Nuprl Lemma : swap-exists
∀n:ℕ. ∀AType:array{i:l}(ℤ;n). ∃prog:ℕn ⟶ ℕn ⟶ (A-map Unit). ∀[i,j:ℕn]. alt-swap-spec(AType;n;prog)
Proof
Definitions occuring in Statement :
alt-swap-spec: alt-swap-spec(AType;n;prog),
A-map: A-map,
array-model: array-model(AType),
array: array{i:l}(Val;n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
nat: ℕ,
alt-swap-spec: alt-swap-spec(AType;n;prog),
and: P ∧ Q,
implies: P ⇒ Q,
array: array{i:l}(Val;n),
simple-swap: simple-swap(AModel;i;j),
A-pre-val: A-pre-val(AType;A;i),
A-post-val: A-post-val(AType;prog;A;i),
idx: idx(AType),
array-model: array-model(AType),
A-fetch': A-fetch'(AModel),
A-coerce: A-coerce(AModel),
A-assign: A-assign(AModel),
A-bind: A-bind(AModel),
A-eval: A-eval(AModel),
Arr: Arr(AType),
pi1: fst(t),
pi2: snd(t),
upd: upd(AType),
array-monad: array-monad(AType),
M-bind: M-bind(Mnd),
let: let,
mk_monad: mk_monad(M;return;bind),
cand: A c∧ B,
not: ¬A,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
false: False,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P)
Lemmas referenced :
array_wf,
nat_wf,
simple-swap_wf,
int_seg_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-void,
istype-universe,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
istype-int,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_properties,
nat_properties,
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_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
equal-wf-base,
assert_wf,
decidable__equal_int,
bnot_wf,
not_wf,
assert_elim,
eq_int_eq_true,
bfalse_wf,
btrue_neq_bfalse,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
subtype_rel-equal,
base_wf,
alt-swap-spec_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
hypothesis,
dependent_pairFormation_alt,
lambdaEquality_alt,
inhabitedIsType,
natural_numberEquality,
setElimination,
rename,
isect_memberFormation_alt,
sqequalRule,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
because_Cache,
isectIsType,
independent_pairFormation,
productIsType,
functionIsType,
equalityIsType4,
equalityTransitivity,
equalitySymmetry,
applyEquality,
closedConclusion,
independent_isectElimination,
unionElimination,
equalityElimination,
equalityIsType2,
baseApply,
baseClosed,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
approximateComputation,
int_eqEquality,
equalityIsType1,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}AType:array\{i:l\}(\mBbbZ{};n).
\mexists{}prog:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n {}\mrightarrow{} (A-map Unit). \mforall{}[i,j:\mBbbN{}n]. alt-swap-spec(AType;n;prog)
Date html generated:
2019_10_15-AM-10_59_39
Last ObjectModification:
2018_10_11-PM-06_53_12
Theory : monads
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