Nuprl Lemma : flip_symmetry
∀[k:ℕ]. ∀[i,j:ℕk]. ((i, j) = (j, i) ∈ (ℕk ⟶ ℕk))
Proof
Definitions occuring in Statement :
flip: (i, j),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
flip: (i, j),
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
guard: {T},
lelt: i ≤ j < k,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
equal_wf,
eqtt_to_assert,
assert_of_eq_int,
int_seg_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
bnot_wf,
not_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
sqequalRule,
because_Cache,
hypothesis,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
extract_by_obid,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
baseClosed,
intEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
impliesFunctionality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[i,j:\mBbbN{}k]. ((i, j) = (j, i))
Date html generated:
2017_04_17-AM-08_05_52
Last ObjectModification:
2017_02_27-PM-04_35_15
Theory : list_1
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