hcju/isabelleps huggingface dataset:
assms: shows "sorted_sublist_wrt R xs 0 (length xs - 1)"
fun_ord_parametric: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool" assumes "bi_total C" shows "rel_fun (rel_fun A (rel_fun B (=))) (rel_fun (rel_fun C A) (rel_fun (rel_fun C B) (=))) fun_ord fun_ord"
intersect_tsD1: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and k :: "'b" assumes "c \<in> (A \<Rightarrow>\<inter>  B) k" shows "c \<in> A k"
group_by_sound: fixes c :: "'a"   and cs :: "'a set"   and b :: "'b"   and f :: "'c \<Rightarrow> 'b \<times> 'a"   and as :: "'c set" assumes "c \<in> cs"   and "(b, cs) \<in> group_by f as" shows "\<exists>a\<in>as. f a = (b, c)"
brouwer_ball: fixes e :: "real"   and a :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "0 < e"   and "continuous_on (cball a e) f"   and "f ` cball a e \<subseteq> cball a e" obtains x :: "'a" where "x \<in> cball a e" and "f x = x"
freshEnv_idEnv: fixes xs :: "'b"   and x :: "'c" shows "freshEnv xs x idEnv"
poly_mapping_eqI_proj: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair"   and "\<And>k. term_powerprod.proj_poly term_of_pair k p = term_powerprod.proj_poly term_of_pair k q" shows "p = q"
law_restrict_iso_imp_eq: fixes A :: "('c, 'd) ring_scheme"   and B :: "('c, 'e) ring_scheme" assumes "id \<in> ring_iso (law_restrict A) (law_restrict B)"   and "ring A"   and "ring B" shows "law_restrict A = law_restrict B"
min_height_le_height: fixes kdt :: "'b kdt" shows "min_height kdt \<le> height kdt"
get_disconnected_nodes_is_l_get_disconnected_nodes: shows "l_get_disconnected_nodes DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs"
assms: fixes x :: "'a" assumes "x \<in> set xs"   and "\<not> Q x" shows "filter P (f x) = []"
wf_design: shows "design \<V> (\<B> + mset_set \<G>)"
simps_3: fixes x :: "'a"   and y :: "'a" shows "(Val x \<le> Val y) = (x \<le> y)"
inducts: fixes x :: "'c"   and A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> reachable A I"   and "\<And>s. s \<in> init A \<Longrightarrow> P s"   and "\<And>s a s'. \<lbrakk>s \<in> reachable A I; P s; (s, a, s') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P s'" shows "P x"
exhaust: fixes y :: "('a, 'b) efsym" obtains x1 :: "'a" where "y = Oldf x1"   | x2 :: "'b" where "y = Tag x2"   | x3 :: "'b" where "y = Wit x3"
continuous_at_within_log: fixes a :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "continuous (at a within s) f"   and "continuous (at a within s) g"   and "0 < f a"   and "f a \<noteq> 1"   and "0 < g a" shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
sorted_sort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sort_key less_eq (\<lambda>x. x) xs)"
rel_S_pos_distr_cond_def: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and uu :: "('o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't \<times> 'u) itself" shows "rel_S_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1' L2 L2'. rel_S L1 L2 Co1 Co2 Contra1 Contra2 OO rel_S L1' L2' Co1' Co2' Contra1' Contra2' \<le> rel_S (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2'))"
hyps: fixes ys :: "['a]" shows "foldl\<cdot>(flip\<cdot>Cons)\<cdot>ys\<cdot>xs = foldl\<cdot>(flip\<cdot>Cons)\<cdot>[]\<cdot>xs ++ ys"
simps: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.canonic_int_fun_ordered atom_ordering A = (\<lambda>S. \<exists>C. C \<in> S \<and> propositional_atoms.strictly_maximal_literal atom_ordering C (Pos A) \<and> (\<forall>B. Pos B \<in> C \<longrightarrow> (B, A) \<in> atom_ordering \<longrightarrow> \<not> propositional_atoms.canonic_int_fun_ordered atom_ordering B S) \<and> (\<forall>B. Neg B \<in> C \<longrightarrow> (B, A) \<in> atom_ordering \<longrightarrow> propositional_atoms.canonic_int_fun_ordered atom_ordering B S))"
assms_6: shows "us = us1 @ berlekamp_factorization_main d divs vs2 n2"
init_s: fixes t :: "real" assumes "t \<le> 0" shows "ego2.s t = ego.q \<delta>"
initial_def: fixes cfg :: "('p, 'v, 's) configuration" shows "initial cfg \<equiv> (\<forall>p. \<exists>v. msgs cfg <p, inM v> = 1) \<and> (\<forall>p m1 m2. m1 \<in># msgs cfg \<and> m2 \<in># msgs cfg \<and> isReceiverOf p m1 \<and> isReceiverOf p m2 \<longrightarrow> m1 = m2) \<and> (\<forall>v. msgs cfg <\<bottom>, outM v> = 0) \<and> (\<forall>p v. msgs cfg <p, v> = 0) \<and> states cfg = start"
ltl_models_equiv_prop_entailment: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl" shows "w \<Turnstile> \<phi> = {\<chi>. w \<Turnstile> \<chi>} \<Turnstile>\<^sub>P \<phi>"
IH: shows "\<forall>ys. length ys < length xs \<longrightarrow> True \<in> set ys \<longrightarrow> harry_meas ys = Suc (harry_meas (harry_step ys))"
1: shows "\<forall>x y z. (- z \<le> - f (x \<squnion> y)) = (- z \<le> - f x \<and> - z \<le> - f y)"
assms: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
simps_2: fixes t :: "ty"   and e :: "expr" shows "assignsE (New t[e]) = assignsE e"
sat_cnf_def: fixes \<sigma> :: "nat \<Rightarrow> bool"   and cnf :: "lit uprod set" shows "sat_cnf \<sigma> cnf = (\<forall>cl\<in>cnf. sat_clause \<sigma> cl)"
OPT_cost: assumes "a \<noteq> b"   and "qs \<in> pattern" shows "real (T\<^sub>p [a, b] qs (OPT2 qs [a, b])) = opt"
FOREACHci_refine_rcg': fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; c \<sigma>; c' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> R (f' x' \<sigma>')" shows "FOREACH\<^sub>C S c f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>C\<^bsup>\<Phi>'\<^esup> S' c' f' \<sigma>0')"
power_not_zero: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero"   and "a \<noteq> zero" shows "power.power one times a n \<noteq> zero"
lnil_def: shows "lnil \<equiv> llist_abs\<cdot>(sinl\<cdot>ONE)"
trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
typed_secure: fixes c :: "('Var, 'AExp, 'BExp) Stmt" assumes "\<turnstile> \<Gamma>\<^sub>0 {c} \<Gamma>\<^sub>0" shows "com_sifum_secure c"
synthetic_divmod_def: fixes p :: "'b poly"   and c :: "'b" shows "synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0::'b)"
atLeastatMost_subset_iff: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word"   and d :: "'a Word.word" shows "(signed.atLeastAtMost a b \<subseteq> signed.atLeastAtMost c d) = (\<not> a \<le>s b \<or> c \<le>s a \<and> b \<le>s d)"
bst_of_list_map: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and xs :: "'a list" assumes "strict_mono_on f A"   and "set xs \<subseteq> A" shows "bst_of_list (map f xs) = map_tree f (bst_of_list xs)"
estimate_by_abs: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a + b \<le> c" shows "a \<le> c + \<bar>b\<bar>"
unitary_embed: fixes P :: "'c \<Rightarrow> bool" shows "unitary \<guillemotleft> P \<guillemotright>"
tail_invar_cons_up_invar: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list"   and t' :: "('c, 'd) SkewBinomialTree" assumes "tail_invar (t # bq)"   and "rank t' \<le> rank t"   and "tree_invar t'" shows "invar (t' # t # bq)"
line_integral_exists_on_chain2: shows "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>"
consistent_pres_op'_def: fixes op :: "'a \<times> (nat \<times> 'b) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> 'c" shows "consistent_pres_op' op \<equiv> case op of (name, pres, effs, cost) \<Rightarrow> consistent_map_lists pres (implicit_pres effs)"
stateinterpol_def: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and S :: "variable set" shows "stateinterpol \<nu> \<omega> S = (\<lambda>x. if x \<in> S then \<nu> x else \<omega> x)"
induct: fixes P :: "'a list \<Rightarrow> 'a set \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a set" assumes "\<And>todo fin. \<lbrakk>\<And>x xa y. \<lbrakk>set todo \<noteq> {}; x = (case todo of h # t \<Rightarrow> (h, t)); (xa, y) = x; \<exists>b\<in>fin. subsumes b xa\<rbrakk> \<Longrightarrow> P y fin; \<And>x xa y. \<lbrakk>set todo \<noteq> {}; x = (case todo of h # t \<Rightarrow> (h, t)); (xa, y) = x; \<not> (\<exists>b\<in>fin. subsumes b xa)\<rbrakk> \<Longrightarrow> P (r xa @ y) (insert xa fin)\<rbrakk> \<Longrightarrow> P todo fin" shows "P a0 a1"
simps_6: fixes f1 :: "'g option \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('i \<Rightarrow> ('g, 'h, 'i) Resumption.resumption) \<Rightarrow> 'f"   and x21 :: "'h"   and x22 :: "'i \<Rightarrow> ('g, 'h, 'i) Resumption.resumption" shows "(case Resumption.resumption.Pause x21 x22 of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
antisym_conv3: fixes x :: "'b"   and y :: "'b" assumes "\<not> x < y" shows "(\<not> y < x) = (x = y)"
ide_is_iso: fixes a :: "'a" assumes "Right.ide a" shows "Right.iso a"
wf_digraph_axioms: shows "wf_digraph (with_proj (mk_symmetric G))"
more_arith_simps_9: fixes a :: "'b"   and b :: "'b" shows "- (a + b) = - a + - b"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<Z> \<alpha>"
has_bottom_pow: fixes A :: "'b set" shows "order.has_bottom (Pow A)"
mult_left_cancel: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times c a = times c b) = (a = b)"
vbrelation_vcomp_vid_on_right: fixes A :: "V" assumes "\<AA>\<lparr>Arr\<rparr> \<subseteq>\<^sub>\<circ> A" shows "\<FF>\<lparr>ArrMap\<rparr> \<circ>\<^sub>\<circ> vid_on A = \<FF>\<lparr>ArrMap\<rparr>"
dims2_def: shows "st.dims2 = nths dims (- vars1)"
AL_emp1: fixes a :: "'b" shows "emp\<down>a = None"
assms_5: assumes "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"   and "\<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)"   and "\<not> lt dir c (\<langle>\<V> s\<rangle> x)" shows "P (update\<B>\<I> (UBI_upd dir) i x c s)"
lu: shows "Some (l, u) = bnds_powr prec l1 u1 l2 u2"
card_points: fixes a :: "'s" assumes "ide a" shows "|hom unity a| <o \<AA>"
boundedI: 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> a" shows "b \<oplus> c \<preceq> a"
qf1_prime_exists: fixes k :: "nat" assumes "prime (4 * k + 1)" shows "is_sum2sq_nat (4 * k + 1)"
borel_singleton: fixes A :: "'b set"   and x :: "'b" assumes "A \<in> sets borel" shows "insert x A \<in> sets borel"
emeasure_mono: fixes a :: "'b set"   and b :: "'b set"   and M :: "'b measure" assumes "a \<subseteq> b"   and "b \<in> sets M" shows "emeasure M a \<le> emeasure M b"
semiconnex_axioms: shows "semiconnex A (\<sqsubseteq>)"
finite: fixes f :: "'a \<Rightarrow> nat" assumes "finite S"   and "bij_betw f S {..<card S}" shows "thesis"
\<alpha>: shows "\<alpha> = Derive a (take (N - 1) D)"
swap_cols_rows_carrier_3: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n n" shows "swap_cols_rows k l A \<in> carrier_mat n n"
smc_in_ArrE: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> smc_0\<lparr>Arr\<rparr>" obtains a :: "V" and b :: "V" where "f : a \<mapsto>\<^bsub>smc_0\<^esub> b" and "a \<in>\<^sub>\<circ> smc_0\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> smc_0\<lparr>Obj\<rparr>"
interp_msg_def: fixes msg :: "'a \<times> ('a, 'b) operation"   and state :: "('a \<times> 'b \<times> bool) list" shows "interp_msg msg state \<equiv> \<langle>snd msg\<rangle> state"
image_Pow_surj: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set" assumes "f ` A = B" shows "f \<turnstile> Pow A = Pow B"
one_arctic_delta_def: shows "1 = Num_arc_delta (0::'a)"
qtables: shows "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<phi>)) \<phi>s rels"
rep_not_zero: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and design_rep_number :: "int" assumes "constant_rep_design point_set block_collection design_rep_number" shows "design_rep_number \<noteq> 0"
cmult_in_iff: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "((\<lambda>x. c * f x) \<in> O[F](g)) = (f \<in> O[F](g))"
ok_add_ok: fixes e :: "'b environment"   and x :: "char list"   and X :: "'b" assumes "ok e"   and "x \<notin> env_dom e" shows "ok (e\<lparr>x:X\<rparr>)"
XX: shows "cmod (inner_prod v (X N *\<^sub>v v) - inner_prod v (A *\<^sub>v v)) = cmod (inner_prod v (X N *\<^sub>v v)) - cmod (inner_prod v (A *\<^sub>v v))"
tllist_all2_trans: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "('g, 'h) tllist"   and ys :: "('g, 'h) tllist"   and zs :: "('g, 'h) tllist" assumes "tllist_all2 P Q xs ys"   and "tllist_all2 P Q ys zs"   and "transp P"   and "transp Q" shows "tllist_all2 P Q xs zs"
eval_expression_no_jump': fixes G :: "prog"   and s0 :: "state"   and e :: "expr"   and v :: "val"   and s1 :: "state"   and j :: "jump"   and C :: "qtname"   and L :: "lenv"   and T :: "ty" assumes "G\<turnstile>s0 \<midarrow>e-\<succ>v\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>e\<Colon>-T"   and "wf_prog G" shows "fst s1 \<noteq> Some (Jump j)"
simps_1: fixes s :: "('c, 'd) fmap" shows "state_list s [] = [s]"
D_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "linordered_monoid.D m i j k \<equiv> linordered_monoid.Min {linordered_monoid.len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
Rep_formula': fixes x :: "('e, 'f, 'g) formula" shows "hereditarily_fs (Rep_formula x)"
finite_Collect_conjI: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" assumes "finite {x. P x} \<or> finite {x. Q x}" shows "finite {x. P x \<and> Q x}"
simps_1: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysis.ABinds Aexp [] = \<bottom>"
assms: shows "rs\<^sub>1 @ rs\<^sub>2 = [x]"
pref_profile_from_tableI: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list" assumes "prefs_from_table_wf agents alts xss" shows "pref_profile_wf agents alts (prefs_from_table xss)"
constrains_imp_lift_constrains: fixes F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and i :: "nat"   and A :: "'c set"   and B :: "'c set"   and j :: "nat" assumes "F i \<in> A \<times> UNIV co B \<times> UNIV"   and "F j \<in> preserves snd" shows "lift j (F j) \<in> lift_set i (A \<times> UNIV) co lift_set i (B \<times> UNIV)"
msubstltneg: fixes vs :: "'b list"   and c :: "poly"   and a :: "poly"   and b :: "tm"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> < (0::'b)"   and "islin (Lt (CNP 0 a b))" shows "Ifm vs (x # bs) (msubstltneg c t a b) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) (Lt (CNP 0 a b))"
list_all_fusion_def: fixes g :: "('b, 'c) generator"   and P :: "'b \<Rightarrow> bool"   and s :: "'c" shows "list_all_fusion g P s = list_all P (list.unfoldr g s)"
equi_desc_descA_1: fixes v :: "i"   and x :: "\<nu>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<equiv> x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<^bold>\<A>\<phi> x) in v]"
prems_2: shows "vs \<in> L M2 \<inter> L M1"
lspasl_magicr_inv: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Gamma :: "bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h2 :: "'a"   and Delta :: "bool" assumes "class.heap_sep_algebra plus zero sep_disj"   and "Gamma \<longrightarrow> sep_algebra.sep_impl plus sep_disj A B h2 \<or> Delta" shows "(\<exists>h1 h0. Gamma \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> A h1 \<and> \<not> B h0) \<longrightarrow> Delta"
representsD_2: fixes p :: "int poly"   and x :: "'b" assumes "p represents x" shows "ipoly p x = (0::'b)"
sets_restrict_space_subset: fixes S :: "'c set"   and M :: "'c measure" assumes "S \<in> sets (completion M)" shows "sets (restrict_space (completion M) S) \<subseteq> sets (completion M)"
map_nth_eq_conv: fixes xs :: "'c list"   and ys :: "'d list"   and f :: "'c \<Rightarrow> 'd" assumes "length xs = length ys" shows "(map f xs = ys) = (\<forall>i<length ys. f (xs ! i) = ys ! i)"
cset2seq_inj: shows "inj_on cset2seq {A. A \<noteq> {}\<^sub>c}"
bounded_linear: shows "bounded_linear (if x \<in> S then f' x else (\<lambda>x. 0::'b))"
mset_neg_singleton: fixes x :: "'a" shows "mset_neg {#x#}\<^sub>z = {#}"
EF_lem2c: fixes x :: "'a"   and f :: "'a set" assumes "x \<notin> f" shows "x \<in> EF (- f)"
mb_super: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and CDef :: "classDef"   and m :: "nat"   and D :: "nat"   and xs :: "nat list"   and e :: "exp" assumes "CT C = Some CDef"   and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = None"   and "cSuper CDef = D"   and "mbody(CT,m,D) = xs . e" shows "mbody(CT,m,C) = xs . e"
hadamard_on_zero: shows "hadamard *\<^sub>v ket_zero = ket_plus"
ext_ids_def: fixes s :: "('a, 'b option) insert_message list" shows "ext_ids s = \<turnstile> # map (\<lambda>x. \<lbrakk>I x\<rbrakk>) s @ [\<stileturn>]"
pow_eq_if_list: fixes p :: "'b list"   and m :: "nat" shows "p \<^sup>@ m = (if m = 0 then \<epsilon> else p \<cdot> p \<^sup>@ (m - 1))"
fps_lr_inverse_gp_ring1'_2: shows "fps_right_inverse (Abs_fps (\<lambda>n. 1::'b)) (1::'b) = 1 - fps_X"
squarefree_unit: fixes n :: "'b" assumes "is_unit n" shows "squarefree n"
dist_Pair_Pair: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and d :: "'b" shows "dist (a, b) (c, d) = sqrt ((dist a c)\<^sup>2 + (dist b d)\<^sup>2)"
\<I>_def: shows "\<I> \<equiv> {(x, b). b \<in># \<B> \<and> x \<in> b}"
UP_one_not_zero: shows "\<one>\<^bsub>P\<^esub> \<noteq> \<zero>\<^bsub>P\<^esub>"
favorites_altdef': fixes i :: "'agent" shows "favorites R i = {x \<in> alts. \<forall>y\<in>alts. R i y x}"
UGroupHomI: fixes T :: "'a \<Rightarrow> 'b" assumes "\<And>g g'. T (g + g') = T g + T g'" shows "UGroupHom T"
distinct_1: fixes x11 :: "agent"   and x12 :: "agent"   and x13 :: "msg"   and x21 :: "agent"   and x22 :: "msg" shows "Says x11 x12 x13 \<noteq> Gets x21 x22"
midpoint_existence_aux: fixes A :: "'p"   and B :: "'p"   and Q :: "'p"   and P :: "'p"   and T :: "'p" assumes "A \<noteq> B"   and "A B Perp Q B"   and "A B Perp P A"   and "Col A B T"   and "Bet Q T P"   and "A P Le B Q" shows "\<exists>X. X Midpoint A B"
rel_pmf_False: fixes x :: "'c pmf"   and y :: "'d pmf" shows "rel_pmf (\<lambda>x y. False) x y = False"
rep_eq: fixes x :: "mini_alg_unique" shows "mau_is_rat x = ma_is_rat (Rep_mini_alg_unique x)"
simps_4: fixes jmps :: "jump set"   and c1 :: "stmt"   and c2 :: "stmt" shows "jumpNestingOkS jmps (c1;; c2) = (jumpNestingOkS jmps c1 \<and> jumpNestingOkS jmps c2)"
atom_in_p_Vs: fixes i :: "name"   and V :: "name set" assumes "i \<in> p \<bullet> V"   and "V \<subseteq> Vs" shows "i \<in> p \<bullet> Vs"
lens_indep_sym: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c" assumes "x \<bowtie> y" shows "y \<bowtie> x"
represents_inverse: fixes x :: "'a"   and p :: "int poly" assumes "x \<noteq> (0::'a)"   and "p represents x" shows "reflect_poly p represents inverse x"
equiv_eq_app_right: fixes L :: "'b list set" shows "equiv UNIV (eq_app_right L)"
cfg_m: fixes r :: "('a Standard_Constant, nat) labeled_graph \<times> ('a Standard_Constant, nat) labeled_graph" assumes "r \<in> Rs" shows "maintained r cfg"
superclasses_mono: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and c :: "class"   and x :: "qtname" assumes "G\<turnstile>C\<prec>\<^sub>C D"   and "ws_prog G"   and "class G C = Some c"   and "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk> \<Longrightarrow> \<exists>sc. class G (super c) = Some sc"   and "x \<in> superclasses G D" shows "x \<in> superclasses G C"
mfinal_def2: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set" shows "mfinal s = (dom (thr s) \<subseteq> final_threads s)"
f_Exec_Stream_drop: fixes n :: "nat"   and trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun xs c \<up> n = f_Exec_Comp_Stream trans_fun (xs \<up> n) (f_Exec_Comp trans_fun (xs \<down> n) c)"
homotopic_with_trans: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "homotopic_with P X Y f g"   and "homotopic_with P X Y g h" shows "homotopic_with P X Y f h"
states_subset: shows "set (q1 # states p q1) \<subseteq> nodes A \<times> nodes B"
simple_incidence_system_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" shows "simple_incidence_system point_set block_collection \<equiv> incidence_system point_set block_collection \<and> simple_incidence_system_axioms block_collection"