Nuprl Lemma : assoced_nelim
∀a,b:ℕ. (a ~ b ⇐⇒ a = b ∈ ℤ)
Proof
Definitions occuring in Statement :
assoced: a ~ b,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ
Lemmas referenced :
assoced_elim,
istype-int,
set_subtype_base,
le_wf,
int_subtype_base,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMinus_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
assoced_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_functionElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_pairFormation,
promote_hyp,
because_Cache,
sqequalRule,
Error :unionIsType,
Error :equalityIsType4,
applyEquality,
isectElimination,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
minusEquality,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :universeIsType,
Error :inlFormation_alt,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a,b:\mBbbN{}. (a \msim{} b \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2019_06_20-PM-02_21_15
Last ObjectModification:
2018_10_02-PM-11_35_06
Theory : num_thy_1
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