hcju/isabelleps huggingface dataset:
RS1End: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" assumes "x \<noteq> DenyAll" shows "removeShadowRules1 (xs @ [x]) = removeShadowRules1 xs @ [x]"
comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
rsided_bres: fixes x :: "'a" shows "rsd x = (\<top> \<le> x \<rightarrow> x)"
injective: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A"   and "x \<noteq> y" shows "f x \<noteq> f y"
inject_2: fixes x2 :: "('a, 'b) Term.term"   and y2 :: "('a, 'b) Term.term" shows "(Decomp x2 = Decomp y2) = (x2 = y2)"
c_graph_lm_1: fixes x :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "c_pair x y \<in> c_graph f" shows "y = f x"
simps_1: fixes n :: "nat" shows "paramst (Var n) = {}"
satisfy_equation_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and subset_list :: "(nat list \<times> nat list) list"   and sign_list :: "rat list list" shows "satisfy_equation_R p qs subset_list sign_list = (M_mat_R sign_list subset_list *\<^sub>v w_vec_R p qs sign_list = v_vec_R p qs subset_list)"
run_return_nondet: fixes return :: "'f \<Rightarrow> 'd"   and single :: "'e \<Rightarrow> 'f"   and x :: "'e" shows "run_nondet (nondetM_base.return_nondet return single x) = return (single x)"
nAct_same_not_active: fixes c :: "'id"   and n' :: "enat"   and t :: "nat \<Rightarrow> cnf"   and n :: "enat" assumes "\<langle>c #\<^bsub>n'\<^esub>inf_llist t\<rangle> = \<langle>c #\<^bsub>n\<^esub>inf_llist t\<rangle>" shows "\<forall>k. n \<le> enat k \<longrightarrow> enat k < n' \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>t k\<^esub>"
three_fixed_points_01inf: fixes M :: "moebius" assumes "moebius_pt M 0\<^sub>h = 0\<^sub>h"   and "moebius_pt M 1\<^sub>h = 1\<^sub>h"   and "moebius_pt M \<infinity>\<^sub>h = \<infinity>\<^sub>h" shows "M = id_moebius"
simps_2: fixes i :: "nat" shows "hadamard_n (Suc i) = hadamard_n i;; Utrans (tensor_P (hadamard_on_i i) (1\<^sub>m K))"
deg_leE: fixes p :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "p \<in> carrier P"   and "deg R p < k" shows "p k = \<zero>"
mem_ord_of_nat_iff: fixes x :: "V"   and n :: "nat" shows "(x \<in>\<^sub>\<circ> n\<^sub>\<nat>) = (\<exists>m<n. x = m\<^sub>\<nat>)"
ev_alt_def: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and w :: "'b stream" shows "ev \<phi> w = (\<exists>u v. w = u @- v \<and> \<phi> v)"
h_def: shows "h = (if sum_list (filter (coll 0 x) (x # xs)) \<noteq> 0 then rec_nat f' (\<lambda>i _. g' i) else g')"
right_unique_rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_unique R" shows "right_unique (stream_all2 R)"
intersect_def: fixes xs :: "'b odlist"   and ys :: "'b odlist" shows "intersect xs ys = ODList (intersection (toList xs) (toList ys))"
nested_propos_5: fixes \<phi> :: "'b ltl"   and \<nu> :: "'b set" shows "nested_propos (af_G_letter_opt \<phi> \<nu>) \<subseteq> nested_propos \<phi>"
set_pmf_eq: fixes M :: "'a pmf" shows "set_pmf M = {x. pmf M x \<noteq> 0}"
term_variants_pred_const: fixes b :: "'e"   and P :: "'e \<Rightarrow> 'e list"   and a :: "'e" assumes "b \<in> set (P a)" shows "term_variants_pred P (Fun a []) (Fun b [])"
elims: fixes x :: "('a, 'c) Term.term"   and xa :: "'c \<Rightarrow> ('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" assumes "x \<cdot> xa = y" obtains xb :: "'c" and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" where "x = Var xb" and "xa = \<sigma>__" and "y = \<sigma>__ xb"   | f :: "'a" and ss :: "('a, 'c) Term.term list" and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" where "x = Fun f ss" and "xa = \<sigma>" and "y = Fun f (ss \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<sigma>)"
\<theta>_simps_3: shows "trg \<theta> = src r"
monoid_epimorphism_axioms: shows "monoid_epimorphism \<eta> G (\<cdot>) \<one> G' (\<cdot>') \<one>'"
retraction_strict: fixes f :: "'b \<rightarrow> 'a"   and g :: "'a \<rightarrow> 'b" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x" shows "f\<cdot>\<bottom> = \<bottom>"
l_new_document_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_new_document_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs"
simps_6: fixes \<rho> :: "(nat \<times> exp) list"   and e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp" shows "psubst \<rho> (EIf e1 e2 e3) = EIf (psubst \<rho> e1) (psubst \<rho> e2) (psubst \<rho> e3)"
stable_no_blocking_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Singleton_for_d.stable_no_blocking_on ds X = (\<forall>h X'. \<not> Singleton_for_d.blocking_on ds X h X')"
exists_wt_sym: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and \<zeta> :: "('b, 'a) hd" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "\<exists>f\<in>ground_heads.ground_heads ground_heads_var \<zeta>. kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym (Hd \<zeta>) = wt_sym f + the_enat (enat \<delta> * arity_sym f)"
even_signed_take_bit_iff: fixes m :: "nat"   and a :: "'a" shows "even (signed_take_bit m a) = even a"
map_poly_hom_smult: fixes c :: "real"   and p :: "real poly" shows "map_poly of_real (smult c p) = smult (of_real c) (map_poly of_real p)"
open_right: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.linorder_topology less_eq less open"   and "open S"   and "x \<in> S"   and "less x y" shows "\<exists>b. less x b \<and> ord.atLeastLessThan less_eq less x b \<subseteq> S"
sum_in_Bounded_vecI: fixes x :: "'b vec"   and B1 :: "'b"   and y :: "'b vec"   and B2 :: "'b"   and n :: "nat" assumes "x \<in> Bounded_vec B1"   and "y \<in> Bounded_vec B2"   and "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "x + y \<in> Bounded_vec (B1 + B2)"
parent_child_rel_child_in_heap: fixes h :: "(_) heap"   and parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptr parent"   and "(parent, child_ptr) \<in> Shadow_DOM.parent_child_rel h" shows "child_ptr |\<in>| object_ptr_kinds h"
that: fixes as'' :: "'edge list"   and nx :: "'node" assumes "as = as' @ as''"   and "sourcenode a -{V}a # as'\<rightarrow>\<^sub>d\<^sub>d nx"   and "nx -as''\<rightarrow>\<^sub>d* n'" shows "thesis"
independent_UnI: fixes A :: "'b set"   and B :: "'b set" assumes "local.independent A"   and "local.independent B"   and "local.span A \<inter> local.span B = {0::'b}" shows "local.independent (A \<union> B)"
le_unat_uoi: fixes y :: "nat"   and z :: "'b word" assumes "y \<le> unat z" shows "unat (word_of_nat y) = y"
asset_ii_init: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset n, \<lbrace>Num 2, PubKey A\<rbrace>) \<in> s" shows "PriKey A \<notin> spied s\<^sub>0"
when_distrib: fixes b :: "bool"   and f :: "'c \<Rightarrow> 'b"   and a :: "'c" assumes "\<not> b \<Longrightarrow> f (0::'c) = (0::'b)" shows "f (a when b) = (f a when b)"
has_field_is_class: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "P \<turnstile> C has M:T (fm) in D" shows "is_class P C"
des2_is_proper: assumes "\<B>' \<noteq> {#}" shows "proper_design \<V>' \<B>'"
fix_def: shows "fix = (\<Lambda> F. \<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
DJ_qe: fixes qe :: "fm \<Rightarrow> fm" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bs (qe p) = Ifm bs (E p)" shows "\<forall>bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)"
subspace_UNIV: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "module scale" shows "module.subspace scale UNIV"
res_def: fixes key_gen :: "'h spmf" shows "key.res key_gen = RES (key.key_oracle key_gen \<oplus>\<^sub>O key.key_oracle key_gen) None"
awalk_not_distinct_decomp: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v"   and "\<not> distinct (awalk_verts u p)" shows "\<exists>q r s. p = q @ r @ s \<and> distinct (awalk_verts u q) \<and> 0 < length r \<and> (\<exists>w. awalk u q w \<and> awalk w r w \<and> awalk w s v)"
absorbent: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and x :: "'c"   and y :: "'c" assumes "pr_ideal R I addition multiplication zero unit"   and "x \<in> R"   and "y \<in> R"   and "multiplication x y \<in> I" shows "x \<in> I \<or> y \<in> I"
And: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula"   and \<psi> :: "'a mformula"   and \<psi>' :: "'a MFOTL.formula"   and pos :: "bool"   and \<chi> :: "'a MFOTL.formula"   and buf :: "'a option list set list \<times> 'a option list set list" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j n R \<psi> \<psi>'"   and "if pos then \<chi> = MFOTL.And \<phi>' \<psi>' \<and> \<not> (safe_formula (formula.Neg \<psi>') \<and> fv \<psi>' \<subseteq> fv \<phi>') else \<chi> = MFOTL.And_Not \<phi>' \<psi>' \<and> safe_formula \<psi>' \<and> fv \<psi>' \<subseteq> fv \<phi>'"   and "wf_mbuf2' \<sigma> j n R \<phi>' \<psi>' buf" shows "wf_mformula \<sigma> j n R (MAnd \<phi> pos \<psi> buf) \<chi>"
induct: fixes x1 :: "string_binop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'b OCL_Types.type"   and x4 :: "'c OCL_Types.type"   and P :: "string_binop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'b OCL_Types.type \<Rightarrow> 'c OCL_Types.type \<Rightarrow> bool" assumes "string_binop_type x1 x2 x3 x4"   and "P ConcatOp String[1] String[1] String[1]"   and "P EqualsIgnoreCaseOp String[1] String[1] Boolean[1]"   and "P string_binop.LessOp String[1] String[1] Boolean[1]"   and "P string_binop.LessEqOp String[1] String[1] Boolean[1]"   and "P string_binop.GreaterOp String[1] String[1] Boolean[1]"   and "P string_binop.GreaterEqOp String[1] String[1] Boolean[1]"   and "P IndexOfOp String[1] String[1] Integer[1]"   and "\<And>\<tau>. \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1] \<Longrightarrow> P AtOp String[1] \<tau> String[1]" shows "P x1 x2 x3 x4"
boundedTreeInduction': fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool"   and m :: "nat" assumes "fans subs"   and "\<forall>delta. \<not> terminal subs delta \<longrightarrow> (\<forall>sigma\<in>subs delta. P sigma) \<longrightarrow> P delta" shows "\<forall>Gamma. boundedBy m (tree subs Gamma) \<longrightarrow> founded subs P (tree subs Gamma) \<longrightarrow> P Gamma"
irreducible_field: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "comm_semiring_1.irreducible times one zero x = False"
pochhammer_0: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a 0 = one"
isGuard_not_satL_intT: fixes l :: "('fsym, 'psym) lit"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" assumes "wtL l"   and "\<not> I.satL \<xi> l"   and "isGuard x l"   and "I.wtE \<xi>" shows "Ik_intT (tpOfV x) (\<xi> x)"
Q: fixes f :: "'c \<Rightarrow> 'a" assumes "continuous_on U f"   and "f ` U \<subseteq> s"   and "P f" shows "Q (h \<circ> f)"
of_int_of_nat_eq: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (int n) = semiring_1.of_nat one plus zero n"
M1_def: shows "M1 = mat K K (\<lambda>(i, j). if i = j \<and> i < R then 1 else 0)"
distinct_37: shows "ToBooleanOp \<noteq> string_unop.ToIntegerOp"
choice_cases_def: fixes t1 :: "transition"   and t2 :: "transition" shows "choice_cases t1 t2 = (if \<exists>(x, y)\<in>set (List.product (Guards t1) (Guards t2)). mutex x y then False else if Guards t1 = Guards t2 then satisfiable (fold gAnd (rev (Guards t1)) (Bc True)) else satisfiable (fold gAnd (rev (Guards t1 @ Guards t2)) (Bc True)))"
cl_latticeof: fixes r :: "('b \<times> 'b) set"   and X :: "'b set" assumes "refl r"   and "trans r" shows "cl (latticeof r) X = {t. \<exists>s. s \<in> X \<and> (s, t) \<in> r}"
simps: 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 a1 :: "char list"   and a2 :: "char list" shows "subclsSp P a1 a2 = (\<exists>C Bs rest D. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(Bs, rest)\<rfloor> \<and> Shares D \<in> set Bs)"
PartNum_def: shows "PartNum = length \<circ> Rep_dataspace"
set_tag_name_get_tag_name_different_pointers: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and ptr :: "(_) element_ptr"   and ptr' :: "(_) element_ptr"   and w :: "((_) heap, exception, unit) prog"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "l_set_tag_name_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"   and "ptr \<noteq> ptr'"   and "w \<in> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name_locs ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name_locs ptr'" shows "r h h'"
simplify_f: fixes p :: "'a"   and e :: "'a"   and f :: "'a" assumes "regular p"   and "regular e" shows "f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> f \<sqinter> - e\<^sup>T \<sqinter> p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> - p)\<^sup>T \<squnion> e\<^sup>T \<squnion> e = f \<squnion> f\<^sup>T \<squnion> e \<squnion> e\<^sup>T"
elims_3: fixes x :: "nat \<times> nat"   and xa :: "nat \<times> nat" assumes "\<not> x \<approx> xa" obtains xb :: "nat" and y :: "nat" and u :: "nat" and v :: "nat" where "x = (xb, y)" and "xa = (u, v)" and "xb + v \<noteq> u + y"
join_def: fixes x :: "int"   and y :: "int" shows "int.join x y = (THE sup. int.is_sup x y sup)"
add_suc: fixes m :: "nat"   and ns :: "nat list" shows "foldl (+) (Suc m) ns = Suc (foldl (+) m ns)"
has_codomain_iff_arr: fixes f :: "'a \<times> 'a" shows "(CC.codomains f \<noteq> {}) = CC.arr f"
fresh_3: fixes a :: "W.var"   and x2 :: "ty"   and x1 :: "ty" shows "a \<sharp> x2\<rightarrow>x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
arr_char: fixes f :: "'i" shows "J.arr f = (f \<in> I)"
poly_rel_inj: fixes x :: "'i list"   and y :: "'a mod_ring poly"   and z :: "'a mod_ring poly" assumes "poly_rel x y"   and "poly_rel x z" shows "y = z"
tendsto_matrix_vector_mult: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "('a, 'b) Finite_Cartesian_Product.vec"   and S :: "('a, 'b) Finite_Cartesian_Product.vec set" shows "((*v) A \<longlongrightarrow> A *v v) (at v within S)"
sdp_repeat: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "sub_distrib_pconj a"   and "healthy (wp a)"   and "nearly_healthy (wlp a)" shows "sub_distrib_pconj (repeat n a)"
prod_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less zero (f a)" shows "less zero (comm_monoid_mult.prod times one f A)"
assms_1: shows "\<Gamma> (Fun f T) = Fun g (map \<Gamma> S)"
v_nS12: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg"   and "state cfg \<notin> S1"   and "state cfg \<notin> S2" shows "v cfg = 0"
second_vertices_new_path: fixes xs :: "'a list" assumes "hd (tl xs) \<notin> Q'.second_vertices" shows "xs \<notin> Q"
rel_fun_mono: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "(X ===> A) f g"   and "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "(Y ===> B) f g"
id_fcomp: fixes g :: "'a \<Rightarrow> 'b" shows "fcomp id g = g"
R_def: shows "R = ((I, {''l'', ''r''}), S, E, {''l'', ''r''}, Es)"
append_eq_conv_conj: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(xs @ ys = zs) = (xs = List.take (length xs) zs \<and> ys = drop (length xs) zs)"
one_add_one: shows "II \<sqinter> II = numeral.numeral II (\<sqinter>) (num.Bit0 num.One)"
seqE: fixes g :: "'a"   and f :: "'a" assumes "Right_a.seq g f" obtains "Right_a.arr f" and "Right_a.arr g" and "Right_a.dom g = Right_a.cod f"
prems: shows "\<forall>x\<in>vars\<^sub>l (Neg p ts). \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
vsv_ex1_app2: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(r\<lparr>a\<rparr> = b) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
not_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "not one = uminus (numeral.numeral one plus (num.Bit0 num.One))"
compose_parametric: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "(rel_dds A B ===> rel_dds B C ===> rel_dds A C) (\<bullet>) (\<bullet>)"
sampled_stutter_equiv: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "stutter_sampler f \<sigma>" shows "\<sigma> \<circ> f \<approx> \<sigma>"
unstream_stream: fixes xs :: "'a LList" shows "unstream\<cdot>(stream\<cdot>xs) = xs"
gSwapGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSwapGOp MOD \<equiv> \<forall>zs z1 z2 delta inp' inp binp' binp. wlsInp delta inp' \<and> gWlsInp MOD delta inp \<and> wlsBinp delta binp' \<and> gWlsBinp MOD delta binp \<longrightarrow> gSwap MOD zs z1 z2 (Op delta inp' binp') (gOp MOD delta inp' inp binp' binp) = gOp MOD delta (inp' %[z1 \<and> z2]_zs) (gSwapInp MOD zs z1 z2 inp' inp) (binp' %%[z1 \<and> z2]_zs) (gSwapBinp MOD zs z1 z2 binp' binp)"
IH2: assumes "\<exists>pn. p2 \<in> subterms (\<Gamma> pn)"   and "q \<in> dterms \<Gamma> p2" shows "\<exists>pn. q \<in> subterms (\<Gamma> pn)"
rep_eq: fixes x :: "'a dense"   and xa :: "'a dense" shows "Rep_dense (x \<sqinter> xa) = Rep_dense x \<sqinter> Rep_dense xa"
Sup_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. less_eq a b" shows "less_eq (Sup A) (Sup B)"
fin: fixes A :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) set" assumes "finite A" shows "finite {h. (h, {}) \<in> A \<and> poly_deg h = z}"
punit_monom_mult_monomial_idx_pm_of_pm: fixes t :: "'c"   and c :: "'d"   and bs :: "('t \<Rightarrow>\<^sub>0 'e) list"   and s :: "('t \<Rightarrow>\<^sub>0 'e) \<Rightarrow>\<^sub>0 'c \<Rightarrow>\<^sub>0 'd" shows "MPoly_Type_Class.punit.monom_mult (monomial c t) 0 (idx_pm_of_pm bs s) = idx_pm_of_pm bs (MPoly_Type_Class.punit.monom_mult (monomial c t) 0 s)"
loop_def: shows "dwa.loop \<equiv> while (dwa_cond (hs_dfs_dwa \<Sigma>i post)) (dwa_step (hs_dfs_dwa \<Sigma>i post)) (dwa_initial (hs_dfs_dwa \<Sigma>i post))"
card_order_dir_image: fixes f :: "'d \<Rightarrow> 'e"   and r :: "('d \<times> 'd) set" assumes "bij f"   and "card_order r" shows "card_order (dir_image r f)"
monotone_on_def: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" shows "monotone_on X r s f \<equiv> \<forall>x y. x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> r x y \<longrightarrow> s (f x) (f y)"
match_rews_5: fixes k :: "'s \<rightarrow> 's tree \<rightarrow> 's tree \<rightarrow> 'r match" shows "match_Node\<cdot>Null\<cdot>k = Fixrec.fail"
Max_eq_iff: fixes A :: "'d DBMEntry set"   and m :: "'d DBMEntry" assumes "finite A"   and "A \<noteq> {}" shows "(linordered_monoid.Max A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<preceq> m))"
left_diff_distrib': fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b - c) * a = b * a - c * a"
distinct_remdups_id: fixes xs :: "'d list" assumes "distinct xs" shows "remdups xs = xs"
extr_insert_bad: fixes A :: "agent"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set" shows "extr (insert A bad) IK CH \<subseteq> extr bad IK CH \<union> {M. \<exists>B. Confid A B M \<in> CH \<or> Confid B A M \<in> CH \<or> Secure A B M \<in> CH \<or> Secure B A M \<in> CH}"
wf_fdecl_def: fixes P :: "'b prog" shows "wf_fdecl P \<equiv> \<lambda>(F, b, T). is_type P T"