Nuprl Lemma : qeq-functionality
∀[r,s,x:ℤ ⋃ (ℤ × ℤ-o)]. qeq(r;x) = qeq(s;x) supposing qeq(r;s) = tt
Proof
Definitions occuring in Statement :
qeq: qeq(r;s),
int_nzero: ℤ-o,
b-union: A ⋃ B,
btrue: tt,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_nzero: ℤ-o,
btrue: tt,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
true: True,
squash: ↓T
Lemmas referenced :
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
int_nzero_wf,
product-valueall-type,
set-valueall-type,
nequal_wf,
bool_wf,
qeq_wf,
btrue_wf,
iff_imp_equal_bool,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
iff_weakening_uiff,
assert_wf,
equal-wf-base,
int_subtype_base,
istype-assert,
set_subtype_base,
subtype_base_sq,
int_nzero_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
mul_cancel_in_eq,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mul_com,
subtype_rel_self,
iff_weakening_equal,
intformand_wf,
int_formula_prop_and_lemma,
mul-associates,
mul-commutes,
mul-swap,
mul_assoc,
int_entire,
itermConstant_wf,
int_term_value_constant_lemma,
zero-mul,
zero_ann_a,
intformor_wf,
int_formula_prop_or_lemma,
mul_nzero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
unionElimination,
equalityElimination,
sqequalRule,
extract_by_obid,
isectElimination,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
callbyvalueReduce,
because_Cache,
productEquality,
lambdaEquality_alt,
inhabitedIsType,
independent_functionElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_pairEquality,
equalityIstype,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
isintReduceTrue,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyEquality,
promote_hyp,
multiplyEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
productIsType,
applyLambdaEquality,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
instantiate,
cumulativity,
dependent_functionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
voidElimination,
universeEquality,
imageMemberEquality,
inlFormation_alt
Latex:
\mforall{}[r,s,x:\mBbbZ{} \mcup{} (\mBbbZ{} \mtimes{} \mBbbZ{}\msupminus{}\msupzero{})]. qeq(r;x) = qeq(s;x) supposing qeq(r;s) = tt
Date html generated:
2020_05_20-AM-09_12_48
Last ObjectModification:
2020_01_28-PM-02_40_46
Theory : rationals
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