Nuprl Lemma : qeq-qrep
∀[r:ℚ]. qeq(r;qrep(r)) = tt
Proof
Definitions occuring in Statement :
qrep: qrep(r),
rationals: ℚ,
qeq: qeq(r;s),
btrue: tt,
bool: 𝔹,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
rationals: ℚ,
member: t ∈ T,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
qrep: qrep(r),
qeq: qeq(r;s),
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
btrue: tt,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
bfalse: ff,
spreadn: spread3,
it: ⋅,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
le: A ≤ B,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
nequal: a ≠ b ∈ T
Lemmas referenced :
bool_wf,
b-union_wf,
int_nzero_wf,
equal_wf,
equal-wf-base,
equal-wf-T-base,
qeq_wf,
rationals_wf,
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
product-valueall-type,
evalall-sqequal,
int_subtype_base,
set-valueall-type,
nequal_wf,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
set_subtype_base,
eq_int_eq_true,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
btrue_wf,
iff_weakening_equal,
gcd_reduce_property,
gcd_reduce_wf,
nat_wf,
squash_wf,
true_wf,
equal-wf-base-T,
coprime_wf,
nat_properties,
int_nzero_properties,
intformand_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
equalitySymmetry,
isectElimination,
intEquality,
productEquality,
lambdaFormation,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
baseClosed,
imageElimination,
unionElimination,
equalityElimination,
independent_isectElimination,
callbyvalueReduce,
isintReduceTrue,
lambdaEquality,
independent_pairEquality,
natural_numberEquality,
baseApply,
closedConclusion,
applyEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
minusEquality,
multiplyEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageMemberEquality,
universeEquality,
setElimination,
rename,
independent_pairFormation
Latex:
\mforall{}[r:\mBbbQ{}]. qeq(r;qrep(r)) = tt
Date html generated:
2018_05_21-PM-11_47_34
Last ObjectModification:
2017_07_26-PM-06_43_10
Theory : rationals
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