Nuprl Lemma : flip-conjugation
∀[n:ℕ]. ∀[k:ℕn]. ∀[j:ℕk]. ∀[l:ℕn - k]. ((j, k + l) = ((k, k + l) o ((j, k) o (k, k + l))) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
flip: (i, j),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
flip: (i, j),
compose: f o g,
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
nequal: a ≠ b ∈ T ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
ge: i ≥ j ,
decidable: Dec(P)
Lemmas referenced :
int_seg_wf,
subtract_wf,
nat_wf,
eq_int_wf,
bool_wf,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
itermAdd_wf,
intformle_wf,
itermConstant_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
ifthenelse_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformnot_wf,
int_formula_prop_not_lemma,
int_seg_properties,
nat_properties,
decidable__le,
decidable__lt,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
because_Cache,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
dependent_set_memberEquality,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
addEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbN{}n]. \mforall{}[j:\mBbbN{}k]. \mforall{}[l:\mBbbN{}n - k]. ((j, k + l) = ((k, k + l) o ((j, k) o (k, k + l))))
Date html generated:
2018_05_21-PM-00_41_12
Last ObjectModification:
2018_05_19-AM-06_48_36
Theory : list_1
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