hcju/isabelleps huggingface dataset:
I: shows "\<not> eta_reducible (decr (lev + 1) body)"
list_of_pdevs_all_nonzero: fixes xs :: "'b pdevs" shows "list_all (\<lambda>x. x \<noteq> (0::'b)) (map snd (list_of_pdevs xs))"
rep_eq: fixes x :: "('c, 'b) sq_matrix" shows "to_fun (cofactor x) = (\<lambda>i j. Square_Matrix.det (minor x i j))"
degree_msum_pdevs_ge1: fixes f :: "'b pdevs"   and n :: "nat"   and g :: "'b pdevs" assumes "degree f \<le> n" shows "degree f \<le> degree (msum_pdevs n f g)"
allDefs_def: fixes defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and phis :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option"   and g :: "'d"   and n :: "'e" shows "CFG_SSA_base.allDefs defs phis g n \<equiv> defs g n \<union> CFG_SSA_base.phiDefs phis g n"
assms_2: fixes P :: "ccs"   and Q :: "ccs" assumes "Prop P Q" shows "Prop Q P"
Fb0: fixes i :: "nat" shows "(local.F i = 0 \<longrightarrow> local.F (Suc i) = 0) \<and> local.b i \<noteq> 0 \<and> local.h i \<noteq> 0"
strict_lower_solution: fixes y :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and T :: "real set"   and X :: "real set"   and v :: "real \<Rightarrow> real"   and v' :: "real \<Rightarrow> real"   and t0 :: "real"   and t :: "real" assumes "(y solves_ode f) T X"   and "(v has_vderiv_on v') T"   and "\<And>t. \<lbrakk>t \<in> T; t0 < t\<rbrakk> \<Longrightarrow> v' t < f t (v t)"   and "v t0 \<le> y t0"   and "v t0 = y t0 \<Longrightarrow> v' t0 < f t0 (y t0)"   and "t0 < t"   and "t0 \<in> T"   and "t \<in> T"   and "is_interval T" shows "v t < y t"
CompDefinedI: fixes f :: "'a"   and C :: "('b, 'a, 'c) Category_scheme"   and g :: "'a" assumes "f \<in> mor\<^bsub>C\<^esub>"   and "g \<in> mor\<^bsub>C\<^esub>"   and "cod\<^bsub>C\<^esub> f = dom\<^bsub>C\<^esub> g" shows "f \<approx>>\<^bsub>C\<^esub> g"
a2: shows "f = (\<lambda>a (t, u). ([], a # u))"
assms_2: shows "(x, y) \<in> r\<^sup>*"
boxL: fixes A :: "(C, BD) form" shows "([ \<LM> A  \<RM> \<Rightarrow>* \<Empt>],  \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>) \<in> g3mod2"
a: shows "((x, B) # \<Gamma>) \<turnstile> Ax x a \<turnstile> \<Delta>"
in_sccsI: fixes c :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "induced_subgraph c G"   and "strongly_connected c"   and "\<nexists>c'. induced_subgraph c' G \<and> strongly_connected c' \<and> verts c \<subset> verts c'" shows "c \<in> pre_digraph.sccs G"
tree_map'_strict: fixes f :: "'s \<rightarrow> 'r" shows "tree_map'\<cdot>f\<cdot>\<bottom> = \<bottom>"
Mapping_RBT_def: fixes t :: "('c, 'd) rbt" shows "Mapping_RBT t = Mapping_RBT' (if ord.is_rbt cless t \<or> ID ccompare = None then t else RBT_Impl.fold (ord.rbt_insert cless) t rbt.Empty)"
FDERIV: shows "einterpret xs vs \<in> Csafe"
inverse_closed'': shows "\<forall>x\<in>U. inverse x \<in> U"
dual_top: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual top = bot"
derived_blocks_nempty: fixes bld :: "'a set" assumes "\<And>b. b \<in># remove1_mset bl \<B> \<Longrightarrow> 0 < bl |\<inter>| b"   and "bld \<in># \<B>\<^sup>D" shows "bld \<noteq> {}"
to_nat_def: shows "to_nat = (SOME f. inj f)"
WhileF: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and c :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,s\<^sub>1\<rangle>"
dyn_influence_source_notin_tl_edges: fixes n :: "'node"   and V :: "'var"   and n' :: "'node"   and a :: "'edge"   and as :: "'edge list" assumes "n influences V in n' via a # as" shows "n \<notin> set (sourcenodes as)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.in_hom f a b" shows "S.in_hom (M.\<ll>' f) (S.map a) (M\<^sub>\<omega>.tensor M.unity b)"
rep_eq: fixes x :: "'l word" shows "unsigned x = semiring_1_class.of_nat (nat (take_bit LENGTH('l) (Word.rep x)))"
remap_accs1: fixes \<delta> :: "('a, 'b) ta_rule set"   and n :: "'b tree"   and q :: "'a"   and f :: "'a \<Rightarrow> 'c" assumes "accs \<delta> n q" shows "accs (remap_rule f ` \<delta>) n (f q)"
invertible_inverse_closed: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> G" shows "sub.inverse u \<in> G"
continuity_lem_lt0: fixes c :: "real"   and b :: "real"   and a :: "real"   and r :: "real" assumes "poly [:c, b, a:] r < 0" shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. poly [:c, b, a:] x < 0"
ideD_1: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VxVxV.ide a" shows "VxVxV.arr a"
INum_int_2: shows "INum 0\<^sub>N = (0::'b)"
Cs_def: shows "Cs = partition_ev_blocks (step_2 (step_1 A)) []"
assms_2: shows "point_measurable (fct_gen_subalgebra M N Y) (space N) Y"
E_sup_bot: fixes s :: "'s" shows "E_sup s \<bottom> = 0"
hd_replicate: fixes n :: "nat"   and x :: "'c" assumes "n \<noteq> 0" shows "hd (replicate n x) = x"
nonPrincipalInvertLeft: fixes R1 :: "(sequent list \<times> sequent) set"   and R2 :: "(sequent list \<times> sequent) set"   and R3 :: "(sequent list \<times> sequent) set"   and R :: "(sequent list \<times> sequent) set"   and r :: "sequent list \<times> sequent"   and ps :: "sequent list"   and c :: "sequent"   and n :: "nat"   and F :: "char list"   and x :: "NominalSequents.var"   and A :: "form"   and \<Gamma>' :: "form multiset"   and \<Delta>' :: "form multiset"   and \<Gamma> :: "form multiset"   and \<Delta> :: "form multiset"   and S :: "sequent"   and Ps :: "sequent list"   and n' :: "nat" assumes "R1 \<subseteq> upRules"   and "R2 \<subseteq> nprovRules"   and "R3 \<subseteq> provRules"   and "R = Ax \<union> R1 \<union> R2 \<union> R3"   and "r \<in> R1 \<or> r \<in> R2 \<or> r \<in> R3"   and "r = (ps, c)"   and "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Delta>, m) \<in> derivable  R* \<longrightarrow> (\<forall>r'\<in>R. leftPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> \<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>' \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable  R*)"   and "( \<Gamma> \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Delta>, n) \<in> derivable  R*"   and "\<forall>r'\<in>R. leftPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"   and "\<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>'"   and "\<not> leftPrincipal r (F \<nabla> [x].A)"   and "extendRule S r = (Ps,  \<Gamma> \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Delta>)"   and "n = n' + 1"   and "\<forall>p\<in>set Ps. \<exists>n\<le>n'. (p, n) \<in> derivable  R*"   and "Ps \<noteq> []" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable  R*"
select_convs_2: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getM \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = getM"
eqI_snth: fixes u :: "'c stream"   and v :: "'c stream" assumes "\<And>i. u !! i = v !! i" shows "u = v"
in_generate': fixes x :: "nat list \<times> nat list"   and m :: "nat"   and n :: "nat"   and c :: "nat list"   and b :: "nat list" assumes "x \<in> set (generate' m n c b)" shows "x \<in> set (Algorithm.gen2 m n c b)"
CMU_def: shows "CMU \<equiv> CM \<and> CMU_axioms"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G"   and "monoid_cancel_axioms G" shows "monoid_cancel G"
set_rec: fixes xs :: "'a list" shows "set xs = rec_list {} (\<lambda>x _. insert x) xs"
assms_5: shows "kKS k \<notin> LocalSecrets P"
Pareto_strict_iff: fixes dom :: "'c set"   and carrier :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "preorder_family dom carrier R" shows "x \<prec>[Pareto R] y = ((\<forall>i\<in>dom. R i x y) \<and> (\<exists>i\<in>dom. x \<prec>[R i] y))"
less_linear: fixes x :: "'c word"   and y :: "'c word" shows "x <s y \<or> x = y \<or> y <s x"
invh_baliR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (baliR l a r)"
ch_by_ord_def: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" shows "ch_by_ord f X \<equiv> short_ch X \<or> long_ch_by_ord f X"
inv_chart_restrict_chart: fixes S :: "'c set"   and c :: "('c, 'b) chart" shows "inv_chart (restrict_chart S c) = inv_chart c"
wpt_dual_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "mbt_algebra.wpt one times inf top (dual x) = one"
targets_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "weak_composition V H" shows "weak_composition.targets V H \<mu> \<equiv> {b. weak_composition.weak_unit V H b \<and> H b \<mu> \<noteq> partial_magma.null V}"
fin': fixes t :: "'t" assumes "r1.final_thread s1 t" shows "r2.final_thread (ls2, (ts2', m2), ws2, is2) t"
FAssThrow\<^sub>2\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
Cons_3: shows "non_volatile_owned_or_read_only True \<S> \<O> (x # sb)"
sigma_sets_top: fixes sp :: "'d set"   and A :: "'d set set" shows "sp \<in> sigma_sets sp A"
hb_elim: fixes x :: "'a"   and y :: "'a" assumes "hb x y" obtains i :: "nat" where "Broadcast x \<sqsubset>\<^sup>i__ Broadcast y"   | i :: "nat" where "Deliver x \<sqsubset>\<^sup>i Broadcast y"   | m2 :: "'a" where "hb x m2" and "hb m2 y"
range_disjointed_sets: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M" shows "range (disjointed A) \<subseteq> sets M"
uint16_set_bit_def: fixes x :: "uint16"   and n :: "integer"   and b :: "bool" shows "uint16_set_bit x n b = (if n < 0 \<or> 15 < n then undefined set_bit_class.set_bit x n b else set_bit_class.set_bit x (nat_of_integer n) b)"
simps_1: fixes CatDom :: "('a, 'b, 'c) Category_scheme"   and CatCod :: "('d, 'e, 'f) Category_scheme"   and MapM :: "'b \<Rightarrow> 'e"   and more :: "'g" shows "CatDom \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = CatDom"
Extend_subset: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm" shows "S \<subseteq> Extend S f"
f3_def: fixes uu :: "nat set llist" shows "f3 uu = ()"
pair_resTy_simp: fixes x :: "'a"   and m :: "'b" shows "resTy (x, m) = resTy m"
deriv: fixes y :: "real" assumes "a \<le> y" shows "(G has_real_derivative g y) (at y within {a..})"
of_int_uint: fixes x :: "'a word" shows "word_of_int (uint x) = x"
root_int_ceiling_pos_def: fixes p :: "nat"   and x :: "int" shows "root_int_ceiling_pos p x = (if p = 0 then 0 else case root_int_main p x of (y, b) \<Rightarrow> if b then y else y + 1)"
product_closure: fixes x :: "i"   and y :: "i" assumes "O x y" shows "\<exists>z. \<forall>v. P v z = (P v x \<and> P v y)"
list_emb_Nil2_cases: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" assumes "list_emb P xs []" obtains "xs = []"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and f4 :: "'c"   and acc_modi :: "acc_modi" shows "h (case acc_modi of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = (case acc_modi of Private \<Rightarrow> h f1 | Package \<Rightarrow> h f2 | Protected \<Rightarrow> h f3 | Public \<Rightarrow> h f4)"
natfun_bigo_1E: fixes f :: "nat \<Rightarrow> 'b"   and lb :: "real" assumes "f \<in> O(\<lambda>_. 1::'b)" obtains C :: "real" where "lb \<le> C" and "\<And>n. norm (f n) \<le> C"
rm_vars_ident: fixes t :: "('d, 'c) Term.term"   and vs :: "'c set"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "fv t \<inter> vs = {}" shows "t \<cdot> rm_vars vs \<theta> = t \<cdot> \<theta>"
affine_hull_subset_span: fixes s :: "'b set" shows "affine hull s \<subseteq> span s"
induced_arrow_eqI: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c"   and f :: "'c" assumes "D.cone a' \<chi>'"   and "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>"   and "D.cones_map f \<chi> = \<chi>'" shows "induced_arrow a' \<chi>' = f"
simps_37: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "String.literal \<Rightarrow> 'b"   and f6 :: "ty \<Rightarrow> 'b"   and x5 :: "String.literal" shows "(case Class x5 of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x | x\<lfloor>\<rceil> \<Rightarrow> f6 x) = f5 x5"
integrable_small_integral_on_small_sets: fixes M :: "'a measure"   and h :: "'a \<Rightarrow> real"   and delta :: "real" assumes "integrable M h"   and "0 < delta" shows "\<exists>epsilon>0. \<forall>U\<in>sets M. emeasure M U < ennreal epsilon \<longrightarrow> \<bar>set_lebesgue_integral M U h\<bar> < delta"
tmp_2: fixes t :: "nat" assumes "t \<in> set (linearize tss)" shows "t \<le> cur"
pathin_def: fixes X :: "'a topology"   and g :: "real \<Rightarrow> 'a" shows "pathin X g \<equiv> continuous_map (top_of_set {0..1}) X g"
minus_plus_minus: fixes s :: "'a"   and t :: "'a"   and u :: "'a"   and v :: "'a" assumes "s adds t"   and "u adds v" shows "t - s + (v - u) = t + v - (s + u)"
walls_betw_subset_wall_crossings: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "gallery (C # Cs @ [D])" shows "order.greater_eq (set (wall_crossings (C # Cs @ [D]))) (walls_betw C D)"
EG_def: fixes p :: "'b set" shows "\<^bold>E\<^bold>G p = gfp (\<lambda>s. p \<inter> \<^bold>E\<^bold>X s)"
entropy_simple_distributed: fixes X :: "'a \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real" assumes "simple_distributed M X f" shows "\<H>(X) = - (\<Sum>x\<in>X ` space M. f x * log b (f x))"
process_names_finite: fixes ss :: "(integer \<times> edge list) iarray"   and p :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list" shows "finite (process_names ss p)"
igSubst_termMOD: shows "igSubst termMOD = subst"
set_concat: fixes xs :: "'c list list" shows "set (concat xs) = \<Union> (set ` set xs)"
consequence_relation_def: fixes Bot :: "'b set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool" shows "consequence_relation Bot entails \<equiv> (Bot \<noteq> {} \<and> (\<forall>B N1. B \<in> Bot \<longrightarrow> entails {B} N1)) \<and> (\<forall>N2 N1. N2 \<subseteq> N1 \<longrightarrow> entails N1 N2) \<and> (\<forall>N2 N1. (\<forall>C\<in>N2. entails N1 {C}) \<longrightarrow> entails N1 N2) \<and> (\<forall>N1 N2 N3. entails N1 N2 \<longrightarrow> entails N2 N3 \<longrightarrow> entails N1 N3)"
translation_composition_eq: fixes a :: "'a"   and b :: "'a" assumes "a \<in> G"   and "b \<in> G" shows "compose G (a)\<^sub>R (b)\<^sub>R = (b \<cdot> a)\<^sub>R"
right_idem: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) \<sqinter> b = a \<sqinter> b"
borel_measurable_ennreal_iff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "((\<lambda>x. ennreal (f x)) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
convert_substInfinitesimalLinear: fixes var :: "nat"   and a :: "atom"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and a' :: "atomUni"   and b :: "real mpoly"   and B :: "real"   and c :: "real mpoly"   and C :: "real" assumes "convert_atom var a (xs' @ x # xs) = Some a'"   and "insertion (nth_default 0 (xs' @ x # xs)) b = B"   and "insertion (nth_default 0 (xs' @ x # xs)) c = C"   and "B \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "length xs' = var" shows "eval (substInfinitesimalLinear var (- c) b a) (xs' @ x # xs) = evalUni (substInfinitesimalLinearUni B C a') x"
shift_degree0: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "n < deg R f" shows "Suc (deg R (shift (Suc n) f)) = deg R (shift n f)"
confluentE: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and y' :: "'b" assumes "confluent R"   and "(x, y) \<in> R"   and "(x, y') \<in> R" obtains z :: "'b" where "(y, z) \<in> R" and "(y', z) \<in> R"
invar_initial: shows "cinvar {r \<in> \<delta>. q \<in> set (rhsq r)} \<Sigma>h"
path_component_of_sym: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" assumes "path_component_of X x y" shows "path_component_of X y x"
hyps: fixes t :: "('x, nat) pp" assumes "deg_pp t \<le> n"   and "keys_pp t \<subseteq> set xs" shows "t \<in> keys (deg_le_sect_pp_aux xs n)"
part_equivI: fixes R :: "('a \<times> 'a) set" assumes "sym R"   and "trans R" shows "part_equiv R"
sset_shift: fixes xs :: "'a list"   and s :: "'a stream" shows "sset (xs @- s) = set xs \<union> sset s"
cases: fixes S :: "'node SDG_node \<Rightarrow> bool"   and a :: "'node SDG_node" assumes "combine_SDG_slicesp S a" obtains n :: "'node SDG_node" where "a = n__" and "S n__"   | n' :: "'node SDG_node" and n'' :: "'node SDG_node" and p :: "'pname" and n :: "'node SDG_node" where "a = n" and "S n'" and "n'' s-p\<rightarrow>\<^bsub>ret\<^esub> CFG_node (parent_node n')" and "sum_SDG_slice2p n' n"
inj_map: fixes f :: "'g \<Rightarrow> 'h" assumes "inj f" shows "inj (map f)"
B: shows "\<forall>x\<in>insert x' A. (xs, F x) \<in> seq_comp_failures P Q"
is_pureI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P' :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x x'. P x x' = \<up> (P' x x')" shows "is_pure P"
wf_less_hmultiset: shows "wf {(X, Y). X < Y}"
fminus_cancel: fixes A :: "'b fset" shows "A |-| A = {||}"
dgrad_p_set_exhaust_expl: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat" assumes "finite F" shows "F \<subseteq> punit.dgrad_p_set d (Max (d ` local.punit.pp_of_term ` Keys F))"
length_i: assumes "\<exists>d. (l, i) = (STR ''select'', [d])" shows "length i = 1"
aadd_le_mono: fixes x :: "ant"   and y :: "ant"   and z :: "ant" assumes "x \<le> y" shows "x + z \<le> y + z"
added_ids_Deliver_Expunge_same_collapse: fixes i :: "'d"   and e :: "'b"   and mo :: "'d"   and j :: "'d" shows "added_ids [Deliver (i, Expunge e mo j)] e = [j]"
continuous_map_eq_topcontinuous_at: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X Y f = (\<forall>x\<in>topspace X. topcontinuous_at X Y f x)"
pmf_bernoulli_None: fixes p :: "real" shows "pmf (bernoulli p) None = 0"
lincomb_sum: fixes A :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a"   and b :: "'a vec \<Rightarrow> 'a" assumes "finite A"   and "A \<subseteq> carrier_vec n" shows "lincomb (\<lambda>v. a v + b v) A = lincomb a A + lincomb b A"
subterm_wpo_s_arg: fixes i :: "nat"   and f :: "'f"   and ss :: "('f, 'v) Term.term list" assumes "i \<in> set (\<sigma> (f, length ss))" shows "Fun f ss \<succ> ss ! i"
insert_subset_eq_iff: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(add_mset a A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># remove1_mset a B)"
preserves_arr: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "arr (L f)"
homeomorphic_maps_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "homeomorphic_maps X Y f g = (homeomorphic_map X Y f \<and> homeomorphic_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y))"
prems_1: shows "snd ` set (tx # txs) \<subseteq> var"
fundchamber_W_image_ker: fixes w :: "'a permutation" assumes "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"   and "w `\<rightarrow> C0 = C0" shows "w = 0"
Limsup_ereal_mult_right: fixes F :: "'b filter"   and c :: "real"   and f :: "'b \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "0 \<le> c" shows "Limsup F (\<lambda>n. f n * ereal c) = Limsup F f * ereal c"
iso_group_mul: fixes G :: "('b, 'c) monoid_scheme"   and A :: "'b set"   and B :: "'b set" assumes "group_disjoint_sum G A B"   and "comm_group G" shows "((\<lambda>(x, y). x \<otimes>\<^bsub>G\<^esub> y) \<in> Group.iso (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (A \<inter> B \<subseteq> {\<one>\<^bsub>G\<^esub>} \<and> A <#>\<^bsub>G\<^esub> B = carrier G)"
Hoare_plus_unhalt: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and A :: "(action \<times> nat) list"   and Q :: "cell list \<times> cell list \<Rightarrow> bool"   and B :: "(action \<times> nat) list" assumes "{P} A {Q}"   and "{Q} B \<up>"   and "tm_wf0 A" shows "{P} A |+| B \<up>"
top_le_conv: fixes x :: "'a" shows "(\<top> \<sqsubseteq>\<^bsub>r\<^esub> x) = (x = \<top>)"
subgroup_kernel: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "subgroup (kernel G H h) G"
induct: fixes x1 :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and x2 :: "('c, 'e) flow_entry_match list"   and x3 :: "'d"   and x4 :: "'e option"   and P :: "('c set \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('c, 'e) flow_entry_match list \<Rightarrow> 'd \<Rightarrow> 'e option \<Rightarrow> bool" assumes "guha_table_semantics x1 x2 x3 x4"   and "\<And>\<gamma> fe p ft1 ft2. \<lbrakk>\<gamma> (ofe_fields fe) p = True; \<forall>fe'\<in>set (ft1 @ ft2). ofe_prio fe < ofe_prio fe' \<longrightarrow> \<gamma> (ofe_fields fe') p = False\<rbrakk> \<Longrightarrow> P \<gamma> (ft1 @ fe # ft2) p (Some (ofe_action fe))"   and "\<And>ft \<gamma> p. \<forall>fe\<in>set ft. \<gamma> (ofe_fields fe) p = False \<Longrightarrow> P \<gamma> ft p None" shows "P x1 x2 x3 x4"
rref_first_element: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "reduced_row_echelon_form A"   and "column (0::'c) A \<noteq> 0" shows "A $ (0::'d) $ (0::'c) = (1::'b)"