hcju/isabelleps huggingface dataset:
comp_subst_assoc: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and \<sigma> :: "'c"   and \<tau> :: "'c"   and \<gamma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "comp_subst \<sigma> (comp_subst \<tau> \<gamma>) = comp_subst (comp_subst \<sigma> \<tau>) \<gamma>"
content_dvd: fixes p :: "'a poly" shows "[:content p:] dvd p"
T_add: fixes m1 :: "'c"   and m2 :: "'c" assumes "m1 \<in> carrier V"   and "m2 \<in> carrier V" shows "T (m1 \<oplus>\<^bsub>V\<^esub> m2) = T m1 \<oplus>\<^bsub>W\<^esub> T m2"
simps_2: fixes D :: "ast_domain"   and ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option" shows "ast_domain.wf_fmla D ty_ent \<bottom> = True"
hyps_1: shows "While_CFG.path prog n'' as (_ l _)"
upf_key_gen_lossless: shows "lossless_spmf upf_key_gen"
locally_path_connected_sphere: fixes a :: "'a"   and r :: "real" shows "locally path_connected (sphere a r)"
elims: fixes x :: "'b nres"   and xa :: "'b nres"   and y :: "'b nres" assumes "sup x xa = y" obtains uu_ :: "'b nres" where "x = uu_" and "xa = FAILi" and "y = FAILi"   | v :: "'b set" where "x = FAILi" and "xa = RES v" and "y = FAILi"   | a :: "'b set" and b :: "'b set" where "x = RES a" and "xa = RES b" and "y = RES (a \<union> b)"
inject: fixes x21 :: "'g"   and x22 :: "'g list"   and y21 :: "'g"   and y22 :: "'g list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
abs_eq: fixes x :: "'a \<times> 'a" assumes "fractrel x x" shows "inverse (abs_fract x) = abs_fract (if fst x = (0::'a) then (0::'a, 1::'a) else (snd x, fst x))"
nn_integral_0_1_ln_times_power: fixes n :: "nat" shows "\<integral>\<^sup>+x\<in>{0<..<1}. ennreal (- ln x * x ^ n)\<partial>lborel = ennreal (1 / (real n + 1)\<^sup>2)"
hom_between_subgroups: fixes A :: "'a set"   and B :: "'c set" assumes "h ` A \<subseteq> B" shows "h \<in> hom (subgroup_generated G A) (subgroup_generated H B)"
cs_done: shows "snd (cs (S t (length t)) cid) = Done"
assms_2: shows "q \<notin> \<int> \<or> (\<forall>x\<in>A. f x \<noteq> 1)"
axioms_1: fixes x :: "'d \<Longrightarrow> 'e" assumes "pbij_lens x" shows "weak_lens x"
distinct_succ_tran: fixes G :: "'b list_graph"   and v :: "'b" assumes "wf_list_graph G" shows "distinct (FiniteListGraph.succ_tran G v)"
deriv_in_subterms_3: fixes p1 :: "('e, 'f, 'g, 'h) seqp"   and p2 :: "('e, 'f, 'g, 'h) seqp"   and p :: "('e, 'f, 'g, 'h) seqp" assumes "p1 \<oplus> p2 \<in> subterms p" shows "p1 \<in> subterms p"
Smith_zero_imp_zero: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat m n"   and "Smith_normal_form_mat A"   and "A $$ (i, i) = (0::'b)"   and "j < min m n"   and "i \<le> j" shows "A $$ (j, j) = (0::'b)"
F_n_integral_prob_comp: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "f \<in> borel_measurable (F n)" shows "has_bochner_integral M f (\<Sum>w\<in>range (pseudo_proj_True n). prod (prob_component p w) {0..<n} * f w)"
ring_simprules_21: shows "\<ominus> \<zero> = \<zero>"
size_image_mset: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" shows "size (image_mset f M) = size M"
below_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
sorted_le_last: fixes xs :: "'b list"   and x :: "'b" assumes "sorted xs"   and "x \<in> list.set xs" shows "x \<le> last xs"
right_totalE: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c" assumes "right_total A" obtains x :: "'b" where "A x y"
map_poly_mult_1: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "map_mat (\<lambda>a. [:a:]) (A * B) = map_mat (\<lambda>a. [:a:]) A * map_mat (\<lambda>a. [:a:]) B"
res_function_def: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.res_function times plus minus f = (\<forall>c. (\<forall>a b. ring_1.cong times minus a b c = (f c a = f c b)) \<and> pairwise (\<lambda>a b. \<not> ring_1.cong times minus a b c) (range (f c)) \<and> (\<forall>a. \<exists>k. f c a = plus a (times k c)))"
cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf" assumes "rel_pmf R p q" obtains pq :: "('c \<times> 'd) pmf" where "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" and "map_pmf fst pq = p" and "map_pmf snd pq = q"
lookup_env_restr: fixes x :: "'d"   and S :: "'d set"   and m :: "'d \<Rightarrow> 'e" assumes "x \<in> S" shows "(m f|` S) x = m x"
Typed_in_\<R>: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"   and "tyenv_wellformed mds \<Gamma> \<S> P"   and "\<forall>x. type_max (to_total \<Gamma> x) mem\<^sub>1 = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"   and "pred P mem\<^sub>1"   and "pred P mem\<^sub>2"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c, mds, mem\<^sub>2\<rangle>"
finite_Collect_disjI: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "finite {x. P x \<or> Q x} = (finite {x. P x} \<and> finite {x. Q x})"
permute_set_eq_vimage: fixes p :: "perm"   and X :: "'a set" shows "p \<bullet> X = permute (- p) -` X"
add_left: fixes a :: "'a"   and a' :: "'a"   and b :: "'d" shows "(a + a') ** g b = a ** g b + a' ** g b"
leI: fixes y :: "'b"   and x :: "'b" assumes "\<not> y < x" shows "x \<le> y"
meridian_def: shows "meridian \<equiv> proj2_line_abs (vector [1, 0, 0])"
add_zmset_commute: fixes x :: "'b"   and y :: "'b"   and M :: "'b zmultiset" shows "add_zmset x (add_zmset y M) = add_zmset y (add_zmset x M)"
one_stalk_def: fixes V :: "'a set" shows "stx.one_stalk V \<equiv> stx.class_of V \<one>\<^bsub>V\<^esub>"
Expr_lemma3: fixes s :: "Var \<Rightarrow> Val"   and h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and e :: "Expr"   and X :: "Var set"   and l :: "Location"   and C :: "Class"   and Flds :: "(Field \<times> Val) list"   and \<beta> :: "(Location \<times> Location) set"   and t :: "Var \<Rightarrow> Val"   and k :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and f :: "Field" assumes "noLowDPs (s, h)"   and "EVars e \<subseteq> X"   and "Expr_noLoc e"   and "lookup h l = Some (C, Flds)"   and "\<And>x. x \<in> X \<Longrightarrow> CONTEXT x = low"   and "s  \<approx>\<^sub>\<beta> t"   and "twiddleHeap \<beta> h k" shows "noLowDPs (s, (l, C, (f, evalE e s) # Flds) # h)"
card_of_empty: fixes A :: "'b set" shows "|{}| \<le>o |A|"
not_Null_eq: fixes x :: "'b ref" shows "(x \<noteq> Null) = (\<exists>y. x = Ref y)"
scp_slice: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "(\<pi>, i) \<in> scp" shows "\<pi> i \<in> slice"
HoVH_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "H.HoVH \<mu> \<equiv> if H.VVV.arr \<mu> then \<a> (fst \<mu>) (fst (snd \<mu>)) (snd (snd \<mu>)) else V.null"
xyz: shows "\<forall>w. P w ((x \<otimes> y) \<otimes> z) = (P w (x \<otimes> y) \<and> P w z)"
LeftDerives1_unique: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list"   and j :: "nat"   and s :: "symbol \<times> symbol list" assumes "LeftDerives1 a i r b"   and "LeftDerives1 a j s b" shows "i = j \<and> r = s"
nz: shows "of_nat n + complex_of_real a \<noteq> 0"
RSpanD_lincomb: fixes ms :: "'m list" shows "RSpan ms = {rs \<bullet>\<cdot> ms |rs. set rs \<subseteq> R \<and> length rs = length ms}"
suminf_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'd" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'d)" shows "suminf f = sum f N"
has_derivative_componentwise_within: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and a :: "'a"   and S :: "'a set" shows "(f has_derivative f') (at a within S) = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. f' x \<bullet> i)) (at a within S))"
set: fixes x1 :: "'a"   and x2 :: "'a stream" shows "i.R (x1 ## x2) = insert x1 (i.R x2)"
distinct_11: fixes x11 :: "nat set"   and x12 :: "'d" shows "x11:*cast(x12) \<noteq> \<tau>\<^sub>n"
nat_filtration_info: fixes f :: "bool stream \<Rightarrow> 'c"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "f \<circ> pseudo_proj_True n = f"
pw_nres_rel_iff: fixes a :: "'e nres"   and b :: "'f nres"   and A :: "('e \<times> 'f) set" shows "((a, b) \<in> \<langle>A\<rangle>nres_rel) = (nofail (\<Down> A b) \<longrightarrow> nofail a \<and> (\<forall>x. inres a x \<longrightarrow> inres (\<Down> A b) x))"
tranclp_induct: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r a y \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r\<^sup>+\<^sup>+ a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
step_extend: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "\<Gamma>\<turnstile> \<langle>c',s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
simps_2: fixes f :: "'d"   and ss :: "('d, 'f) Term.term list"   and \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "Fun f ss \<cdot> \<sigma> = Fun f (map (\<lambda>t. t \<cdot> \<sigma>) ss)"
outU: shows "\<forall>j. j \<notin> net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2) \<longrightarrow> U (\<sigma> j) (\<sigma>' j)"
valid_slide_append: fixes w :: "nat \<times> nat"   and as :: "'b list"   and t :: "'b tree"   and as' :: "'b list" assumes "0 < fst w"   and "fst w \<le> snd w"   and "valid as t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "snd w \<le> length as + length as'" shows "valid (as @ as') (slide (as @ as') t w) \<and> l (slide (as @ as') t w) = fst w \<and> r (slide (as @ as') t w) = snd w"
expect_prob_comp: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "f \<in> borel_measurable (F n)" shows "expectation f = (\<Sum>w\<in>range (pseudo_proj_True n). prod (prob_component p w) {0..<n} * f w)"
a: shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
sds_S_wf: shows "sds S \<in> lowered.lotteries"
SOC_iff_strong_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(((\<forall>S S'. S \<longmapsto>Source S' \<longrightarrow> (\<exists>T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)) \<and> (\<forall>S T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<longrightarrow> (\<exists>S'. S \<longmapsto>Source S' \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*))) \<and> strong_reduction_simulation (TRel\<^sup>+) Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel\<^sup>+ \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel\<^sup>+))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*) \<and> strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel)))"
strict_sorted_sorted_wrt: shows "signed.strict_sorted = sorted_wrt (<s)"
cINF_greatest: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and m :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq m (f x)" shows "less_eq m (Inf (f ` A))"
components_eq_empty: fixes s :: "'a set" shows "(components s = {}) = (s = {})"
ramanujan_sum_conv_gauss_sum: fixes k :: "nat" assumes "\<chi> = principal_dchar n" shows "ramanujan_sum n k = gauss_sum k"
zip_Cons: fixes v :: "'b list"   and w :: "'c list"   and a :: "'b"   and b :: "'c" assumes "length v = length w" shows "zip (a # v) (b # w) = (a, b) # zip v w"
notSpecKeysSecretsExpr_L2: fixes P :: "specID"   and a :: "Expression"   and l :: "Expression list" assumes "notSpecKeysSecretsExpr P (a # l)" shows "notSpecKeysSecretsExpr P l"
jumpF_polyL_coprime: fixes p :: "real poly"   and q :: "real poly"   and x :: "real" assumes "coprime p q" shows "jumpF_polyL q p x = (if p \<noteq> 0 \<and> q \<noteq> 0 \<and> poly p x = 0 then if even (order x p) = (sign_r_pos p x = (0 < poly q x)) then 1 / 2 else - 1 / 2 else 0)"
int_simps_4: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> \<not> \<not> P = P"
isGuard_simps_2: fixes x :: "Preliminaries.var"   and p :: "'psym"   and Tl :: "'fsym trm list" shows "isGuard x (Neg (Pr p Tl)) = (x \<in> \<Union> (set (map nv2T Tl)) \<and> polC p = Fext)"
simps_5: fixes var :: "nat" shows "freeIn var TrueF = True"
nth_append: fixes xs :: "'c list"   and ys :: "'c list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))"
invar_abs': fixes s :: "'b" assumes "s \<in> wa_invar WB" shows "\<alpha>2 s \<in> wa_invar WA"
distinct_25: fixes term1 :: "'d"   and term2 :: "'d"   and psi :: "('d, 'e, 'f) psi"   and psi' :: "('d, 'e, 'f) psi" shows "term1\<langle>term2\<rangle>.psi \<noteq> !psi'"
prv_fls: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c" assumes "Deduct_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv" shows "prv (imp fls \<phi>)"
map_def: fixes f :: "'c" shows "local.map f = (if C.arr f then J_C.MkArr (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>) (C.dom f)) (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>) (C.cod f)) (constant_transformation.map (\<cdot>\<^sub>J) (\<cdot>) f) else J_C.null)"
add_leD1: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m + k \<le> n" shows "m \<le> n"
split_ids: fixes xs :: "('c \<times> ('c, 'b) operation) event list"   and ys :: "('c \<times> ('c, 'b) operation) event list"   and e :: "'b" shows "added_ids (xs @ ys) e = added_ids xs e @ added_ids ys e"
fresh_list_cons: fixes a :: "'v"   and x :: "'w"   and xs :: "'w list" shows "a \<sharp> (x # xs) = (a \<sharp> x \<and> a \<sharp> xs)"
intros: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "d_OUT f x = 0" shows "x \<in> MFMC_Network.SINK f"
elims: fixes x :: "nat"   and y :: "'c bro set" assumes "B x = y" obtains "x = 0" and "y = {N0}"   | h :: "nat" where "x = Suc h" and "y = {N2 t1 a t2 |t1 a t2. t1 \<in> T h \<and> t2 \<in> B h \<or> t1 \<in> B h \<and> t2 \<in> T h}"
fv_disj_Fun_subterm_param_cases: fixes t :: "('b, 'a) Term.term"   and X :: "'a set"   and f :: "'b"   and T :: "('b, 'a) Term.term list" assumes "fv t \<inter> X = {}"   and "Fun f T \<sqsubseteq> t" shows "T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` X)"
lim_def: fixes X :: "nat \<Rightarrow> 'b" shows "lim X = (THE L. X \<longlonglongrightarrow> L)"
runStateT_def: shows "runStateT \<equiv> stateT_rep"
assms: fixes x :: "'a" shows "f x = term_cases if_const if_free if_app otherwise x"
freq_swapSyms: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a" assumes "consistent t"   and "a \<in> alphabet t"   and "b \<in> alphabet t" shows "freq (swapSyms t a b) = freq t"
simps_2: fixes t :: "'c" shows "dbm_entry_bound (Lt t) = t"
ap_specI: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "\<And>gs bs ps hs data. set (ap gs bs ps hs data) \<subseteq> set ps \<union> set hs \<times> (set gs \<union> set bs \<union> set hs)"   and "\<And>gs bs ps hs data B d m h g. \<lbrakk>dickson_grading d; set gs \<union> set bs \<union> set hs \<subseteq> B; fst ` B \<subseteq> dgrad_p_set d m; h \<in> set hs; g \<in> set gs \<union> set bs \<union> set hs; set ps \<subseteq> set bs \<times> (set gs \<union> set bs); unique_idx (gs @ bs @ hs) data; is_Groebner_basis (fst ` set gs); h \<noteq> g; fst h \<noteq> 0; fst g \<noteq> 0; \<And>a b. \<lbrakk>(a, b) \<in>\<^sub>p set (ap gs bs ps hs data); fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b); \<And>a b. \<lbrakk>a \<in> set gs \<union> set bs; b \<in> set gs \<union> set bs; fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)"   and "\<And>gs bs ps hs data B d m h g. \<lbrakk>dickson_grading d; set gs \<union> set bs \<union> set hs \<subseteq> B; fst ` B \<subseteq> dgrad_p_set d m; set ps \<subseteq> set bs \<times> (set gs \<union> set bs); (set gs \<union> set bs) \<inter> set hs = {}; unique_idx (gs @ bs @ hs) data; is_Groebner_basis (fst ` set gs); h \<noteq> g; fst h \<noteq> 0; fst g \<noteq> 0; (h, g) \<in> set ps -\<^sub>p set (ap gs bs ps hs data); \<And>a b. \<lbrakk>(a, b) \<in>\<^sub>p set (ap gs bs ps hs data); (a, b) \<in>\<^sub>p set hs \<times> (set gs \<union> set bs \<union> set hs); fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)" shows "ap_spec ap"
Lim_transform_bound: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and net :: "'b filter" assumes "\<forall>\<^sub>F n in net. norm (f n) \<le> norm (g n)"   and "(g \<longlongrightarrow> (0::'d)) net" shows "(f \<longlongrightarrow> (0::'c)) net"
dghm_def: shows "\<HH> = [\<HH>\<lparr>ObjMap\<rparr>, \<HH>\<lparr>ArrMap\<rparr>, \<HH>\<lparr>HomDom\<rparr>, \<HH>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
comm_monoid_add_ow_axioms: shows "comm_monoid_add_ow S (\<oplus>\<^sub>o\<^sub>w) z"
dma_\<C>: fixes mem\<^sub>1 :: "'Var\<^sub>A \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "\<forall>x\<in>\<C>\<^sub>A. mem\<^sub>1 x = mem\<^sub>2 x" shows "dma\<^sub>A mem\<^sub>1 = dma\<^sub>A mem\<^sub>2"
simps_37: shows "subcomp sS12 = {sA81, sA91}"
preserves_lunit_2: fixes f :: "'c" assumes "C.ide f" shows "F \<l>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "y \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "z \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> y \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> z = x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> (y \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> z)"
pivot_positions_gen_def: fixes zer :: "'a"   and A :: "'a mat" shows "pivot_positions_gen zer A \<equiv> pivot_positions_main_gen zer A (dim_row A) (dim_col A) 0 0"
app: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T1 :: "Syntax.type"   and T2 :: "Syntax.type"   and s :: "trm" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T1 \<rightarrow> T2"   and "\<Gamma> , \<Delta> \<turnstile>\<^sub>T s : T1" shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t \<degree> s : T2"
cindex_pathE_part_circlepath: fixes z :: "complex"   and z0 :: "complex"   and r :: "real"   and st :: "real"   and tt :: "real" assumes "cmod (z - z0) \<noteq> r"   and "0 < r"   and "0 \<le> st"   and "st < tt"   and "tt \<le> 2 * pi" shows "cindex_pathE (part_circlepath z r st tt) z0 = (if \<bar>Re z - Re z0\<bar> < r then let \<theta> = arccos ((Re z0 - Re z) / r); \<beta> = 2 * pi - \<theta> in jumpF_pathstart (part_circlepath z r st tt) z0 + (if st < \<theta> \<and> \<theta> < tt then if Im z0 < r * sin \<theta> + Im z then - 1 else 1 else 0) + (if st < \<beta> \<and> \<beta> < tt then if Im z0 < r * sin \<beta> + Im z then 1 else - 1 else 0) - jumpF_pathfinish (part_circlepath z r st tt) z0 else if \<bar>Re z - Re z0\<bar> = r then jumpF_pathstart (part_circlepath z r st tt) z0 - jumpF_pathfinish (part_circlepath z r st tt) z0 else 0)"
nth_append: fixes xs :: "'b List.list"   and ys :: "'b List.list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < List.length xs then xs ! n else ys ! (n - List.length xs))"
pp_of_term_of_pair: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and t :: "'d"   and k :: "'e" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.pp_of_term pair_of_term (term_of_pair (t, k)) = t"
semigroup_axioms: shows "Type_Semigroups.semigroup (\<oplus>\<^sub>a)"
uniformity_trans: fixes E :: "'a \<times> 'a \<Rightarrow> bool" assumes "eventually E uniformity" shows "\<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))"
CJ_12p: fixes A :: "c \<Rightarrow> w \<Rightarrow> bool" shows "\<forall>c. (\<^bold>\<box>\<^sub>pA) c \<sqsubseteq> (\<lambda>w. (\<^bold>\<not>\<^bold>O\<^sub>iA) c w \<and> \<not> (ob (pv w) ((\<^bold>\<not>A) c) \<and> \<I>pv w \<sqinter> (\<^bold>\<not>\<^bold>\<not>A) c))"
tensor_lookup: fixes A :: "'b tensor"   and e :: "nat list \<Rightarrow> 'b" assumes "\<And>is. is \<lhd> dims A \<Longrightarrow> lookup A is = e is" shows "tensor_from_lookup (dims A) e = A"