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,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
or: P ∨ Q,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top
Lemmas referenced :
type-equating-some,
or_wf,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
not_wf,
decidable__equal_int,
full-omega-unsat,
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,
equal_functionality_wrt_subtype_rel2,
intformand_wf,
int_formula_prop_and_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
intEquality,
productElimination,
rename,
dependent_pairFormation,
independent_pairFormation,
independent_functionElimination,
inlFormation,
inrFormation,
productEquality,
unionElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry
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:
2018_05_21-PM-01_12_23
Last ObjectModification:
2018_05_03-PM-02_23_46
Theory : num_thy_1
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