hcju/isabelleps huggingface dataset:
flow_trans: fixes s :: "real"   and x0 :: "'a"   and t :: "real" assumes "s \<in> existence_ivl0 x0"   and "t \<in> existence_ivl0 (flow0 x0 s)" shows "flow0 x0 (s + t) = flow0 (flow0 x0 s) t"
\<delta>_nemp: shows "\<delta> \<noteq> {#}"
direct_decomp_some_basis_pairwise_disjnt: fixes A :: "'b set"   and ss :: "'b set list" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> local.subspace s" shows "pairwise (\<lambda>s1 s2. disjnt (some_basis s1) (some_basis s2)) (set ss)"
rec_comb: fixes \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma>' :: "(name, value) fmap"   and cs :: "(Term_Class.term \<times> sterm) list"   and u :: "sterm"   and u' :: "value"   and env :: "(name, value) fmap"   and uv :: "Term_Class.term"   and rhs :: "sterm"   and val :: "value" assumes "\<Gamma> \<turnstile>\<^sub>v t \<down> Vrecabs css name \<Gamma>'"   and "fmlookup css name = Some cs"   and "\<Gamma> \<turnstile>\<^sub>v u \<down> u'"   and "vfind_match cs u' = Some (env, uv, rhs)"   and "\<Gamma>' ++\<^sub>f mk_rec_env css \<Gamma>' ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> val" shows "\<Gamma> \<turnstile>\<^sub>v t $\<^sub>s u \<down> val"
wellformed_def: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp" shows "wellformed \<Gamma> = wf {(q, p). p \<leadsto>\<^bsub>\<Gamma>\<^esub> q}"
Sigma: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a" assumes "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "F (\<lambda>x. F (g x) (B x)) A = F (\<lambda>(x, y). g x y) (Sigma A B)"
add_leE: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m + k \<le> n" obtains "m \<le> n" and "k \<le> n"
qeq: fixes x :: "'a['b, 'c]"   and y :: "'a['d, 'c]" assumes "x \<cong>\<^sub>Q y" shows "QD('b) = QD('d)"
graeffe_poly_impl_mahler: fixes f :: "int poly"   and m :: "nat" shows "mahler_measure (graeffe_poly_impl f m) = mahler_measure f ^ 2 ^ m"
simps_63: fixes x8 :: "expr"   and x31 :: "jump"   and x32 :: "stmt" shows "Throw x8 \<noteq> x31\<bullet> x32"
periodic_orbit_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.periodic_orbit f X x = (auto_ll_on_open.closed_orbit f X x \<and> 0 < auto_ll_on_open.period f X x)"
dual_neg_top: fixes x :: "'a" shows "x ^ o * \<bottom> \<squnion> x * \<top> = \<top>"
eval_val: fixes t :: "lam" assumes "val t" shows "t \<Down> t"
current_current_of_bipartite: fixes f :: "('v + 'v) \<times> ('v + 'v) \<Rightarrow> ennreal" assumes "current (bipartite_web_of \<Gamma>) f"   and "wave (bipartite_web_of \<Gamma>) f" shows "current \<Gamma> (current_of_bipartite f)"
Q': shows "P' = Polynomial.monom 1 (A - 1) * Q'"
Rep2: fixes x :: "'d" shows "T2 (Rep2 x) x"
quot_dbtm_inject: fixes t :: "dbtm"   and u :: "dbtm" shows "(quot_dbtm t = quot_dbtm u) = (t = u)"
case_sum_KK: fixes a :: "'c" shows "case_sum (\<lambda>x. a) (\<lambda>x. a) = (\<lambda>x. a)"
assms_2: fixes x' :: "'b" assumes "P' x'" shows "\<exists>x. P x"
measure_notin_sets: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<notin> sets M" shows "Sigma_Algebra.measure M A = 0"
simps: fixes a :: "('a, 'b) trm" shows "dfree a = ((\<exists>i. a = Var i) \<or> (\<exists>r. a = Const r) \<or> (\<exists>args i. a = $f i args \<and> (\<forall>x. dfree (args x))) \<or> (\<exists>\<theta>\<^sub>1 \<theta>\<^sub>2. a = Plus \<theta>\<^sub>1 \<theta>\<^sub>2 \<and> dfree \<theta>\<^sub>1 \<and> dfree \<theta>\<^sub>2) \<or> (\<exists>\<theta>\<^sub>1 \<theta>\<^sub>2. a = Times \<theta>\<^sub>1 \<theta>\<^sub>2 \<and> dfree \<theta>\<^sub>1 \<and> dfree \<theta>\<^sub>2))"
1: shows "fmlookup (fmrestrict_set vs s) \<subseteq>\<^sub>m fmlookup s"
is_bound_f_def: fixes Q\<^sub>b :: "'a"   and Q :: "'a set"   and f :: "nat \<Rightarrow> 'a" shows "is_bound_f Q\<^sub>b Q f \<equiv> \<forall>i j. [f[f 0 ..]Q] \<and> (i < j \<longrightarrow> [[f i f j Q\<^sub>b]])"
assms: shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
simps_3: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "qfree (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = (qfree \<phi>\<^sub>1 \<and> qfree \<phi>\<^sub>2)"
has_derivative: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and g' :: "'e \<Rightarrow> 'c"   and F :: "'e filter" assumes "bounded_linear f"   and "(g has_derivative g') F" shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
assms_1: shows "m icd\<^bsup>\<pi>\<^esup>\<rightarrow> n"
mset_empty: shows "mset_tree \<langle>\<rangle> = {#}"
FactGroup_the_elem_mem: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and X :: "'c set" assumes "group_hom G H h"   and "X \<in> carrier (G Mod kernel G H h)" shows "the_elem (h ` X) \<in> carrier H"
Diff1_foldSetD: fixes A :: "'e set"   and x :: "'e"   and y :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" assumes "(A - {x}, y) \<in> foldSetD D f e"   and "x \<in> A"   and "f x y \<in> D" shows "(A, f x y) \<in> foldSetD D f e"
5: shows "Ana_invar_subst (ik\<^sub>s\<^sub>t (unlabel \<A>) \<union> assignment_rhs\<^sub>s\<^sub>t (unlabel \<A>))"
id: shows "{0..<n} = insert (n - 1) {0..<n - 1}"
upset_prop: fixes x :: "'b" shows "\<up> x = {y. x \<le> y}"
rat_to_normalized_int_poly_3: fixes p :: "rat poly"   and d :: "rat"   and q :: "int poly" assumes "rat_to_normalized_int_poly p = (d, q)"   and "p \<noteq> 0" shows "content q = 1"
wf_UN: fixes I :: "'b set"   and r :: "'b \<Rightarrow> ('c \<times> 'c) set" assumes "\<And>i. i \<in> I \<Longrightarrow> wf (r i)"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I; r i \<noteq> r j\<rbrakk> \<Longrightarrow> Domain (r i) \<inter> Range (r j) = {}" shows "wf (\<Union> (r ` I))"
nth_in: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set"   and i :: "nat" assumes "length xs = n"   and "set xs \<subseteq> A"   and "i < n" shows "xs ! i \<in> A"
mod_add_right_eq: fixes divide :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g"   and modulo :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and euclidean_size :: "'g \<Rightarrow> nat"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (modulo b c)) c = modulo (plus a b) c"
simps_1: fixes n :: "nat" shows "[] \<down> n = []"
sync_commute: fixes P :: "'b process"   and A :: "'b set"   and Q :: "'b process" shows "(P \<lbrakk>A\<rbrakk> Q) = (Q \<lbrakk>A\<rbrakk> P)"
distinct_13: fixes x41 :: "'a formula"   and x42 :: "'a formula" shows "\<bottom> \<noteq> x41 \<^bold>\<and> x42"
RS3subset: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "set (rm_MT_rules Cp p) \<subseteq> set p"
Cond: fixes G :: "prog"   and s0 :: "st"   and e0 :: "expr"   and n :: "nat"   and b :: "val"   and s1 :: "state"   and e1 :: "expr"   and e2 :: "expr"   and v :: "val"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<midarrow>n\<rightarrow> s2"
rel_interior_affine: fixes S :: "'b set" assumes "affine S" shows "rel_interior S = S"
FV_Fvar: fixes x :: "char list" shows "FV (Fvar x) = {x}"
alphaInp_def: shows "alphaInp \<equiv> {(qinp, qinp'). sameDom qinp qinp' \<and> liftAll2 (#=) qinp qinp'}"
real_of_nat_div2: fixes n :: "nat"   and x :: "nat" shows "0 \<le> real n / real x - real (n div x)"
Sep_def: fixes A :: "ZF"   and p :: "ZF \<Rightarrow> bool" shows "Sep A p \<equiv> if \<forall>x. Elem x A \<longrightarrow> \<not> p x then Empty else let z = SOME x. Elem x A \<and> p x in Let (\<lambda>x. if p x then x else z) (Repl A)"
assms_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x + y ; z \<le> z" shows "rtc y ; x \<le> z"
VA': shows "Vagree \<nu> \<omega> (\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF \<phi>)))"
evex: shows "\<forall>\<^sub>F x in at x0 within S. h \<in> existence_ivl0 x"
fps_integral_nth_0_Suc_1: fixes a :: "'a fps"   and a0 :: "'a" shows "fps_nth (fps_integral a a0) 0 = a0"
IH: assumes "i < length xs" shows "Abstract_Linear_Poly.coeff (list_to_lpoly xs) i = xs ! i"
wens_single_eq_Union: fixes act :: "('b \<times> 'b) set"   and B :: "'b set" shows "wens_single act B = \<Union> (range (wens_single_finite act B))"
Fix_map_dual: shows "Fix \<circ> \<partial>\<^sub>F = (`) \<partial> \<circ> Fix"
Abs_formula_inverse: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "Rep_formula (Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
rel_G_conversep: fixes L1 :: "'u \<Rightarrow> 'n \<Rightarrow> bool"   and L2 :: "'v \<Rightarrow> 'o \<Rightarrow> bool"   and Co1 :: "'w \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'x \<Rightarrow> 'q \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'r \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 's \<Rightarrow> bool" shows "rel_G L1\<inverse>\<inverse> L2\<inverse>\<inverse> Co1\<inverse>\<inverse> Co2\<inverse>\<inverse> Contra1\<inverse>\<inverse> Contra2\<inverse>\<inverse> = (rel_G L1 L2 Co1 Co2 Contra1 Contra2)\<inverse>\<inverse>"
divide_le_cancel: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a c) (divide b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
from_list_correct: fixes xs :: "('fun \<times> ('b, 'c, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef) list"   and k :: "'fun" shows "map_of xs k = Fubx_get (Subx.Fenv.from_list xs) k"
dist_suc: fixes v :: "nat"   and d :: "nat"   and v' :: "nat"   and v'' :: "nat" assumes "dist v d v'"   and "(v', v'') \<in> E" shows "dist v (Suc d) v''"
N: fixes n :: "nat"   and s :: "loc \<Rightarrow> nat" shows "(N n, s) -|-> (n, s)"
init_sim: fixes a0 :: "'a" assumes "a0 \<in> a.V0" shows "\<exists>b0. b0 \<in> b.V0 \<and> (a0, b0) \<in> R"
gState_progress_rel_trans: fixes x :: "'b gState_scheme"   and y :: "'b gState_scheme"   and prog :: "program"   and z :: "'b gState_scheme" assumes "(x, y) \<in> gState_progress_rel prog"   and "(y, z) \<in> gState_progress_rel prog" shows "(x, z) \<in> gState_progress_rel prog"
iff: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (Cooper.iff p q) = Ifm bbs bs (Iff p q)"
simps: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "Bochner_Integration.simple_bochner_integrable M f = (simple_function M f \<and> emeasure M {y \<in> space M. f y \<noteq> (0::'d)} \<noteq> \<infinity>)"
simps_2: fixes s :: "'a list"   and q :: "'a"   and qs :: "'a list"   and a :: "nat \<times> nat list"   and as :: "(nat \<times> nat list) list" shows "Partial_Cost_Model.steps s (q # qs) (a # as) = Partial_Cost_Model.steps (step s q a) qs as"
intros: fixes u :: "'a"   and r :: "'a list"   and ce :: "'a list multiset"   and p :: "'b"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and u' :: "'a"   and w :: "('b, 'c) label list"   and v :: "'a"   and ce' :: "'a list multiset" assumes "((u # r, ce), LCall p, entry fg p # u' # r, ce) \<in> trss fg"   and "(([entry fg p], ce), w, [v], ce') \<in> trcl (trss fg)" shows "((u # r, ce), LCall p # w, v # u' # r, ce') \<in> ntrs fg"
span_gso: shows "local.span (gso ` {0..<length fs}) = local.span (set fs)"
eq_shift: fixes w :: "'a stream"   and u :: "'a list"   and v :: "'a stream" shows "(w = u @- v) = (stake (length u) w = u \<and> sdrop (length u) w = v)"
vs: fixes K :: "('e, 'f) ring_scheme"   and W :: "'g set"   and V :: "('e, 'g, 'h) module_scheme" assumes "subspace K W V" shows "vectorspace K V"
Bag_def: fixes xs :: "('a, nat) alist" shows "Bag xs = Abs_multiset (count_of (alist.impl_of xs))"
lappend_inf: fixes xs :: "'f llist"   and ys :: "'f llist" assumes "\<not> lfinite xs" shows "lappend xs ys = xs"
at_ide_is_diagram: fixes a :: "'a" assumes "A.ide a" shows "diagram (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at a D)"
induct: fixes P :: "stopType \<Rightarrow> bool"   and stopType :: "stopType" assumes "P stop_vc" shows "P stopType"
eq_TT_dest: fixes x :: "'a"   and y :: "'a" assumes "eq\<cdot>x\<cdot>y = TT" shows "x = y"
Pow_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_fun (rel_set A) (rel_set (rel_set A)) Pow Pow"
Vr_0_f_0: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<zero>\<^bsub>Vr K v\<^esub> = \<zero>"
Ball_code_1: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "Ball (Set_Monad xs) P = list_all P xs"
nneg: fixes x :: "'a" assumes "x \<in> A" shows "0 \<le> f x"
PSuf_Union: fixes f :: "'b list \<Rightarrow> 'a set"   and s :: "'b list"   and c :: "'b" shows "\<Union> (f ` (PSuf s @@ {[c]})) = (\<Union>v\<in>PSuf s. f (v @ [c]))"
assms: fixes Q :: "pi" shows "(Q, Q \<parallel> \<zero>) \<in> Rel"
reduction_3: assumes "4 / 3 < \<alpha>" shows "reduction < 1"
injects_2: fixes discr :: "bool discr"   and discr' :: "bool discr" shows "(CB\<cdot>discr = CB\<cdot>discr') = (discr = discr')"
delete_update_1: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "delete k (update k v m) = delete k m"
is_UP_monom_def: shows "is_UP_monom = (\<lambda>f. f \<in> carrier (UP Zp) \<and> f = monom (UP Zp) (f (deg Zp f)) (deg Zp f))"
singular_simplex_def: fixes p :: "nat"   and X :: "'b topology"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" shows "singular_simplex p X f \<equiv> continuous_map (subtopology (powertop_real UNIV) (standard_simplex p)) X f \<and> f \<in> extensional (standard_simplex p)"
f_minus_ctrm: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "zcf (f \<ominus>\<^bsub>P\<^esub> monom P (f 0) 0) = \<zero>"
simps: fixes x :: "complex"   and y :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" shows "(x, y) *\<^sub>v\<^sub>m (a, b, c, d) = (x * a + y * c, x * b + y * d)"
assms_1: shows "fst (root (red prb)) = init (black prb)"
sorted_list_of_set_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
CDPlayer_CTRL_def: shows "CDPlayer_CTRL = Abs_seqauto (CDPlayer_CTRL_States, CDPlayer_CTRL_Init, CDPlayer_CTRL_Labels, CDPlayer_CTRL_Delta)"
periodic_fun_simple_axioms: shows "periodic_fun_simple sin ((2::'b) * of_real pi)"
conj_nb0: fixes p :: "fm"   and q :: "fm" assumes "bound0 p"   and "bound0 q" shows "bound0 (Parametric_Ferrante_Rackoff.conj p q)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_after_clear (as, lm) (s, l, r) ires = (\<exists>rn m lm1 lm2 r'. lm = lm1 @ m # lm2 \<and> (if lm1 = [] then l = Oc \<up> Suc m @ Bk # Bk # ires else l = Oc \<up> Suc m @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> r = Bk # r' \<and> Oc # r' = <lm2> @ Bk \<up> rn)"
simps_2: fixes m :: "nat"   and p :: "('a, 'b) form" shows "put_unis (Suc m) p = Forall (put_unis m p)"
ABS_in: fixes a :: "'a"   and x :: "'b" shows "[a].x \<in> ABS_set"
subint1_induct: fixes G :: "prog"   and P :: "qtname \<Rightarrow> bool"   and a :: "qtname" assumes "ws_prog G"   and "\<And>x. \<forall>y. (x, y) \<in> subint1 G \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
function_ring_car_closed: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F" shows "f a \<in> carrier R"
assms_2: fixes e :: "'b" assumes "e \<in> arcs G" shows "0 \<le> c e"
simps_1: fixes s :: "name"   and env :: "(name, nterm) fmap" shows "subst (Nvar s) env = (case fmlookup env s of None \<Rightarrow> Nvar s | Some t \<Rightarrow> t)"
size_cnj: fixes A :: "fmla"   and B :: "fmla" shows "size (A AND B) = size A + size B + 4"
additionTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> nat" assumes "\<forall>f h. f \<in> {j. j \<le> Suc n} \<rightarrow> carrier A \<and> h \<in> {j. j \<le> Suc n} \<rightarrow> {j. j \<le> Suc n} \<and> inj_on h {j. j \<le> Suc n} \<longrightarrow> \<Sigma>\<^sub>e A cmp f h Suc n = \<Sigma>\<^sub>e A f Suc n"   and "f \<in> {j. j \<le> Suc (Suc n)} \<rightarrow> carrier A"   and "h \<in> {j. j \<le> Suc (Suc n)} \<rightarrow> {j. j \<le> Suc (Suc n)}"   and "inj_on h {j. j \<le> Suc (Suc n)}"   and "h (Suc (Suc n)) = Suc (Suc n)" shows "\<Sigma>\<^sub>e A cmp f h Suc (Suc n) = \<Sigma>\<^sub>e A f Suc (Suc n)"
remove: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "x \<in> A" shows "prod g A = g x * prod g (A - {x})"
pafj: fixes k :: "nat" assumes "k < n" shows "positive (adjoint (f k) * f k)"
ord_\<Gamma>_saturated_upto_complete: fixes Sts :: "(('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat) llist"   and N :: "'a literal multiset set" assumes "full_chain (\<leadsto>\<^sub>w) Sts"   and "wrp.P_of_wstate (lhd Sts) = {}"   and "wrp.Q_of_wstate (lhd Sts) = {}"   and "redundancy_criterion.saturated_upto (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts)) Rf (standard_redundancy_criterion.Ri (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts))) N" shows "(\<not> satisfiable N) = ({#} \<in> N)"
Red_I_def: fixes N :: "'f set" shows "Red_I N = (\<Inter>q\<in>Q. Red_I_q q N)"
iter_nth: fixes w :: "'b list"   and n :: "nat" assumes "0 < length w" shows "w\<^sup>\<omega> n = w ! (n mod length w)"
Inf_eq_bot_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Inf A = bot) = (\<forall>x. less bot x \<longrightarrow> (\<exists>i\<in>A. less i x))"
right_unique_rel_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "right_unique A"   and "right_unique B" shows "right_unique (rel_gpv A B)"
alpha_imp_asTerm_equal: fixes qX :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and qY :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "qGood qX"   and "qX #= qY" shows "asTerm qX = asTerm qY"
path_cons_conv: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and u' :: "'b"   and l :: "'b list"   and w :: "'b" shows "path E u (u' # l) w = (u' = u \<and> (\<exists>v. (u, v) \<in> E \<and> path E v l w))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>B) F' (FG.GF.map \<circ> F') (FG.\<eta> \<circ> F')"
tranclp_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "r\<^sup>+\<^sup>+ x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
line_integral_on_reverse_path_1: fixes \<gamma> :: "real \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a"   and basis :: "'a set" assumes "valid_path \<gamma>"   and "line_integral_exists F basis \<gamma>" shows "line_integral F basis \<gamma> = - line_integral F basis (reversepath \<gamma>)"
p: shows "v \<in> carrier_vec n \<and> fs ! i - v \<in> local.span (local.span (gso ` {0..<i})) \<and> (\<forall>x. x \<in> local.span (gso ` {0..<i}) \<longrightarrow> v \<bullet> x = (0::'a))"
cases_scheme: fixes r :: "('e, 'f) lasso_scheme" obtains lasso_reach :: "'e list" and lasso_va :: "'e" and lasso_cysfx :: "'e list" and more :: "'f" where "r = \<lparr>lasso_reach = lasso_reach, lasso_va = lasso_va, lasso_cysfx = lasso_cysfx, \<dots> = more\<rparr>"
sin_minus: fixes x :: "'a" shows "sin (- x) = - sin x"
sup_mono: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a \<le> c"   and "b \<le> d" shows "a + b \<le> c + d"
inv_into_f_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'b" assumes "inj_on f A"   and "x \<in> A"   and "f x = y" shows "inv_into A f y = x"
SkipSem: shows " \<Turnstile> Skip : Sec (\<lambda>(s, t, \<beta>). s \<equiv>\<^sub>\<beta> t) "
a': shows "emb a A \<in> A \<union> B \<rightarrow> carrier K"
leg1_is_map: shows "is_left_adjoint r\<^sub>1"
update_def: fixes f :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'k \<Rightarrow> 'a option"   and m :: "'m" shows "update f m = get (\<lambda>s. put (f s) m)"
diamond_binder: fixes L1 :: "char list set"   and t :: "sterm"   and ta :: "sterm"   and L2 :: "char list set"   and tb :: "sterm" assumes "finite L1"   and "\<forall>s p. s \<notin> L1 \<and> p \<notin> L1 \<and> s \<noteq> p \<longrightarrow> (\<exists>t'. (t\<^bsup>[Fvar s,Fvar p]\<^esup> => t' \<and> (\<forall>z. t\<^bsup>[Fvar s,Fvar p]\<^esup> => z \<longrightarrow> (\<exists>u. t' => u \<and> z => u))) \<and> ta = \<sigma>[s,p] t')"   and "finite L2"   and "\<forall>s p. s \<notin> L2 \<and> p \<notin> L2 \<and> s \<noteq> p \<longrightarrow> (\<exists>t'. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t' \<and> tb = \<sigma>[s,p] t')" shows "\<exists>L'. finite L' \<and> (\<exists>t''. (\<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>u. ta\<^bsup>[Fvar s,Fvar p]\<^esup> => u \<and> t'' = \<sigma>[s,p] u)) \<and> (\<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>u. tb\<^bsup>[Fvar s,Fvar p]\<^esup> => u \<and> t'' = \<sigma>[s,p] u)))"
ts_okD: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and ts :: "'f \<Rightarrow> ('g \<times> 'i \<Rightarrow>f nat) option"   and m :: "'h"   and t :: "'f"   and x :: "'g"   and ln :: "'i \<Rightarrow>f nat" assumes "ts_ok P ts m"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "P t x m"
continuous_on_norm: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. norm (f x))"
check_poly_weak_anti_mono_def: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" shows "check_poly_weak_anti_mono p v \<equiv> list_all (\<lambda>(m, c). c \<le> (0::'d) \<or> v \<notin> monom_vars m) p"
symD: fixes r :: "('a \<times> 'a) set"   and b :: "'a"   and a :: "'a" assumes "sym r"   and "(b, a) \<in> r" shows "(a, b) \<in> r"
weak_lens_def: fixes x :: "'b \<Longrightarrow> 'c" shows "weak_lens x \<equiv> \<forall>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
Span_contains_Span_take: fixes ms :: "'v list"   and k :: "nat" assumes "set ms \<subseteq> V" shows "RSpan (take k ms) \<subseteq> RSpan ms"
antisym: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x"   and "x \<le> y" shows "x = y"
abs_eq: fixes xa :: "'b"   and x :: "'b poly" shows "pCons xa x = Abs_poly (case_nat xa (coeff x))"
WN_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "WN_on r A = (\<forall>a\<in>A. \<exists>b. (a, b) \<in> r\<^sup>!)"
strict_implies_not_eq: fixes a :: "'a list"   and b :: "'a list" assumes "strict_prefix a b" shows "a \<noteq> b"