Nuprl Lemma : rat-int-part_wf2
∀q:ℚ. (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,
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],
rationals: ℚ,
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
quotient: x,y:A//B[x; y],
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_apply: x[s],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
sq_type: SQType(T)
Lemmas referenced :
rationals_wf,
qle_wf,
int-subtype-rationals,
qless_wf,
equal_wf,
qadd_wf,
equal-wf-base,
b-union_wf,
int_nzero_wf,
equal-wf-T-base,
bool_wf,
qeq_wf,
rat-int-part_wf,
set_wf,
subtype_quotient,
qeq-equiv,
quotient-member-eq,
spread_wf,
decidable__le,
qle-int,
qadd-add,
qadd_preserves_qless,
qmul_wf,
squash_wf,
true_wf,
qadd_ac_1_q,
qadd_comm_q,
qadd_inv_assoc_q,
qinverse_q,
mon_ident_q,
iff_weakening_equal,
qadd_preserves_qle,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformle_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
setEquality,
productEquality,
intEquality,
thin,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
natural_numberEquality,
applyEquality,
sqequalRule,
hypothesisEquality,
because_Cache,
spreadEquality,
productElimination,
independent_pairEquality,
setElimination,
rename,
dependent_set_memberEquality,
pertypeElimination,
baseClosed,
dependent_functionElimination,
lambdaEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
instantiate,
cumulativity,
universeEquality,
addEquality,
unionElimination,
independent_pairFormation,
minusEquality,
imageElimination,
imageMemberEquality,
voidElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
applyLambdaEquality
Latex:
\mforall{}q:\mBbbQ{}. (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_46
Last ObjectModification:
2017_07_26-PM-06_56_48
Theory : rationals
Home
Index