Nuprl Lemma : tswap_eval_3
∀n:ℕ. ∀i,j,k:ℕn. ((¬(k = i ∈ ℕn)) ⇒ (¬(k = j ∈ ℕn)) ⇒ ((swap{n}(i;j) k) = k ∈ ℕn))
Proof
Definitions occuring in Statement :
tswap: swap{n}(i;j),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
tswap: swap{n}(i;j),
all: ∀x:A. B[x],
implies: P ⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ
Lemmas referenced :
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
not_wf,
equal_wf,
int_seg_wf,
nat_wf,
swap_eval_3,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
dependent_set_memberEquality,
hypothesis,
equalitySymmetry,
independent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j,k:\mBbbN{}n. ((\mneg{}(k = i)) {}\mRightarrow{} (\mneg{}(k = j)) {}\mRightarrow{} ((swap\{n\}(i;j) k) = k))
Date html generated:
2018_05_22-AM-07_44_31
Last ObjectModification:
2018_05_19-AM-08_33_38
Theory : perms_1
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