Nuprl Lemma : test23
∀[a,b,c:ℚ]. (False) supposing (0 < c and ((b + c) ≤ a) and ((a + c) ≤ b))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qadd: r + s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
false: False,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
isl: isl(x),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
btrue: tt,
true: True,
q-constraints: q-constraints(k;A;y),
all: ∀x:A. B[x],
top: Top,
cand: A c∧ B,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
sq_type: SQType(T),
guard: {T},
select: L[n],
cons: [a / b],
pi2: snd(t),
pi1: fst(t),
q-rel: q-rel(r;x),
eq_int: (i =z j),
squash: ↓T,
nat_plus: ℕ+,
less_than: a < b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
select?: as[i]?a,
lt_int: i <z j,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
qless_wf,
int-subtype-rationals,
qle_wf,
qadd_wf,
rationals_wf,
decidable__q-constraints-opt,
false_wf,
le_wf,
cons_wf,
nat_wf,
select?_wf,
nil_wf,
outr_wf,
sq_exists_wf,
list_wf,
q-constraints_wf,
not_wf,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
squash_wf,
true_wf,
q-linear-unroll,
less_than_wf,
subtype_rel_self,
iff_weakening_equal,
qmul_wf,
q-linear-base,
int_seg_subtype,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
qadd_preserves_qle,
qadd_preserves_qless,
mon_ident_q,
qmul_one_qrng,
mon_assoc_q,
qadd_ac_1_q,
qadd_comm_q,
qinverse_q,
qadd_inv_assoc_q,
qmul_zero_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
because_Cache,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
applyEquality,
hypothesisEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
instantiate,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
productEquality,
functionEquality,
intEquality,
independent_pairEquality,
lambdaEquality,
minusEquality,
computeAll,
independent_isectElimination,
independent_functionElimination,
dependent_set_memberFormation,
dependent_functionElimination,
voidEquality,
setElimination,
rename,
unionElimination,
cumulativity,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
hypothesis_subsumption,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. (False) supposing (0 < c and ((b + c) \mleq{} a) and ((a + c) \mleq{} b))
Date html generated:
2018_05_22-AM-00_26_37
Last ObjectModification:
2018_05_19-PM-04_08_33
Theory : rationals
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