hcju/isabelleps huggingface dataset:
hyps_15: fixes thesis :: "bool" assumes "\<And>t' xa. \<lbrakk>t' \<in> .[X]; xa \<in> X; V \<subseteq> U - {x}; 0 \<notin> (\<lambda>s. s - t') ` S; s = t' + t + monomial 1 xa; (monomial (1::'a) s, V) \<in> set (fst (split (t' + t) V ((\<lambda>s. s - t') ` S))); set (snd (split (t' + t) V ((\<lambda>s. s - t') ` S))) \<subseteq> set (snd (ps0, qs0))\<rbrakk> \<Longrightarrow> thesis"   and "0 \<notin> S"   and "(monomial (1::'a) s, V) \<in> set (fst (ps0, qs0))" shows "thesis"
mode_acquire_refinement_helper: fixes mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and SomeMode :: "Mode"   and x :: "Eg1.var" shows "mds\<^sub>A_of (mds\<^sub>C(SomeMode := insert (Eg2_var\<^sub>C_of_Eg1 x) (mds\<^sub>C SomeMode))) = (mds\<^sub>A_of mds\<^sub>C)(SomeMode := insert x (mds\<^sub>A_of mds\<^sub>C SomeMode))"
list_of_oalist_reorder: fixes ko :: "'o"   and xs :: "'x" shows "list_of_oalist (reorder ko xs) = (sort_oalist_aux ko (list_of_oalist xs), ko)"
int_intv_frac_gt_0': fixes a :: "real"   and b :: "real" assumes "a \<in> \<int>"   and "b \<in> \<int>"   and "a \<le> b"   and "a \<noteq> b" shows "a \<le> b - 1"
sds_impossibility_axioms: shows "sds_impossibility agents alts sds"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_parent_locs. r h h')" shows "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs"
concat_append: fixes xs :: "'c list list"   and ys :: "'c list list" shows "concat (xs @ ys) = concat xs @ concat ys"
polyhedra_are_convex: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and b :: "'b vec"   and P :: "'b vec set" assumes "A \<in> carrier_mat nr n"   and "b \<in> carrier_vec nr"   and "P = gram_schmidt.polyhedron n A b" shows "gram_schmidt.convex n P"
h_le_f: fixes x :: "(real, 'n) vec"   and n :: "nat" assumes "x \<in> S" shows "h n (g x) \<le> f (g x)"
Lim_emeasure_incseq: fixes A :: "nat \<Rightarrow> 'c set"   and M :: "'c measure" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Union> (range A))"
induct: fixes x :: "('d, 'e, 'f) pre_RedBlack"   and P :: "('d, 'e, 'f) pre_RedBlack \<Rightarrow> bool" assumes "RedBlack x"   and "\<And>prb. \<lbrakk>fst (root (red prb)) = init (black prb); edges (red prb) = {}; subs prb = {}; confs prb (root (red prb)) = init_conf prb; marked prb = (\<lambda>rv. False); strengthenings prb = (\<lambda>rv \<sigma>. True)\<rbrakk> \<Longrightarrow> P prb"   and "\<And>prb re p' prb'. \<lbrakk>RedBlack prb; P prb; ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) p' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := p'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>\<rbrakk> \<Longrightarrow> P prb'"   and "\<And>prb rv prb'. \<lbrakk>RedBlack prb; P prb; mark_extends prb rv prb'\<rbrakk> \<Longrightarrow> P prb'"   and "\<And>prb sub prb'. \<lbrakk>RedBlack prb; P prb; (subsumee sub \<noteq> subsumer sub \<and> fst (subsumee sub) = fst (subsumer sub) \<and> subsumee sub \<in> red_vertices prb \<and> subsumee sub \<notin> subsumers (subs prb) \<and> subsumee sub \<notin> subsumees (subs prb) \<and> subsumer sub \<in> red_vertices prb \<and> subsumer sub \<notin> subsumees (subs prb) \<and> out_edges (red prb) (subsumee sub) = {} \<and> subs prb' = subs prb \<union> {sub}) \<and> \<not> marked prb (subsumer sub) \<and> \<not> marked prb (subsumee sub) \<and> confs prb (subsumee sub) \<sqsubseteq> confs prb (subsumer sub) \<and> prb' = \<lparr>red = red prb, black = black prb, subs = insert sub (subs prb), init_conf = init_conf prb, confs = confs prb, marked = marked prb, strengthenings = strengthenings prb, \<dots> = pre_RedBlack.more prb\<rparr>\<rbrakk> \<Longrightarrow> P prb'"   and "\<And>prb rv c\<^sub>a prb'. \<lbrakk>RedBlack prb; P prb; abstract_extends prb rv c\<^sub>a prb'\<rbrakk> \<Longrightarrow> P prb'"   and "\<And>prb rv e prb'. \<lbrakk>RedBlack prb; P prb; strengthen_extends prb rv e prb'\<rbrakk> \<Longrightarrow> P prb'" shows "P x"
simps: fixes uu :: "'d list"   and uv :: "('a, 'b, 'c, 'd, 'e) qTerm set list"   and AL :: "('a, 'b, 'c, 'd, 'e) qAbs set list"   and uw :: "('c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set option) list" shows "absOf (Par uu uv AL uw) = set AL"
lsorted_lprefixD: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "xs \<sqsubseteq> ys"   and "lsorted ys" shows "lsorted xs"
GreenThm_typeII_divisible_region_boundary: fixes \<gamma> :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "two_chain_vertical_boundary two_chain_typeII \<subseteq> \<gamma>"   and "\<gamma> \<subseteq> two_chain_boundary two_chain_typeII" shows "integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} \<gamma>"
is_ground_subst_cls_iff: fixes C :: "'a literal multiset" shows "is_ground_cls C = (\<forall>\<sigma>. C = C \<cdot> \<sigma>)"
cod_simp: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VV.arr f" shows "VV.cod f = VxV.cod f"
simps: fixes a1 :: "'action stream"   and a2 :: "'state" shows "R.run a1 a2 = (\<exists>a p r. a1 = a ## r \<and> a2 = p \<and> ren a p \<and> R.run r (ex a p))"
simps_2: fixes x :: "'b"   and \<theta> :: "('a, 'b) trm" shows "SIGO (OSing x \<theta>) = {Inl x |x. x \<in> SIGT \<theta>}"
renaming_set_minus_subset: fixes Sel :: "'at set"   and C :: "'at Literal set"   and L :: "'at Literal" shows "rename_clause Sel (C - {L}) \<subseteq> rename_clause Sel C - {rename_literal Sel L}"
Sup_set_fold: 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 xs :: "'a list" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (set xs) = fold sup xs bot"
dnf_to_bool_append: fixes \<gamma> :: "'b \<Rightarrow> bool"   and a1 :: "'b negation_type list list"   and a2 :: "'b negation_type list list" shows "dnf_to_bool \<gamma> (a1 @ a2) = (dnf_to_bool \<gamma> a1 \<or> dnf_to_bool \<gamma> a2)"
power_pos: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and n :: "nat" assumes "class.semiring_1_real_line one times plus zero minus less_eq less"   and "less zero a" shows "less zero (power.power one times a n)"
containtwo: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "chamber D" shows "\<exists>A\<in>\<A>. C \<in> A \<and> D \<in> A"
cont: shows "continuous_on UNIV (\<lambda>v. Blinfun (\<lambda>v'. sterm_sem I \<theta>\<^sub>1 v * frechet I \<theta>\<^sub>2 v v' + frechet I \<theta>\<^sub>1 v v' * sterm_sem I \<theta>\<^sub>2 v))"
inf_prefix_NilL: fixes ys :: "'b list" shows "[] \<sqinter> ys = []"
nprv_clear: fixes \<phi> :: "'fmla"   and F :: "'fmla set" assumes "nprv {} \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla" shows "nprv F \<phi>"
ide_char': fixes a :: "'a" shows "ide a = (arr a \<and> (local.dom a = a \<or> cod a = a))"
make_affine_proj_eq_iff: fixes p :: "int \<times> int \<times> int"   and p' :: "int \<times> int \<times> int" assumes "in_carrierp p"   and "in_carrierp p'" shows "proj_eq p p' = (make_affine p = make_affine p')"
shows_prec_poly_def: fixes p :: "nat"   and x :: "'a poly" shows "shows_prec p x = showsp_poly p x"
simps_5: fixes n :: "nat"   and F :: "'b formula"   and G :: "'b formula" shows "cut_bound' n (F \<^bold>\<or> G) = cut_bound' (cut_bound' n F) G"
fg_start_round_def: fixes rho :: "nat \<Rightarrow> ('e, 'f, 'g, 'h) config_scheme"   and p :: "'f"   and r :: "nat" shows "fg_start_round rho p r \<equiv> LEAST n. round (rho n) p = r"
f: shows "f =m smult c (\<Prod>\<^sub># fs)"
sup_ge1: fixes x :: "'g multiset"   and y :: "'g multiset" shows "x \<subseteq># x \<union># y"
l''_2: shows "(\<pi>' @\<^bsup>m'\<^esup> \<pi>'' \<guillemotleft> Suc k') l'' = \<pi> l"
prems_5: shows "\<forall>s l. PProbs s \<and> PProbl l \<longrightarrow> PProbs (f s l)"
ta_prod_def: fixes TA1 :: "('d, 'f, 'g) tree_automaton_rec_scheme"   and TA2 :: "('e, 'f, 'h) tree_automaton_rec_scheme" shows "ta_prod TA1 TA2 \<equiv> \<lparr>ta_initial = ta_initial TA1 \<times> ta_initial TA2, ta_rules = \<delta>_prod (ta_rules TA1) (ta_rules TA2)\<rparr>"
simps: fixes a1 :: "'a"   and a2 :: "'b list"   and a3 :: "'b list"   and P :: "'b process"   and I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a" shows "((a1, a2, a3) \<in> rel_induct_aux P I D) = ((\<exists>u xs ys. a1 = u \<and> a2 = ys \<and> a3 = xs \<and> (u, xs, ys) \<in> rel_induct_aux P I D) \<or> (\<exists>u xs ys zs. a1 = u \<and> a2 = xs \<and> a3 = zs \<and> (u, xs, ys) \<in> rel_induct_aux P I D \<and> (u, ys, zs) \<in> rel_induct_aux P I D) \<or> (\<exists>u xs ys x. a1 = u \<and> a2 = xs @ [x] \<and> a3 = ys @ [x] \<and> (u, xs, ys) \<in> rel_induct_aux P I D \<and> (D x, xs, ys) \<in> rel_induct_aux P I D \<and> x \<in> next_events P xs \<inter> next_events P ys) \<or> (\<exists>u x xs. a1 = u \<and> a2 = xs \<and> a3 = xs @ [x] \<and> u \<in> range D \<and> (D x, u) \<notin> I \<and> x \<in> next_events P xs))"
partial_equiv_funD: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "f \<sim> g"   and "x \<in> domain"   and "y \<in> domain"   and "x \<sim> y" shows "f x \<sim> g y"
drop_suffix_hd_css_step'': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "('e, 'd, 'f) com"   and ps :: "('e, 'd, 'f) com list"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and pnorm :: "('e, 'd, 'f) com list"   and pabr :: "('e, 'd, 'f) com list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (p # ps @ cs, css, s) \<rightarrow> (cs', (pnorm @ cs, pabr @ cs) # css, t)" shows "\<Gamma>\<turnstile> (p # ps, css, s) \<rightarrow> (cs', (pnorm, pabr) # css, t)"
dim_col_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'd" shows "dim_col (Matrix.mat nr nc f) = nc"
rep_comp: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "seq g f" shows "rep (g \<cdot> f) = \<^bold>\<parallel>rep g \<^bold>\<cdot> rep f\<^bold>\<parallel>"
bicategory_of_spans_axioms: shows "bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg"
Sup_le_iff: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and A :: "'b::type set"   and b :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Sup::'b::type set \<Rightarrow> 'b::type) (A::'b::type set)) (b::'b::type) = (\<forall>a::'b::type\<in>A. less_eq a b)"
iprev_singleton: fixes n :: "nat"   and a :: "nat" shows "iprev n {a} = n"
llist_all2_reflI: fixes xs :: "'e llist"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x. x \<in> lset xs \<Longrightarrow> P x x" shows "llist_all2 P xs xs"
set_disconnected_nodes_locs_def: shows "set_disconnected_nodes_locs \<equiv> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes_locs"
assms_1: shows "\<AA> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
eq_onp_top_eq_eq: shows "eq_onp \<top> = (=)"
not_collinear_angle: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "\<not> collinear {a, b, c}" shows "angle a b c \<in> {0<..<pi}"
concat_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(list_all2 (list_all2 A) ===> list_all2 A) concat concat"
zero_uint_def: shows "0 \<equiv> Abs_uint 0"
find_non0_def: fixes l :: "nat"   and A :: "'a mat" shows "local.find_non0 l A = (let is = [Suc l..<dim_row A]; Ais = filter (\<lambda>(i, Ail). Ail \<noteq> (0::'a)) (map (\<lambda>i. (i, A $$ (i, l))) is) in case Ais of [] \<Rightarrow> None | a # list \<Rightarrow> Some (sel_fun Ais))"
adds_pp_refl: fixes v :: "'t" shows "pp_of_term v adds\<^sub>p v"
case_times_inf: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word"   and P :: "32 Word.word \<Rightarrow> 32 Word.word \<Rightarrow> bool" assumes "w1 = POS_INF \<and> w2 = POS_INF \<Longrightarrow> P w1 w2"   and "w1 = NEG_INF \<and> w2 = POS_INF \<Longrightarrow> P w1 w2"   and "w1 = POS_INF \<and> w2 = NEG_INF \<Longrightarrow> P w1 w2"   and "w1 = NEG_INF \<and> w2 = NEG_INF \<Longrightarrow> P w1 w2"   and "w1 = POS_INF \<and> w2 \<noteq> NEG_INF \<and> w2 <s 0 \<Longrightarrow> P w1 w2"   and "w1 = POS_INF \<and> w2 \<noteq> POS_INF \<and> 0 <s w2 \<Longrightarrow> P w1 w2"   and "w1 = POS_INF \<and> w2 = 0 \<Longrightarrow> P w1 w2"   and "w1 = NEG_INF \<and> w2 \<noteq> NEG_INF \<and> w2 <s 0 \<Longrightarrow> P w1 w2"   and "w1 = NEG_INF \<and> w2 \<noteq> POS_INF \<and> 0 <s w2 \<Longrightarrow> P w1 w2"   and "w1 = NEG_INF \<and> 0 = w2 \<Longrightarrow> P w1 w2"   and "w1 \<noteq> NEG_INF \<and> w1 <s 0 \<and> w2 = POS_INF \<Longrightarrow> P w1 w2"   and "w1 \<noteq> POS_INF \<and> 0 <s w1 \<and> w2 = POS_INF \<Longrightarrow> P w1 w2"   and "0 = w1 \<and> w2 = POS_INF \<Longrightarrow> P w1 w2"   and "w1 \<noteq> NEG_INF \<and> w1 <s 0 \<and> w2 = NEG_INF \<Longrightarrow> P w1 w2"   and "w1 \<noteq> POS_INF \<and> 0 <s w1 \<and> w2 = NEG_INF \<Longrightarrow> P w1 w2"   and "0 = w1 \<and> w2 = NEG_INF \<Longrightarrow> P w1 w2"   and "w1 \<noteq> NEG_INF \<and> w1 \<noteq> POS_INF \<and> w2 \<noteq> NEG_INF \<and> w2 \<noteq> POS_INF \<Longrightarrow> P w1 w2" shows "P w1 w2"
All2_term_Iff_fresh: fixes i :: "name"   and j :: "name"   and j' :: "name"   and A :: "fm" assumes "i \<noteq> j"   and "atom j' \<sharp> (i, j, A)" shows "{} \<turnstile> All2 i (Var j) A IFF SyntaxN.Ex j' (Var j EQ Var j' AND All2 i (Var j') A)"
domain_eq: shows "Domainp (pcr_mapping_rbt (=) (=)) = (\<lambda>t. ord.is_rbt cless t \<or> ID ccompare = None)"
am5_lem: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> y) \<cdot> ad (x \<cdot> ad y) = ad x"
B3: shows "B3 \<in> carrier_mat (Suc m) n"
g_nonsquare: fixes m :: "nat" assumes "\<not> is_square m" shows "g m = 0"
orderE: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" assumes "a \<preceq> b" obtains "a = linordered_monoid.min a b"
hyp: shows "x\<^bsub>0\<^esub>\<^bsup>m + n\<^esup> + x\<^bsub>m\<^esub>\<^bsup>n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>m + n\<^esup>"
map_transfer: fixes Rb :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and Sd :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "((Rb ===> Sd) ===> rel_pmf Rb ===> rel_pmf Sd) map_pmf map_pmf"
normal_vector_4: fixes W :: "'a vec set"   and w :: "'a vec" assumes "finite W"   and "Suc (card W) = n"   and "lin_indpt W"   and "W \<subseteq> carrier_vec n"   and "w \<in> W" shows "normal_vector W \<bullet> w = (0::'a)"
healthy_parts: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> bounded_by b (t P)"   and "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> nneg (t P)"   and "\<And>P Q. \<lbrakk>sound P; sound Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q"   and "\<And>P c x. \<lbrakk>sound P; 0 \<le> c\<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x" shows "healthy t"
sel_1: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "lhd (iterates f x) = x"
trans: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" assumes "b \<preceq> a"   and "c \<preceq> b" shows "c \<preceq> a"
less_imp_le: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
uniformity_prod_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
proots_within_smult: fixes a :: "'a"   and p :: "'a poly"   and s :: "'a set" assumes "a \<noteq> (0::'a)" shows "proots_within (smult a p) s = proots_within p s"
set_3: fixes x1 :: "'c" shows "fv (Var x1) = {x1}"
disjE_Not2: fixes P :: "bool"   and Q :: "bool" assumes "P \<or> Q" obtains "P"   | "\<not> P" and "Q"
arr_char: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "J_C.arr f = (f \<noteq> J_C.Null \<and> J_C.Dom f \<in> Collect (functor (\<cdot>\<^sub>J) (\<cdot>)) \<and> J_C.Cod f \<in> Collect (functor (\<cdot>\<^sub>J) (\<cdot>)) \<and> J_C.Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>J) (\<cdot>) (J_C.Dom f) (J_C.Cod f)))"
distinct_canonical_basis: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and canonical_basis :: "'b list" assumes "class.basis_enum scaleR scaleC plus zero minus uminus canonical_basis" shows "distinct canonical_basis"
matrix_index: fixes m1 :: "'a list list"   and n :: "nat"   and nc :: "nat"   and m2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "mat (row_length m1) n m1"   and "mat n nc m2"   and "i < row_length m1"   and "j < nc" shows "(m1 \<circ> m2) ! j ! i = scalar_product (row m1 i) (col m2 j)"
lsc_sequentially_mem: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal"   and x :: "nat \<Rightarrow> 'b"   and c :: "nat \<Rightarrow> ereal"   and c0 :: "ereal" assumes "lsc_at x0 f"   and "x \<longlonglongrightarrow> x0"   and "c \<longlonglongrightarrow> c0"   and "\<forall>n. f (x n) \<le> c n" shows "f x0 \<le> c0"
cdep_SDG_path_intra_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n cd-ns\<rightarrow>\<^sub>d* n'" shows "n i-ns\<rightarrow>\<^sub>d* n'"
N_1: fixes n :: "nat" assumes "N \<le> n" shows "3 * alpha < Sigma_Algebra.measure (distr P borel (\<lambda>x. (g x + birkhoff_sum f n x) / B n)) {x. 2 * a < \<bar>x\<bar>}"
image_subset_iff_funcset: fixes F :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" shows "(F ` A \<subseteq> B) = (F \<in> funcset A B)"
Abs_int'_inverse: fixes y :: "int" assumes "y \<in> UNIV" shows "Rep_int' (Abs_int' y) = y"
insert_Diff: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
divide_eq_imp: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<noteq> (0::'a)"   and "b = a * c" shows "b / c = a"
subst_fm_OrdP: fixes t :: "tm"   and i :: "name"   and x :: "tm" shows "(OrdP t)(i::=x) = OrdP (subst i x t)"
BU: assumes "t1 \<in> B n"   and "t2 \<in> U n" shows "thesis"
recvmsg_simps_4: fixes P :: "'a \<Rightarrow> bool"   and ip :: "nat" shows "recvmsg P (\<not>unicast ip) = True"
line_integral_exists_def: fixes F :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and \<gamma> :: "real \<Rightarrow> 'a" shows "line_integral_exists F basis \<gamma> \<equiv> (\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
minimal_imp_index_eq_app_right: assumes "minimal" shows "card (dfa.states M) = min_states"
quicksort_correct_case3: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs0 :: "'c list"   and lo :: "nat"   and hi :: "nat"   and xs :: "'c list"   and xs' :: "'c list"   and p :: "nat"   and xs'' :: "'c list" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. x \<noteq> y \<Longrightarrow> R (h x) (h y) \<or> R (h y) (h x)"   and "quicksort_pre R h xs0 lo hi xs"   and "partition_spec R h xs lo hi xs' p"   and "p - Suc 0 \<le> lo"   and "\<not> hi \<le> Suc p"   and "quicksort_post R h (Suc p) hi xs' xs''" shows "quicksort_post R h lo hi xs xs''"
sel: fixes x :: "'d \<Rightarrow> 'e" shows "run_state (StateT x) = x"
conjE2_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "conj_PL p q \<in> Cn A" shows "q \<in> Cn A"
simps: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize" shows "normalization_euclidean_semiring.gcd zero modulo normalize a b = (if b = zero then normalize a else normalization_euclidean_semiring.gcd zero modulo normalize b (modulo a b))"
locally_compact_Int_cball: fixes S :: "'b set" shows "locally compact S = (\<forall>x\<in>S. \<exists>e>0. closed (cball x e \<inter> S))"
assms_6: shows "A1 \<union> A2 \<union> A3 = A"
assms_3: shows "C1 \<in> carrier_mat nr2 n1"
atMost_borel: fixes a :: "'b" shows "{..a} \<in> sets borel"
abs_eq: fixes xa :: "cmsg"   and x :: "cmsg" shows "\<langle>Ab xa, Ab x\<rangle> = Ab (cPair xa x)"
bv_or_length: fixes l :: "bool list"   and r :: "bool list" assumes "length l = length r" shows "length (bv_or l r) = length l"
a1: shows "infsetsum'_converges (\<lambda>x. norm (f x)) A"
a: shows "(SUP n. INF m\<in>{n..}. u (m + 1)) = (SUP n. Inf (u ` {n + 1..}))"
Int_greatest: fixes C :: "'d set"   and A :: "'d set"   and B :: "'d set" assumes "C \<subseteq> A"   and "C \<subseteq> B" shows "C \<subseteq> A \<inter> B"
hyps: fixes P :: "real \<times> real"   and Q :: "real \<times> real"   and Pc :: "real \<times> real" assumes "e \<in> UNIV \<rightarrow> {0<..<1}"   and "ccw'.sortedP 0 (P # Q # Rs)" shows "list_all (\<lambda>a. case a of (xi, xj) \<Rightarrow> ccw' xi xj (Pc + (\<Sum>P\<in>set (P # Q # Rs). e P *\<^sub>R P))) (polychain_of Pc (P # Q # Rs))"