Nuprl Lemma : qtruncate_functionality
∀[a,b:ℚ]. ∀[N:ℕ+]. qtruncate(a;N) ≤ qtruncate(b;N) supposing a ≤ b
Proof
Definitions occuring in Statement :
qtruncate: qtruncate(q;N),
qle: r ≤ s,
rationals: ℚ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
qtruncate: qtruncate(q;N),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat_plus: ℕ+,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
guard: {T},
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
qle-int,
qmul_preserves_qle2,
q-ceil_functionality,
iff_weakening_equal,
qmul-qdiv-cancel,
qmul_comm_qrng,
not_wf,
true_wf,
squash_wf,
qle_witness,
nat_plus_wf,
qle_wf,
qtruncate_wf,
equal_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
nequal_wf,
subtype_rel_sets,
int_nzero-rational,
int-subtype-rationals,
less_than_wf,
rationals_wf,
subtype_rel_set,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
qless-int,
qmul_wf,
q-ceil_wf,
qdiv_wf,
qmul_preserves_qle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
applyEquality,
sqequalRule,
independent_isectElimination,
introduction,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
setEquality,
lambdaFormation,
independent_functionElimination,
isect_memberFormation,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. \mforall{}[N:\mBbbN{}\msupplus{}]. qtruncate(a;N) \mleq{} qtruncate(b;N) supposing a \mleq{} b
Date html generated:
2016_05_15-PM-11_35_34
Last ObjectModification:
2016_01_16-PM-09_12_09
Theory : rationals
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