Nuprl Lemma : square-between
∀a:{a:ℚ| 0 ≤ a} . ∀b:{b:ℚ| a < b} . (∃r:ℚ [(a < r * r < b ∧ 0 < r)])
Proof
Definitions occuring in Statement :
q-between: a < b < c,
qle: r ≤ s,
qless: r < s,
qmul: r * s,
rationals: ℚ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
nat_plus: ℕ+,
sq_type: SQType(T),
guard: {T},
sq_stable: SqStable(P),
squash: ↓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),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
has-value: (a)↓,
has-valueall: has-valueall(a),
qeq: qeq(r;s),
qsub: r - s,
qpositive: qpositive(r),
qmul: r * s,
ifthenelse: if b then t else f fi ,
btrue: tt,
qadd: r + s,
bfalse: ff,
or: P ∨ Q,
uiff: uiff(P;Q),
band: p ∧b q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
mk-rational: mk-rational(a;b),
q_less: q_less(r;s),
cand: A c∧ B,
nat: ℕ,
sq_exists: ∃x:A [B[x]],
let: let,
ge: i ≥ j ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
true: True,
less_than: a < b,
le: A ≤ B,
subtract: n - m,
q-between: a < b < c,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
equals-qrep,
qrep-denom,
sq_exists_wf,
rationals_wf,
q-between_wf,
qrep_wf,
qmul_wf,
qless_wf,
qle_wf,
int-subtype-rationals,
subtype_base_sq,
nat_plus_wf,
product_subtype_base,
int_subtype_base,
set_subtype_base,
less_than_wf,
sq_stable_from_decidable,
decidable__qle,
valueall-type-has-valueall,
product-valueall-type,
int-valueall-type,
set-valueall-type,
evalall-reduce,
assert_wf,
bor_wf,
lt_int_wf,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert_of_lt_int,
bfalse_wf,
eq_int_wf,
equal-wf-base,
istype-assert,
istype-less_than,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformor_wf,
intformless_wf,
itermAdd_wf,
itermMultiply_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_or_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_band,
assert_of_eq_int,
decidable__qless,
assert-q_less-eq,
mk-rational_wf,
nat_plus_inc_int_nzero,
iff_weakening_equal,
mul_bounds_1b,
decidable__lt,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
mul_bounds_1a,
istype-le,
nat_plus_subtype_nat,
isqrt_wf,
nat_properties,
add_nat_plus,
add-is-int-iff,
false_wf,
qdiv_wf,
multiply_nat_wf,
int_nzero-rational,
int_entire_a,
nequal_wf,
mul_nat_plus,
isqrt-property,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
add_functionality_wrt_eq,
istype-nat,
itermMinus_wf,
int_term_value_minus_lemma,
mul_preserves_lt,
nat_wf,
le_wf,
decidable__equal_int,
mul-distributes,
mul-distributes-right,
add-associates,
mul-swap,
mul-commutes,
mul-associates,
one-mul,
two-mul,
add-commutes,
add-swap,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
mul_preserves_le,
mk-rational-qdiv,
qmul_preserves_qless,
qless-int,
subtype_rel_set,
qmul-mul,
qmul_assoc_qrng,
qmul_comm_qrng,
qmul_ac_1_qrng,
qmul-qdiv-cancel3,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
qmul_zero_qrng,
qmul-qdiv-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalitySymmetry,
because_Cache,
equalityTransitivity,
inhabitedIsType,
productElimination,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
sqequalRule,
lambdaEquality_alt,
productEquality,
natural_numberEquality,
applyEquality,
setIsType,
universeIsType,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
callbyvalueReduce,
sqleReflexivity,
isintReduceTrue,
independent_pairEquality,
addEquality,
multiplyEquality,
unionElimination,
unionEquality,
baseApply,
closedConclusion,
isect_memberEquality_alt,
productIsType,
unionIsType,
sqequalBase,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
inlFormation_alt,
promote_hyp,
inrFormation_alt,
functionIsType,
dependent_set_memberEquality_alt,
dependent_set_memberFormation_alt,
pointwiseFunctionality,
universeEquality,
minusEquality
Latex:
\mforall{}a:\{a:\mBbbQ{}| 0 \mleq{} a\} . \mforall{}b:\{b:\mBbbQ{}| a < b\} . (\mexists{}r:\mBbbQ{} [(a < r * r < b \mwedge{} 0 < r)])
Date html generated:
2019_10_16-PM-00_38_33
Last ObjectModification:
2019_06_26-PM-04_14_56
Theory : rationals
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