hcju/isabelleps huggingface dataset:
set_attribute_locs_def: shows "set_attribute_locs = (\<lambda>element_ptr. all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr attrs_update))"
map_iterator_abs_genord_def: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and it :: "('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and m :: "'h \<Rightarrow> 'f option"   and R :: "'h \<times> 'f \<Rightarrow> 'h \<times> 'f \<Rightarrow> bool" shows "map_iterator_abs_genord \<alpha> invar it m R \<equiv> set_iterator_abs_genord (\<lambda>(k, v). (k, \<alpha> v)) (\<lambda>(k, v). invar v) it (map_to_set m) R"
fps_nonzeroI: fixes f :: "'a fps"   and n :: "nat" assumes "f $ n \<noteq> (0::'a)" shows "f \<noteq> 0"
has_derivative_continuous_on: fixes s :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'e \<Rightarrow> 'f" assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x within s)" shows "continuous_on s f"
star_sim4: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y \<le> y \<cdot> x" shows "x\<^sup>\<star> \<cdot> y\<^sup>\<star> \<le> y\<^sup>\<star> \<cdot> x\<^sup>\<star>"
poly_mapping_size_estimation: fixes k :: "'a"   and m :: "'a \<Rightarrow>\<^sub>0 'b"   and y :: "nat" assumes "k \<in> keys m"   and "y \<le> f k + g (lookup m k)" shows "y < local.poly_mapping_size m"
polypoly_poly: fixes p :: "poly"   and n0 :: "nat"   and x :: "'a"   and bs :: "'a list" assumes "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly (x # bs) p) x"
to_nat_less_card: fixes a :: "'b" shows "Bij_Nat.to_nat a < CARD('b)"
sample_Some: fixes gpv :: "('x, 'a, 'b) gpv"   and r :: "'r"   and s :: "'s" shows "exec_gpv (\<lambda>(r, s) x. callee s x \<bind> (\<lambda>(y, s'). if I s' \<and> r = None then map_spmf (\<lambda>r. (y, Some r, s')) (f s') else return_spmf (y, r, s'))) gpv (Some r, s) = map_spmf (\<lambda>(a, s). (a, Some r, s)) (exec_gpv callee gpv s)"
exp_cf7_pos: fixes x :: "real" assumes "0 < numer_cf7 x"   and "0 < numer_cf7 (- x)" shows "0 < exp_cf7 x"
minus_add: fixes a :: "'a"   and b :: "'a" shows "- (a + b) = - b + - a"
P5: shows "C Midpoint B B' \<and> Cong D B D B'"
preserves_reflects_arr: fixes f :: "'a" shows "\<phi>.B.arr (\<phi> f) = \<phi>.A.arr f"
substC_def: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and c :: "('c, 'd) lit list" shows "substC \<pi> c = map (substL \<pi>) c"
ab_1: shows "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
bad_sticky2: fixes s1 :: "'b" assumes "bad1 s1" shows "callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"
CEQ: fixes i :: "nat"   and s :: "'a" shows "c (i, s) = (i < u)"
simps_2: shows "is_lit_plus (\<^bold>\<not> \<bottom>) = True"
steps_redexes: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and r :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and r' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate"   and c :: "('a, 'b, 'c) com" assumes "\<Gamma>\<turnstile> (r, s) \<rightarrow>\<^sup>* (r', s')"   and "r \<in> redexes c" shows "\<exists>c'. \<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s') \<and> r' \<in> redexes c'"
pred_is_pr: shows "(\<lambda>x. x - 1) \<in> PrimRec1"
length_rotate1: fixes xs :: "'a list" shows "|rotate1 xs| = |xs|"
space_restr_to_subalg: fixes M :: "'b measure"   and F :: "'b measure" shows "space (restr_to_subalg M F) = space M"
fst_case: fixes y :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd"   and z :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'e" shows "(\<lambda>x. fst (case x of (t, a1, a2) \<Rightarrow> (t, y t a1 a2, z t a1 a2))) = fst"
Quotient_llist_all2_lmap_Rep: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and a :: "'f llist" assumes "Quotient3 R Abs Rep" shows "llist_all2 R (lmap Rep a) (lmap Rep a)"
single_valued_monom_rel': shows "IS_LEFT_UNIQUE monom_rel"
elem_le_ffold': fixes z :: "'a"   and x :: "'b \<rightharpoonup>\<^sub>f 'a"   and f :: "'a \<Rightarrow> nat" assumes "z \<in> fmran' x" shows "f z < Suc (ffold tcf 0 (fset_of_fmap (fmmap f x)))"
fcomp_id: fixes f :: "'e \<Rightarrow> 'f" shows "fcomp f id = f"
simps_7: fixes c :: "int"   and e :: "MIR.num" shows "plusinf (Gt (CN 0 c e)) = T"
UNIV_rw: shows "UNIV = insert b (insert a (UNIV - {a} - {b}))"
periodic_arithmetic_remove_homothecy: fixes a :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> complex" assumes "coprime a k"   and "periodic_arithmetic f k"   and "0 < k" shows "sum f {1..k} = (\<Sum>l = 1..k. f (l * a))"
less_le_not_le: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" shows "(x \<prec>\<^sub>p y) = strict (\<preceq>\<^sub>p) x y"
simps: fixes f :: "'c \<Rightarrow> 'b val"   and v :: "'c" shows "F_RIGHT f v = val.Right (f v)"
relation_of_state_update_before: fixes sc :: "'d \<times> 'd \<Rightarrow> bool"   and Ac :: "('c, 'd) action" shows "action.relation_of (state_update_before sc Ac) = (R (\<lambda>(A, A'). Pre sc (alpha_rp.more A) \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A') ;; action.relation_of Ac)"
kcomp_klift: fixes f :: "'a \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'b set" shows "f \<circ>\<^sub>K g = g\<^sup>\<dagger> \<circ> f"
emv_subsumeseq_term_imp_num_unique_vars_le: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "emv s t"   and "t \<le>\<cdot> s" shows "num_unique_vars s \<le> num_unique_vars t"
inner_product_space_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> 'a" shows "inner_product_space scale inner \<equiv> vector_space scale \<and> inner_product_space_axioms scale inner"
pp_lub: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and y :: "'c"   and xs :: "'c list" assumes "Semilat A r f"   and "z \<in> A"   and "y \<in> A"   and "set xs \<subseteq> A"   and "\<forall>x\<in>set xs. x \<sqsubseteq>\<^bsub>r\<^esub> z"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z" shows "xs \<Squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z"
vimage_def: fixes D :: "V" shows "CZH_Sets_BRelations.vimage D = (\<lambda>\<langle>r, A\<rangle>\<in>\<^sub>\<circ>D. \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A))"
ex_un_null: shows "\<exists>!n. \<forall>f. S.comp n f = n \<and> S.comp f n = n"
add_mono_thms_linordered_field_3: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i < j \<and> k \<le> l" shows "i + k < j + l"
one_neq_zero: fixes one :: "'c"   and zero :: "'c" assumes "class.zero_neq_one one zero" shows "one \<noteq> zero"
rk_ax_dim: shows "\<exists>A B C D. 4 \<le> rk {A, B, C, D}"
norm_imp_pos_and_ge: fixes x :: "'b"   and n :: "real" assumes "norm x \<equiv> n" shows "0 \<le> norm x \<and> norm x \<le> n"
norm_2: shows "\<lceil>\<lambda>s. True\<rceil> = (\<lambda>s. True)"
must_sync_def2: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and m :: "'heap" shows "exec_mthr.must_sync P t x m = (\<exists>taa x' m' s. exec_mthr.r_syntax P t x m taa x' m' \<and> exec_mthr.actions_ok s t taa)"
arc_distinct_ends: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "pathfinish g \<noteq> pathstart g"
T4: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and w :: "'b \<Rightarrow> 'd" shows "\<turnstile> \<box>[P]_v \<longrightarrow> \<box>[_actrans P v]_w"
form_of_lit_cases: fixes xa :: "'b literal" shows "form_of_lit xa = (case xa of xda\<^sup>+ \<Rightarrow> Atom xda | xea\<inverse> \<Rightarrow> \<^bold>\<not> (Atom xea))"
reachable_redef: shows "reachable {q\<^sub>0} = reach (set \<Sigma>) \<delta> q\<^sub>0"
Discrete_sqrt_eq_newton_sqrt: fixes n :: "nat" shows "Discrete.sqrt n = newton_sqrt n"
interval_subset: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "CLF cl f"   and "a \<in> pset cl"   and "b \<in> pset cl" shows "interval (order cl) a b \<subseteq> pset cl"
exits_def: fixes P :: "instr list" shows "exits P = succs P 0 - {0..<size P}"
simps_1: fixes s :: "trm" shows "ctxt_subst \<diamond> s = s"
exhaust_sel: fixes option :: "'d option" obtains "option = None"   | "option = Some (the option)"
simps: fixes C :: "'a literal multiset" shows "production C = {A. C \<in> N \<and> C \<noteq> {#} \<and> Max_mset C = Pos A \<and> \<not> \<Union> (production ` {D. D < C}) \<Turnstile> C \<and> S C = {#}}"
ex2: assumes "\<exists>v\<in>X. c v \<noteq> (0::'a)" shows "False"
dis_rews_3: shows "is_A\<cdot>A = TT"
simps: fixes i :: "nat"   and n :: "nat"   and dmusa :: "int iarray iarray"   and fsa :: "int vec iarray"   and gsa :: "int vec iarray" shows "gso_array.gsos' i n dmusa fsa gsa = (if n \<le> i then gsa else gso_array.gsos' (i + 1) n dmusa fsa (iarray_append gsa (gso_array.gso' i fsa gsa dmusa)))"
dij_def: fixes i :: "nat"   and j :: "nat" shows "dij i j = lcm (a ! i) (b ! j) div a ! i"
prikey_mpair_snd: fixes Y :: "msg"   and K :: "nat"   and s :: "(agent \<times> msg) set" assumes "(Spy, \<lbrace>Y, AgrKey (PriK K)\<rbrace>) \<in> s"   and "insert (Spy, AgrKey (PubK {})) (insert (Spy, AgrKey (PubK {Rev_PriK})) ({Spy} \<times> ((\<lambda>x. EncKey (SigK x)) ` bad_sigk \<union> (\<lambda>x. AgrKey (PriK (Sec_PriK x))) ` bad_sec_prik \<union> (\<lambda>x. AgrKey (PriK (Tok_PriK x))) ` bad_tok_prik \<union> range (\<lambda>x. EncKey (VerK x)) \<union> range (\<lambda>x. AgrKey (PubK {Sec_PriK x, Rev_PriK})) \<union> range (\<lambda>x. AgrKey (PubK {Tok_PriK x})) \<union> range (\<lambda>(n, m). AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK})) \<union> range (\<lambda>(n, m). Hash (AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK}))) \<union> range (\<lambda>x. \<langle>x, AgrKey (PubK {Tok_PriK x})\<rangle>) \<union> {\<langle>n, AgrKey (PriK (Tok_PriK n))\<rangle> |n. n \<in> bad_tok_prik} \<union> {\<langle>n, AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK})\<rangle> |n m. (n, m) \<in> bad_id} \<union> {\<langle>n, Hash (AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK}))\<rangle> |n m. (n, m) \<in> bad_id}))) \<Turnstile> s" shows "(Spy, AgrKey (PriK K)) \<in> s"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "rotate1 (x # xs) = xs \<cdot> [x]"
simps_3: fixes x5 :: "nat \<times> char list list"   and y5 :: "nat \<times> char list list" shows "(Ref x5 = Ref y5) = (x5 = y5)"
mem_simps_7: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
sopen_FV: fixes n :: "nat"   and s :: "sterm"   and p :: "sterm"   and t :: "sterm" shows "FV ({n \<rightarrow> [s,p]} t) \<subseteq> FV t \<union> FV s \<union> FV p"
de_Morgan_disj: fixes x :: "'a"   and y :: "'a" shows "\<sim> (x \<^bold>\<squnion> y) = \<sim> x \<^bold>\<sqinter> \<sim> y"
subsetD: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A \<subseteq> B"   and "s2p A c" shows "s2p B c"
permutations_of_multiset_doubleton: fixes x :: "'a"   and y :: "'a" shows "permutations_of_multiset {#x, y#} = {[x, y], [y, x]}"
inext_UNIV: fixes n :: "nat" shows "inext n UNIV = Suc n"
is_semifunctorI: fixes \<alpha> :: "V"   and \<FF> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<FF>"   and "semicategory \<alpha> \<AA>"   and "semicategory \<alpha> \<BB>"   and "vcard \<FF> = 4\<^sub>\<nat>"   and "smcf_dghm \<FF> : smc_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> smc_dg \<BB>"   and "\<FF>\<lparr>HomDom\<rparr> = \<AA>"   and "\<FF>\<lparr>HomCod\<rparr> = \<BB>"   and "\<And>b c g a f. \<lbrakk>g : b \<mapsto>\<^bsub>\<AA>\<^esub> c; f : a \<mapsto>\<^bsub>\<AA>\<^esub> b\<rbrakk> \<Longrightarrow> \<FF>\<lparr>ArrMap\<rparr>\<lparr>g \<circ>\<^sub>A\<^bsub>\<AA>\<^esub> f\<rparr> = \<FF>\<lparr>ArrMap\<rparr>\<lparr>g\<rparr> \<circ>\<^sub>A\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
assms_1: shows "x \<in>\<^sub>\<circ> set A"
equiv_subst_ext: fixes \<Gamma> :: "(name \<times> ty) list"   and \<Gamma>' :: "(name \<times> ty) list"   and \<theta> :: "(name \<times> trm) list"   and \<theta>' :: "(name \<times> trm) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and x :: "name" assumes "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"   and "\<Gamma>' \<turnstile> s is t : T"   and "x \<sharp> \<Gamma>" shows "\<Gamma>' \<turnstile> (x, s) # \<theta> is (x, t) # \<theta>' over (x, T) # \<Gamma>"
ars1: fixes x :: "'a" shows "x \<cdot> ar x = (zero_class.zero::'a)"
set_add_def: fixes G :: "('e, 'f) ring_scheme" shows "(<+>\<^bsub>G\<^esub>) = (<#>\<^bsub>add_monoid G\<^esub>)"
IsoIsMor: fixes f :: "'b" assumes "ciso f" shows "f \<in> mor"
term_to_nterm_mono: fixes \<Gamma> :: "name list"   and x :: "Term_Class.term" shows "mono_state (term_to_nterm \<Gamma> x)"
K_gt_0: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat" assumes "grover_state_sig n f dims R K" shows "0 < K"
project_unless2: fixes G :: "'c program"   and C :: "'c set"   and A :: "'a set"   and B :: "'a set" assumes "G \<in> stable C"   and "project h C G \<in> project_set h C \<inter> A unless B" shows "G \<in> C \<inter> extend_set h A unless extend_set h B"
cases: fixes a :: "'a set \<Rightarrow> 'a set" assumes "CK_nf_pos a" obtains "a = id"   | "a = I"   | "a = K"   | "a = I \<circ> K"   | "a = K \<circ> I"   | "a = I \<circ> K \<circ> I"   | "a = K \<circ> I \<circ> K"
assms_1: shows "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
map_raw_def: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> ('e, 'd) step"   and s :: "'d" shows "map_raw f g s = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip s' | Yield a s' \<Rightarrow> Yield (f a) s')"
or_foldl_sem_conv: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'c, 'a) interp"   and \<Gamma> :: "('b, 'c, 'a) formula list" assumes "\<nu> \<in> fml_sem I (foldr (||) \<Gamma> FF)" shows "\<exists>\<phi>. \<nu> \<in> fml_sem I \<phi> \<and> List.member \<Gamma> \<phi>"
assms_4: shows "\<forall>E. Coplanar A B C E \<longrightarrow> Per E D' P"
Inf_aform_le_Sup_aform: fixes X :: "'a \<times> 'a pdevs" shows "Inf_aform X \<le> Sup_aform X"
MWLsSteps_det_cases_5: fixes \<iota> :: "nat"   and b :: "'exp"   and c :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (while\<^bsub>\<iota>\<^esub> b do c od) m \<alpha> p m'" obtains "\<alpha> = []" and "p = Some (c;while\<^bsub>\<iota>\<^esub> b do c od)" and "m' = m" and "BMap (E b m)"   | "\<alpha> = []" and "p = None" and "m' = m" and "\<not> BMap (E b m)"
map1_rel_witness_option: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" assumes "rel_option A x y" shows "map_option fst (rel_witness_option (x, y)) = x"
assms: fixes p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "component_of_term ` keys p \<subseteq> component_of_term ` Keys B" shows "p \<in> pmdl B"
chain_homotopic_simplicial_subdivision3: fixes p :: "nat"   and q :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p (standard_simplex q) c" shows "chain_boundary (Suc p) (subd p c) = simplicial_subdivision p c - c - subd (p - Suc 0) (chain_boundary p c)"
simps: fixes i :: "nat" shows "local.h i = (if i \<le> 1 then 1 else if i = 2 then lead_coeff (local.F 2) ^ (degree (local.F 1) - degree (local.F 2)) else if local.F i = 0 then 1 else lead_coeff (local.F i) ^ (degree (local.F (i - 1)) - degree (local.F i)) / local.h (i - 1) ^ (degree (local.F (i - 1)) - degree (local.F i) - 1))"
to_int_poly_i_def: fixes ops :: "'a arith_ops_record"   and f :: "'a list" shows "to_int_poly_i ops f = poly_of_list (map (arith_ops_record.to_int ops) f)"
dg_Set_components_2: fixes \<alpha> :: "V" shows "dg_Set \<alpha>\<lparr>Arr\<rparr> = ZFC_in_HOL.set {T. arr_Set \<alpha> T}"
single_step: fixes s :: "'b sparc_state" assumes "good_context s" shows "NEXT s = Some (snd (fst (execute_instruction () s)))"
ipurge_fail_aux_t_inv_2_def: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list"   and X :: "'c set"   and Y :: "('c, 'b) ipurge_rec" shows "ipurge_fail_aux_t_inv_2 I D U xs X Y \<equiv> if ListOp Y = None then List Y = xs \<and> Set Y = X else ipurge_ref_aux_less I D U (List Y) (Set Y) = ipurge_ref_aux I D U xs X"
ideI: fixes a :: "'a" assumes "right a a"   and "ide a" shows "S.ide a"
nonempty_path_components_of: fixes C :: "'b set"   and X :: "'b topology" assumes "C \<in> path_components_of X" shows "C \<noteq> {}"
\<theta>_3: shows "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> range_vars \<theta> = {}"
endofunctor_H\<^sub>R: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "weak_composition V H"   and "partial_magma.ide V f"   and "H f f \<noteq> partial_magma.null V" shows "endofunctor (subcategory.comp V (weak_composition.right V H f)) (weak_composition.H\<^sub>R H f)"
assms_6: shows "\<not> distinct [x, y, z, u, v] \<or> poincare_collinear {x, y, z}"
compose_surj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "surj_to f A B"   and "g \<in> B \<rightarrow> C"   and "surj_to g B C" shows "surj_to (compose A g f) A C"
coprime_mult_cross_dvd_1: fixes p :: "'a"   and q :: "'a"   and p' :: "'a"   and q' :: "'a" assumes "comm_monoid_mult_class.coprime p q"   and "p' * p = q' * q" shows "p dvd q'"
closed_FG: fixes \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltl" assumes "closed \<G> w"   and "G \<psi> \<in> \<G>" shows "w \<Turnstile> F G \<psi>"
subs: shows "interp C \<subseteq> \<Union> (production ` CC)"
simp_def: shows "simp \<equiv> \<lambda>q i. oriented_simplex (Suc q) (\<lambda>j. if j \<le> i then vv j else ww (j - 1))"
simps_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and at :: "('d, 'f) atm" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.gL (Pos at) = Pos (ProblemIkTpartG.gA at)"
true_clss_mono_strong: fixes DD :: "'b literal multiset set"   and CC :: "'b literal multiset set"   and I :: "'b set" assumes "\<forall>D\<in>DD. \<exists>C\<in>CC. C \<subseteq># D"   and "I \<TTurnstile>s CC" shows "I \<TTurnstile>s DD"
cBang: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a" assumes "P \<parallel> !P \<longmapsto> a<\<nu>x> \<prec> P'"   and "\<And>C. F C (P \<parallel> !P) a x P'" shows "F C (!P) a x P'"
simps: fixes G :: "('c \<times> ('c, 'd) RHS) list"   and T :: "nat \<times> nat \<Rightarrow> 'c list"   and len :: "nat"   and i :: "nat"   and j :: "nat" shows "main G T len i j = (let T' = T((i, j) := inner G T i 1 j) in if i + j < len then main G T' len (i + 1) j else if j < len then main G T' len 0 (j + 1) else T')"
isPath_tail: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and e :: "nat \<times> nat"   and v :: "nat" shows "Graph.isPath c u (p @ [e]) v = (Graph.isPath c u p (fst e) \<and> e \<in> Graph.E c \<and> snd e = v)"
Var: fixes l :: "sfields"   and V :: "char list"   and v :: "val"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap" assumes "l V = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>Var V,(h, l, sh)\<rangle> \<Rightarrow> \<langle>Val v,(h, l, sh)\<rangle>"
poly_gcd_eq_0I: fixes p :: "'b poly"   and x :: "'b"   and q :: "'b poly" assumes "poly p x = (0::'b)"   and "poly q x = (0::'b)" shows "poly (gcd p q) x = (0::'b)"
not_stable_on_cases: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "\<not> Contracts.stable_on Pd Ch ds X" obtains "\<not> Contracts.individually_rational_on Pd Ch ds X"   | "\<not> Contracts.stable_no_blocking_on Pd Ch ds X"
d_indistinguishable_def: fixes E :: "'c \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and DA :: "'d \<Rightarrow> 'f"   and d :: "'f"   and e1 :: "'c"   and e2 :: "'c" shows "Strongly_Secure_Programs.d_indistinguishable E DA d e1 e2 \<equiv> \<forall>m m'. Strong_Security.d_equal DA d m m' \<longrightarrow> E e1 m = E e2 m'"
z: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<in> carrier_mat nr nc"
abc_comp_second_steps_eq: fixes lm :: "nat list"   and B :: "abc_inst list"   and n :: "nat"   and sa :: "nat"   and lm' :: "nat list"   and A :: "abc_inst list" assumes "abc_steps_l (0, lm) B n = (sa, lm')" shows "abc_steps_l (length A, lm) (A [+] B) n = (sa + length A, lm')"
degree_list_eq_zeroD: fixes xs :: "(nat \<times> 'd) list"   and i :: "nat" assumes "degree_list xs = 0" shows "the_default (0::'d) (map_of xs i) = (0::'d)"
QReachableTrans: fixes cfg2 :: "('p, 'v, 's) configuration"   and Q :: "'p set"   and cfg3 :: "('p, 'v, 's) configuration"   and cfg1 :: "('p, 'v, 's) configuration" assumes "qReachable cfg2 Q cfg3"   and "qReachable cfg1 Q cfg2" shows "qReachable cfg1 Q cfg3"
extension: fixes G :: "('a, 'b) monoid_scheme"   and K :: "'a set"   and H :: "'a set" assumes "solvable_seq G K"   and "K \<lhd> G\<lparr>carrier := H\<rparr>"   and "subgroup H G"   and "comm_group (G\<lparr>carrier := H\<rparr> Mod K)" shows "solvable_seq G H"
IH: shows "(\<lambda>n. emeasure M {x \<in> (T ^^ n) --` A. local_time A n x < k}) \<longlonglongrightarrow> 0"
accessible_on_imp_wfp_on: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>A. accessible_on P A x" shows "wfp_on P A"
OrdP_set_max_lemma: fixes j :: "name"   and i :: "name"   and k :: "name" assumes "atom j \<sharp> i"   and "atom k \<sharp> (i, j)" shows "{} \<turnstile> Neg (Var i EQ Zero) AND All2 j (Var i) (OrdP (Var j)) IMP SyntaxN.Ex j (Var j IN Var i AND All2 k (Var i) (Var k SUBS Var j))"
verts_add_arc_conv: fixes aa :: "'a \<times> nat \<times> 'a"   and a :: "'a \<times> nat \<times> 'a" shows "verts (H.add_arc aa a) = verts (add_arc aa) \<union> {tail G a, head G a}"
invrules_4: shows "\<forall>x>j. T x = 0"
emeasure_uniform_count_measure: fixes A :: "'c set"   and X :: "'c set" assumes "finite A"   and "X \<subseteq> A" shows "emeasure (uniform_count_measure A) X = ennreal (real (card X) / real (card A))"
diff_strict_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b" shows "a - c < b - c"
acyclic_4a_implies_3a: fixes x :: "'a" assumes "acyclic_4a x" shows "acyclic_3a x"
injections_def: fixes X :: "'a set"   and Y :: "'b set" shows "injections X Y = {R. Domain R = X \<and> Range R \<subseteq> Y \<and> runiq R \<and> runiq (R\<inverse>)}"
distinct_zipI2: fixes ys :: "'b list"   and xs :: "'c list" assumes "distinct ys" shows "distinct (zip xs ys)"
IH: shows "set (map fst (insertions ops)) \<subseteq> set (map fst ops)"