Nuprl Lemma : qless-int
∀[x,y:ℤ]. uiff(x < y;x < y)
Proof
Definitions occuring in Statement :
qless: r < s,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
int: ℤ
Definitions unfolded in proof :
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
set_blt: a <b b,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_le: ≤b,
pi2: snd(t),
qadd_grp: <ℚ+>,
grp_le: ≤b,
pi1: fst(t),
infix_ap: x f y,
q_le: q_le(r;s),
qsub: r - s,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
subtype_rel: A ⊆r B,
top: Top,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
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,
bool: 𝔹,
unit: Unit,
it: ⋅,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
band: p ∧b q,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
bor: p ∨bq,
int_term: int_term(),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
qmul-elim,
int-subtype-rationals,
qadd-elim,
isint-int,
istype-void,
qpositive-elim,
qeq-elim,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformor_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_or_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
less_than_wf,
int_subtype_base,
member-less_than,
assert_wf,
bor_wf,
lt_int_wf,
eq_int_wf,
eqtt_to_assert,
iff_transitivity,
or_wf,
equal-wf-base,
iff_weakening_uiff,
assert_of_bor,
assert_of_lt_int,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
bnot_wf,
neg_assert_of_eq_int,
bfalse_wf,
not_wf,
subtype_rel_self,
int_termco_wf,
has-value_wf-partial,
nat_wf,
set-value-type,
le_wf,
int-value-type,
int_termco_size_wf,
btrue_wf,
qless_wf,
qless_witness,
assert_of_band,
assert_of_bnot,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
callbyvalueReduce,
because_Cache,
minusEquality,
natural_numberEquality,
applyEquality,
isect_memberEquality_alt,
voidElimination,
multiplyEquality,
isintReduceTrue,
addEquality,
independent_pairFormation,
isect_memberFormation_alt,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
universeIsType,
productIsType,
unionIsType,
equalityIsType4,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
functionIsType,
rename,
inlFormation_alt,
lambdaFormation_alt,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
inrFormation_alt,
equalityIsType2,
promote_hyp,
instantiate,
cumulativity,
equalityIsType1,
productEquality,
setEquality,
independent_pairEquality
Latex:
\mforall{}[x,y:\mBbbZ{}]. uiff(x < y;x < y)
Date html generated:
2019_10_16-AM-11_48_00
Last ObjectModification:
2018_10_11-PM-03_38_13
Theory : rationals
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