Nuprl Lemma : exists-type-equating-ints
∀x,y,n,m:ℤ.
((¬(x = y ∈ ℤ))
⇒ (¬(n = m ∈ ℤ))
⇒ (¬(x = m ∈ ℤ))
⇒ (¬(y = n ∈ ℤ))
⇒ (∃T:Type. ((x = n ∈ T) ∧ (y = m ∈ T) ∧ (¬(x = y ∈ T)))))
Proof
Definitions occuring in Statement :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
and: P ∧ Q,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
guard: {T},
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pi2: snd(t),
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T
Lemmas referenced :
not_wf,
equal-wf-base,
int_subtype_base,
b-union_wf,
quotient_wf,
or_wf,
true_wf,
equiv_rel_true,
btrue_wf,
quotient-member-eq,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
b-union-equality-disjoint,
isect2_wf,
isect2_decomp,
false_wf,
intformand_wf,
int_formula_prop_and_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
dependent_pairFormation,
setEquality,
because_Cache,
lambdaEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
productEquality,
imageMemberEquality,
dependent_pairEquality,
dependent_functionElimination,
dependent_set_memberEquality,
inlFormation,
unionElimination,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
inrFormation,
independent_functionElimination,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
baseClosed,
pointwiseFunctionality,
pertypeElimination,
applyLambdaEquality,
imageElimination
Latex:
\mforall{}x,y,n,m:\mBbbZ{}.
((\mneg{}(x = y))
{}\mRightarrow{} (\mneg{}(n = m))
{}\mRightarrow{} (\mneg{}(x = m))
{}\mRightarrow{} (\mneg{}(y = n))
{}\mRightarrow{} (\mexists{}T:Type. ((x = n) \mwedge{} (y = m) \mwedge{} (\mneg{}(x = y)))))
Date html generated:
2017_10_01-AM-09_07_31
Last ObjectModification:
2017_07_26-PM-04_46_48
Theory : general
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