Nuprl Lemma : qle-int
∀[x,y:ℤ]. uiff(x ≤ y;x ≤ y)
Proof
Definitions occuring in Statement :
qle: r ≤ s,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qle: r ≤ s,
grp_leq: a ≤ b,
qadd_grp: <ℚ+>,
grp_le: ≤b,
pi2: snd(t),
pi1: fst(t),
infix_ap: x f y,
q_le: q_le(r;s),
qsub: r - s,
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,
le: A ≤ B,
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
le_witness_for_triv,
qle_wf,
qle_witness,
le_wf,
istype-int,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformor_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_or_lemma,
int_formula_prop_less_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,
decidable__or,
equal-wf-base,
decidable__lt,
decidable__equal_int,
assert_wf,
bor_wf,
lt_int_wf,
eq_int_wf,
or_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_lt_int,
assert_of_eq_int,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
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,
productElimination,
independent_pairEquality,
universeIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
inhabitedIsType,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
unionIsType,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
rename,
lambdaFormation_alt,
inlFormation_alt,
inrFormation_alt,
promote_hyp
Latex:
\mforall{}[x,y:\mBbbZ{}]. uiff(x \mleq{} y;x \mleq{} y)
Date html generated:
2019_10_16-PM-00_31_14
Last ObjectModification:
2018_10_10-PM-06_22_59
Theory : rationals
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