hcju/isabelleps huggingface dataset:
restrict_ext: fixes R :: "('a \<times> 'b) set"   and X :: "'a set" shows "R || X = {(x, y) |x y. x \<in> X \<and> (x, y) \<in> R}"
first: assumes "rs\<^sub>1 = [x]"   and "rs\<^sub>2 = []" shows "thesis"
group_action_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set" shows "group_action_axioms G \<phi> M \<equiv> group_hom G (BijGroup M) \<phi>"
right: fixes x :: "'a"   and y :: "'b" assumes "x \<in> carrier \<X>"   and "y \<in> carrier \<Y>"   and "\<pi>\<^sup>* x \<sqsubseteq>\<^bsub>\<Y>\<^esub> y" shows "x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y"
ce_rel_lm_7: fixes r :: "(nat \<times> nat) set" assumes "ce_rel_to_set r \<in> ce_sets" shows "r \<in> ce_rels"
cod_simps_2: shows "g.cod.leg0 = g.leg0"
apex_def: shows "\<epsilon>.dom.apex \<equiv> C.dom \<epsilon>.dom.leg0"
that: fixes p1 :: "('v \<times> 'w \<times> 'v) list"   and p2 :: "('v \<times> 'w \<times> 'v) list"   and u :: "'v" assumes "p = p1 @ p2"   and "is_path v p1 u"   and "is_path u p2 v'"   and "int_vertices p1 \<subseteq> - W"   and "u \<in> W" shows "thesis"
oinvariant_def: fixes A :: "('f \<times> 'g, 'h) automaton"   and S :: "'f \<Rightarrow> 'f \<Rightarrow> 'h \<Rightarrow> bool"   and U :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<times> 'g \<Rightarrow> bool" shows "(A \<Turnstile> (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. P s)"
init_eq_iff_eq_upto: fixes g :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "g \<in> \<R>"   and "f \<in> \<R>" shows "(\<forall>j<Suc n. g j = f j) = (g \<triangleright> n = f \<triangleright> n)"
UNIV_2: shows "UNIV = insert 1 (skip 2)"
succ_in_Field: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "succ r i \<in> Field r"
mC_C_mono: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set"   and theta' :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set" assumes "PL_Indis cval indis"   and "PL_Indis.mC_C aval tval cval indis theta c d s t P F"   and "theta \<subseteq> theta'" shows "PL_Indis.mC_C aval tval cval indis theta' c d s t P F"
steps: fixes i :: "nat" shows "t i \<le> t (Suc i)"
take: shows "take (length (cs\<^bsup>\<pi>\<^esup> k)) (cs\<^bsup>\<pi>\<^esup> l) = cs\<^bsup>\<pi>\<^esup> k"
SeqAppendP_Mem1: fixes f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm"   and x :: "tm"   and y :: "tm" shows "{SeqAppendP f1 k1 f2 k2 g, HPair x y IN f1, x IN k1} \<turnstile> HPair x y IN g"
g_def: shows "g = (\<lambda>x S y. if \<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S then SOME G. geodesic_segment_between G x y \<and> G \<subseteq> S else {x, y})"
meet_shunt: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = \<bottom>) = (x \<le> - y)"
less_prod_def: shows "(\<sqsubset>) = (\<lambda>(x1, x2) (y1, y2). x1 \<sqsubset>\<^sub>a y1 \<or> x1 \<sqsubseteq>\<^sub>a y1 \<and> x2 \<sqsubset>\<^sub>b y2)"
strictly_preserves_\<alpha>: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "F.D.arr f"   and "F.D.arr g"   and "F.D.arr h" shows "G (F.D.assoc f g h) = G.D.assoc (G f) (G g) (G h)"
oracle\<^sub>n_def: fixes key_gen :: "nat \<Rightarrow> ('b \<times> 'c) spmf"   and sign :: "nat \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e spmf"   and valid_message :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and \<eta> :: "nat" shows "suf_cma.oracle\<^sub>n key_gen sign valid_message \<eta> = family_oracle (\<lambda>_. suf_cma.oracle\<^sub>1 key_gen sign valid_message \<eta>)"
arrI: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>C.in_hom f a b" shows "\<F>C.arr f"
ide_char: fixes a :: "'a" shows "C.ide a = (C.arr a \<and> C.dom a = a \<and> C.cod a = a)"
is_FEx_def: fixes f :: "form" shows "is_FEx f = (case f of FEx f \<Rightarrow> True | _ \<Rightarrow> False)"
\<beta>: fixes p :: "fm"   and d :: "int"   and a :: "int"   and bs :: "int list"   and x :: "int"   and bbs :: "bool list" assumes "iszlfm p"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d"   and "\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j)"   and "Ifm bbs (x # bs) p" shows "Ifm bbs ((x - d) # bs) p"
idom_ex: fixes g :: "'g"   and n :: "'node" assumes "invar g"   and "n \<in> set (\<alpha>n g)"   and "n \<noteq> Entry g" shows "\<exists>!m. isIdom g n m"
WT2: shows "\<I> \<turnstile>g gpv' \<surd>"
halfchsys_decomp_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "f \<turnstile> ChamberComplex.chamber_system X \<inter> (g \<turnstile> ChamberComplex.chamber_system X) = {}"
self_append_conv2: fixes ys :: "'f list"   and xs :: "'f list" shows "(ys = xs @ ys) = (xs = [])"
simps: fixes queue :: "'a list"   and n :: "nat"   and T :: "'a set" shows "Round_Robin.round_robin_invar (queue, n) T = (set queue = T \<and> distinct queue)"
intros: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "(a, b) \<in> r" shows "b \<in> Relation.Range r"
dg_Hom_in_Vset: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "Hom \<CC> a b \<in>\<^sub>\<circ> Vset \<alpha>"
homeomorphic_maps_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "homeomorphic_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y)"
seqSubstSimps_5: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list" shows "P \<parallel> Q[<\<sigma>>] = P[<\<sigma>>] \<parallel> Q[<\<sigma>>]"
assms_1: shows "\<forall>k\<le>n. M k k \<le> \<one>"
straighten_topdown_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "straighten_topdown w1 w2" shows "kauff_mat w1 = kauff_mat w2"
v11_vrange_ex1_eq: fixes r :: "V"   and b :: "V" assumes "v11 r" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<and> r\<lparr>a\<rparr> = b)"
Int_assoc: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
join_connection2: fixes x :: "int"   and y :: "int" shows "(x \<le> y) = (int.join x y = y)"
strong_low_bisim_neval': fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and c\<^sub>1 :: "('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)"   and n :: "nat"   and c\<^sub>n :: "('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)"   and c\<^sub>1' :: "('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)"   and \<R>\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of"   and "sifum_security_init.neval eval\<^sub>A c\<^sub>1 n c\<^sub>n"   and "(c\<^sub>1, c\<^sub>1') \<in> \<R>\<^sub>A"   and "snd (fst c\<^sub>1) = snd (fst c\<^sub>1')"   and "sifum_security_init.strong_low_bisim_mm dma\<^sub>A \<C>_vars\<^sub>A eval\<^sub>A \<R>\<^sub>A" shows "\<exists>c\<^sub>n'. sifum_security_init.neval eval\<^sub>A c\<^sub>1' n c\<^sub>n' \<and> (c\<^sub>n, c\<^sub>n') \<in> \<R>\<^sub>A \<and> snd (fst c\<^sub>n) = snd (fst c\<^sub>n')"
lessThan_Suc: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod g {..<Suc n} = prod g {..<n} * g n"
disj': shows "ODE_dom (OsubstFO ODE1 \<sigma>) \<inter> ODE_dom (OsubstFO ODE2 \<sigma>) = {}"
sub3_makes_triangle: fixes A :: "'a mat"   and l :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat" assumes "mute_fun mf"   and "det_selection_fun sel_fun"   and "local.sub3 (dim_row A) l (r, A) = (r', A')"   and "l \<le> dim_row A"   and "l \<le> dim_col A" shows "local.triangular_to l A'"
inj_on_the: fixes A :: "'b option set" assumes "None \<notin> A" shows "inj_on the A"
hd_exp_powser: fixes cs :: "real msllist"   and f :: "('a \<times> real) msllist" shows "ms_aux_hd_exp (powser_ms_aux cs f) = (if cs = MSLNil then None else Some 0)"
lspasl_cs_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a"   and h4 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0"   and "heap_sep_algebra.tern_rel plus sep_disj h3 h4 h0" shows "\<exists>h5 h6 h7 h8. heap_sep_algebra.tern_rel plus sep_disj h5 h6 h1 \<and> heap_sep_algebra.tern_rel plus sep_disj h7 h8 h2 \<and> heap_sep_algebra.tern_rel plus sep_disj h5 h7 h3 \<and> heap_sep_algebra.tern_rel plus sep_disj h6 h8 h4 \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> heap_sep_algebra.tern_rel plus sep_disj h3 h4 h0"
short_im1: shows "\<parallel>gs1.gso (i - 1)\<parallel>\<^sup>2 \<le> B"
Suc: shows "map ((+) n) [i..<j] = [i + n..<j + n]"
substBinp_empBinp: fixes ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" shows "empBinp %%[Y / y]_ys = empBinp"
abs_reach_def: fixes \<Phi> :: "'b ltl_prop_equiv_quotient" shows "af_abs.abs_reach \<Phi> = {foldl af_abs.f_abs \<Phi> w |w. True}"
redundant_inference_clause: fixes E :: "'a eclause"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and C' :: "'a literal set"   and C :: "'a eclause"   and P :: "'a eclause set" assumes "redundant_clause E S \<sigma> C'"   and "derivable C P S \<sigma> Ground C'"   and "grounding_set P \<sigma>"   and "\<forall>x\<in>S. finite (cl_ecl x)" shows "redundant_inference E S P \<sigma>"
red_preserves_defass: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "\<D> e \<lfloor>dom l\<rfloor>" shows "\<D> e' \<lfloor>dom l'\<rfloor>"
quantale_closed_maps_iff: fixes f :: "'b \<Rightarrow> 'c" shows "(f \<in> quantale_closed_maps) = (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y))"
program_inv_def: fixes prog :: "'b program_scheme" shows "program_inv prog = (0 < IArray.length (states prog) \<and> IArray.length (states prog) = IArray.length (processes prog) \<and> (\<forall>s\<in>set (IArray.list_of (states prog)). 0 < IArray.length s) \<and> lm.ball (proc_data prog) (\<lambda>(uu_, sidx). sidx < IArray.length (processes prog) \<and> fst (processes prog !! sidx) = sidx) \<and> (\<forall>(sidx, start, procArgs, args)\<in>set (IArray.list_of (processes prog)). \<exists>s. start = Index s \<and> s < IArray.length (states prog !! sidx)))"
random_serial_dictatorship_singleton: fixes agents' :: "'agent set"   and x :: "'alt" assumes "finite agents"   and "finite alts"   and "agents' \<subseteq> agents"   and "x \<in> alts" shows "random_serial_dictatorship agents' {x} R = return_pmf x"
Y_ide_arr_1: fixes a :: "'c"   and g :: "'c"   and b' :: "'c"   and b :: "'c" assumes "C.ide a"   and "\<guillemotleft>g : b' \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>Y a g : Hom.map (b, a) \<rightarrow>\<^sub>S Hom.map (b', a)\<guillemotright>"
mset_subset_eq_single: fixes a :: "'k"   and B :: "'k multiset" assumes "a \<in># B" shows "{#a#} \<subseteq># B"
simps_12: fixes \<C> :: "t_context"   and ts :: "checker_type" shows "check_single \<C> Nop ts = ts"
preserves_arr: fixes f :: "'a" assumes "hom.arr f" shows "hom.arr (GF.map f)"
agsum_add2p: fixes a :: "'a"   and n :: "int" assumes "a \<in> carrier A"   and "0 \<le> n" shows "\<zero> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> (- n)\<triangleright>a\<^bsub>A\<^esub>"
set_shadow_root_ptr_in_heap: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and shadow_root :: "(_) shadow_root_ptr option" assumes "h \<turnstile> ok set_shadow_root element_ptr shadow_root" shows "element_ptr |\<in>| element_ptr_kinds h"
trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ordering less_eq less"   and "less_eq a b"   and "less_eq b c" shows "less_eq a c"
mcont2mcont: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub' :: "'b set \<Rightarrow> 'b"   and ord' :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'b" assumes "class.ccpo Sup less_eq less"   and "mcont lub' ord' Sup less_eq f"   and "mcont lub ord lub' ord' t" shows "mcont lub ord Sup less_eq (\<lambda>x. f (t x))"
Sup_subset_mono: fixes A :: "('c \<times> 'd) set"   and B :: "('c \<times> 'd) set" assumes "A \<subseteq> B" shows "less_eq_prod (Sup_prod A) (Sup_prod B)"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and sum :: "'d + 'e" shows "h (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (case sum of Inl x \<Rightarrow> h (f1 x) | Inr x \<Rightarrow> h (f2 x))"
encode_all_operator_preconditions_valid: fixes t :: "nat"   and sv :: "sat_plan_variable" assumes "t < Suc h"   and "sv \<in> atoms (encode_all_operator_preconditions (\<phi> prob_with_noop abs_prob ) ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>) t)" shows "valid_state_var sv"
SeqFormP_Ex: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and k1 :: "tm"   and x :: "tm"   and y :: "tm"   and v :: "tm" assumes "atom s \<sharp> (k, s1, k1, x, y, v)"   and "atom k \<sharp> (s1, k1, x, y, v)" shows "{SeqFormP s1 k1 x, AbstFormP v Zero x y, VarP v} \<turnstile> SyntaxN.Ex k (SyntaxN.Ex s (SeqFormP (Var s) (Var k) (Q_Ex y)))"
auxlist0_def: shows "auxlist0 = concat (map (\<lambda>(t, rel). if enat (\<tau> \<sigma> ne - t) \<le> right I then [(t, join rel pos rel1)] else []) auxlist)"
inst_ok_code2: fixes \<Theta> :: "exetheory"   and insts :: "(('a \<times> String.literal set) \<times> typ) list" assumes "exe_wf_theory' \<Theta>" shows "list_all (\<lambda>((idn, S), T). has_sort (osig (sig (translate_theory \<Theta>))) T S) insts = list_all (\<lambda>((idn, S), T). exehas_sort (exesorts (exesig \<Theta>)) T S) insts"
that: fixes TA :: "(nat, 'L) tree_automaton_rec" assumes "L = ta_lang TA"   and "ranked_tree_automaton TA A" shows "thesis"
inj_compose: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c" assumes "inj f"   and "inj g" shows "inj (f \<circ> g)"
cut_greaterE: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>> t" obtains "x \<in> I" and "t < x"
basis_enum_of_vec_def: fixes v :: "complex vec" shows "basis_enum_of_vec v = (if dim_vec v = length canonical_basis then sum_list (map2 (*\<^sub>C) (list_of_vec v) canonical_basis) else (0::'b))"
integrable_subalgebra: fixes f :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and M :: "'c measure" assumes "f \<in> borel_measurable N"   and "sets N \<subseteq> sets M"   and "space N = space M"   and "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" shows "integrable N f = integrable M f"
cyclic_on_alldef: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" shows "cyclic_on f S = (S \<noteq> {} \<and> (\<forall>s\<in>S. S = orbit f s))"
s_def: fixes a :: "real"   and v :: "real"   and s0 :: "real"   and t :: "real" shows "movement.s a v s0 t = (if t \<le> 0 then s0 else if t \<le> movement.t_stop a v then movement.p a v s0 t else movement.p_max a v s0)"
sums_single: fixes i :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "(\<lambda>r. if r = i then f r else (0::'c)) sums f i"
mk_disjoint_insert: fixes a :: "'g"   and A :: "'g set" assumes "a \<in> A" shows "\<exists>B. A = insert a B \<and> a \<notin> B"
CollectionItemTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>P CollectionItem a : \<tau>" obtains "\<Gamma> \<turnstile>\<^sub>E a : \<tau>"
S2_def: fixes p :: "nat"   and a :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)" shows "??.Euler_Criterion.S2 p a = ??.Euler_Criterion.f p a ` ??.Euler_Criterion.S1 p"
r_ifle_def: shows "r_ifle \<equiv> Cn 4 r_ifz [r_dummy 2 r_sub, recf.Id 4 2, recf.Id 4 3]"
IH: assumes "\<forall>x\<le>n. 0 < x \<longrightarrow> s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x)" shows "\<forall>x\<le>Suc n. local.adverse z i j x = local.adverse z' i j x"
law_restrict_one: fixes R :: "('c, 'd) ring_scheme" shows "\<one>\<^bsub>law_restrict R\<^esub> = \<one>\<^bsub>R\<^esub>"
CardinalityPrognosisIfThenElse_axioms: shows "CardinalityPrognosisIfThenElse prognosis"
ltrm_of_X: shows "monom (UP Zp) (X_poly Zp (deg Zp (X_poly Zp))) (deg Zp (X_poly Zp)) = X_poly Zp"
t_stop_zero: assumes "ego2.t_stop = 0" shows "v\<^sub>e = 0"
OBTAIN_refine_iff: fixes P :: "'c \<Rightarrow> bool"   and R :: "('c \<times> 'd) set"   and Q :: "'d \<Rightarrow> bool" shows "(OBTAIN P \<le> \<Down> R (OBTAIN Q)) = (Ex Q \<longrightarrow> Ex P \<and> Collect P \<subseteq> R\<inverse> `` Collect Q)"
dist_z_iff: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and v' :: "nat" shows "Graph.dist c v 0 v' = (v' = v)"
lc_uminus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "lc (- p) = - lc p"
simps_2: fixes a :: "bit"   and b :: "bit list" shows "remzero (a # b) = (if a = \<one> then a # b else remzero b)"
axioms_1: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "ideal I R" shows "additive_subgroup I R"
Exit_source: fixes a :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "src a = NewExit" shows "False"
allDefs_def: fixes defs :: "'c \<Rightarrow> 'd \<Rightarrow> 'b set"   and phis :: "'c \<Rightarrow> 'd \<times> 'b \<Rightarrow> 'b list option"   and g :: "'c"   and n :: "'d" shows "CFG_SSA_base.allDefs defs phis g n \<equiv> defs g n \<union> CFG_SSA_base.phiDefs phis g n"
\<sigma>'_2: shows "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>'"
simps_1: fixes i :: "nat"   and s :: "'c"   and R :: "nat set" shows "net_ips (NodeS i s R) = {i}"
mpoly_to_nested_poly_mult: fixes p :: "'a mpoly"   and q :: "'a mpoly"   and v :: "nat" shows "mpoly_to_nested_poly (p * q) v = mpoly_to_nested_poly p v * mpoly_to_nested_poly q v"
simps: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and guru :: "'b"   and n :: "nat"   and p :: "'b"   and w :: "'b \<Rightarrow> 'a"   and w' :: "'b \<Rightarrow> 'a" assumes "distinct [blue, brown, green, red]"   and "finite UNIV" shows "Blue_Eyes.possible blue guru n p w w' = (Blue_Eyes.valid blue guru w \<and> Blue_Eyes.valid blue guru w' \<and> (\<forall>p'. p' \<noteq> p \<longrightarrow> w p' = w' p') \<and> (\<forall>n'<n. \<forall>p'. Blue_Eyes.leaves blue guru n' p' w = Blue_Eyes.leaves blue guru n' p' w'))"
sub_inter_l1: fixes T1 :: "funty"   and T2 :: "funty" shows "T1 \<sqinter> T2 <:: T1"
epiE: fixes f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "D.epi f"   and "D.seq g f"   and "D.seq g' f"   and "g \<cdot>\<^sub>D f = g' \<cdot>\<^sub>D f" shows "g = g'"
point_indices_alt_def: fixes t :: "int" shows "source.point_indices t = {int (\<B> index ps) |ps. int (card ps) = t \<and> ps \<subseteq> \<V>}"
C_restr_bot_demand: fixes r :: "C"   and f :: "C \<rightarrow> 'a" assumes "C\<cdot>r \<sqsubseteq> demand f" shows "f|\<^bsub>r\<^esub> = \<bottom>"
bit_iff_odd: fixes a :: "'a"   and n :: "nat" shows "bit a n = odd (a div (2::'a) ^ n)"
normalized_fracts_def: shows "normalized_fracts = {(a, b). coprime a b \<and> unit_factor b = (1::'b)}"
Well_Shaped_Graph_axioms: shows "Well_Shaped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"
sup_least: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x"   and "less_eq z x" shows "less_eq (sup y z) x"
ListMem_iff: fixes x :: "'b"   and xs :: "'b list" shows "ListMem x xs = (x \<in> set xs)"
disjoint_def: fixes A :: "'b set set" shows "disjoint A = (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
one_dim_iso_scaleC: fixes c :: "complex"   and \<psi> :: "'c" shows "one_dim_iso (c *\<^sub>C \<psi>) = c *\<^sub>C one_dim_iso \<psi>"
ug_def: shows "ug \<equiv> fls_base_factor g"
induct: fixes P :: "term \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "term"   and a1 :: "'a" assumes "\<And>t\<^sub>1 t\<^sub>2 u. \<lbrakk>\<And>a x y. \<lbrakk>unapp u = Some a; (x, y) = a\<rbrakk> \<Longrightarrow> P t\<^sub>1 x; \<And>a x y aa. \<lbrakk>unapp u = Some a; (x, y) = a; match t\<^sub>1 x = Some aa\<rbrakk> \<Longrightarrow> P t\<^sub>2 y\<rbrakk> \<Longrightarrow> P (t\<^sub>1 $ t\<^sub>2) u"   and "\<And>name u. P (Const name) u"   and "\<And>name u. P (Free name) u"   and "\<And>n u. P (Bound n) u"   and "\<And>t u. P (\<Lambda> t) u" shows "P a0 a1"
preserves_arr: fixes f :: "'b" assumes "D.arr f" shows "D.arr (src\<^sub>D f)"
roots_inclI': fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and m :: "'c multiset" assumes "domain R"   and "p \<in> carrier (poly_ring R)"   and "\<And>a. \<lbrakk>a \<in> carrier R; p \<noteq> []\<rbrakk> \<Longrightarrow> ring.alg_mult R p a \<le> count m a" shows "ring.roots R p \<subseteq># m"
Times_Un_distrib1: fixes A :: "'c set"   and B :: "'c set"   and C :: "'d set" shows "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C"
c_top_welldef: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Group G"   and "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "N \<bullet>\<^bsub>G\<^esub> (a \<cdot>\<^bsub>G\<^esub> b) = c_top G N (N \<bullet>\<^bsub>G\<^esub> a) (N \<bullet>\<^bsub>G\<^esub> b)"