_03-PM-02_10_37
Theory : list_1
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