Nuprl Lemma : rat-int-part_wf
∀q:ℤ ⋃ (ℤ × ℤ-o). (rat-int-part(q) ∈ {p:ℤ × {r:ℚ| (0 ≤ r) ∧ r < 1} | let x,r = p in q = (x + r) ∈ ℚ} )
Proof
Definitions occuring in Statement :
rat-int-part: rat-int-part(q),
qle: r ≤ s,
qless: r < s,
qadd: r + s,
rationals: ℚ,
int_nzero: ℤ-o,
b-union: A ⋃ B,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rat-int-part: rat-int-part(q),
uall: ∀[x:A]. B[x],
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
qdiv: (r/s),
qinv: 1/r,
qmul: r * s,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
btrue: tt,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
sq_type: SQType(T),
guard: {T},
bfalse: ff,
nequal: a ≠ b ∈ T ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
it: ⋅,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
nat_plus: ℕ+,
le: A ≤ B,
int_lower: {...i},
gt: i > j,
ge: i ≥ j
Lemmas referenced :
b-union_wf,
int_nzero_wf,
qle_reflexivity,
int-subtype-rationals,
qless-int,
qle_wf,
qless_wf,
mon_ident_q,
equal-wf-base-T,
rationals_wf,
int_subtype_base,
qadd_wf,
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
set-valueall-type,
nequal_wf,
product-valueall-type,
evalall-sqequal,
set_subtype_base,
subtype_base_sq,
product_subtype_base,
mul-commutes,
one-mul,
div_rem_sum,
int_nzero_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal-wf-base,
value-type-has-value,
int-value-type,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
qmul-preserves-eq,
qdiv_wf,
qmul_wf,
subtype_rel_set,
int_nzero-rational,
qmul-mul,
int-equal-in-rationals,
decidable__equal_int,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
squash_wf,
true_wf,
qmul_over_plus_qrng,
qmul_zero_qrng,
qmul_comm_qrng,
qadd_comm_q,
qmul-qdiv-cancel,
iff_weakening_equal,
lt_int_wf,
assert_of_lt_int,
le_int_wf,
assert_of_le_int,
le_wf,
less_than_wf,
rem_bounds_1,
qdiv-non-neg1,
qle-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
qmul_preserves_qless,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
qmul_one_qrng,
qadd-add,
rem_bounds_2,
subtract_wf,
itermMinus_wf,
int_term_value_minus_lemma,
itermSubtract_wf,
int_term_value_subtract_lemma,
rem_bounds_4,
qmul_preserves_qle,
qmul_over_minus_qrng,
rem_bounds_3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
intEquality,
productEquality,
imageElimination,
productElimination,
unionElimination,
equalityElimination,
sqequalRule,
isintReduceTrue,
hypothesisEquality,
dependent_set_memberEquality,
independent_pairEquality,
natural_numberEquality,
applyEquality,
independent_pairFormation,
independent_isectElimination,
imageMemberEquality,
baseClosed,
because_Cache,
equalitySymmetry,
spreadEquality,
setElimination,
rename,
callbyvalueReduce,
lambdaEquality,
independent_functionElimination,
baseApply,
closedConclusion,
instantiate,
cumulativity,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
equalityTransitivity,
remainderEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
divideEquality,
promote_hyp,
addEquality,
multiplyEquality,
universeEquality,
minusEquality
Latex:
\mforall{}q:\mBbbZ{} \mcup{} (\mBbbZ{} \mtimes{} \mBbbZ{}\msupminus{}\msupzero{}). (rat-int-part(q) \mmember{} \{p:\mBbbZ{} \mtimes{} \{r:\mBbbQ{}| (0 \mleq{} r) \mwedge{} r < 1\} | let x,r = p in q = (x + r)\} )
Date html generated:
2018_05_22-AM-00_27_34
Last ObjectModification:
2017_07_26-PM-06_56_42
Theory : rationals
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