hcju/isabelleps huggingface dataset:
mono_SEQI2b: assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (S `;` P) \<longrightarrow> Ra (S `;` P) s \<subseteq> Ra (S `;` Q) s"
det_rows_mul: fixes c :: "'b \<Rightarrow> 'a"   and a :: "'b \<Rightarrow> ('a, 'b) vec" shows "det (\<chi>i. c i *s a i) = prod c UNIV * det (\<chi>i. a i)"
fmrestrict_set_inter_img: fixes X :: "'d set"   and Y :: "'d set"   and A :: "('d, 'e) fmap set" shows "fmrestrict_set (X \<inter> Y) ` A = (fmrestrict_set X \<circ> fmrestrict_set Y) ` A"
order_iff: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subseteq># b) = (a = a \<inter># b)"
weight_simp: fixes f :: "'f"   and ts :: "('f, 'a) Term.term list" shows "weight (Fun f ts) = w (f, length ts) + sum_list (map weight (scf_list (scf (f, length ts)) ts))"
guard: fixes s :: "'s" shows "m s = 0 \<longrightarrow> \<not> G s"
zle_iff_zadd: fixes w :: "int"   and z :: "int" shows "(w \<le> z) = (\<exists>n. z = w + int n)"
vector_in_orthogonal_spanningset: fixes a :: "'a" obtains S :: "'a set" where "a \<in> S" and "pairwise orthogonal S" and "span S = UNIV"
monic_degree_one_root_condition: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (poly_ring R)" shows "UP.is_root [\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> a] b = (a = b)"
N_N': shows "dom_m N = add_mset C (dom_m N')"
subspace_def: fixes S :: "'c set" shows "m2.subspace S = ((0::'c) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. s2 c x \<in> S))"
matrix_to_iarray_bezout_matrix: fixes A :: "(('a, 'c) vec, 'b) vec"   and a :: "'b"   and b :: "'b"   and j :: "'c"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "matrix_to_iarray (bezout_matrix A a b j bezout) = bezout_matrix_iarrays (matrix_to_iarray A) (mod_type_class.to_nat a) (mod_type_class.to_nat b) (mod_type_class.to_nat j) bezout"
S_fixespointwise_fundchamber_image_int: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "fixespointwise ((\<rightarrow>) s) (C0 \<inter> s `\<rightarrow> C0)"
comm_monoid_add_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" shows "comm_monoid_add_ow U plus zero \<equiv> ab_semigroup_add_ow U plus \<and> zero_ow U zero \<and> comm_monoid_add_ow_axioms U plus zero"
card_insert_disjoint: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
base_k_steps: shows "\<forall>i<k. P (iterate i\<cdot>f\<cdot>\<bottom>)"
fmdom'_fmsubset_restrict_set: fixes X1 :: "'a set"   and X2 :: "'a set"   and s :: "('a, bool) fmap" assumes "X1 \<subseteq> X2"   and "fmdom' s = X2" shows "fmdom' (fmrestrict_set X1 s) = X1"
vlrestriction_vcomp: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<circ>\<^sub>\<circ> s) \<restriction>\<^sup>l\<^sub>\<circ> A = r \<circ>\<^sub>\<circ> s \<restriction>\<^sup>l\<^sub>\<circ> A"
cases: fixes x :: "'c list list" obtains "x = []"   | as :: "'c list" and xs :: "'c list list" where "x = as # xs"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list" assumes "\<And>f. P f []"   and "\<And>f x. P f [x]"   and "\<And>f x y. P f [x, y]"   and "\<And>f v vb va vd. P f (v # vb # va # vd)" shows "P a0 a1"
assms_1: shows "\<guillemotleft>\<phi> : g' \<Rightarrow> g\<guillemotright>"
hyp: fixes x :: "trm"   and z :: "trm" assumes "\<And>y. x \<longrightarrow>\<^sub>a y \<Longrightarrow> SNa y"   and "\<And>y. x \<longrightarrow>\<^sub>a y \<Longrightarrow> P y z"   and "\<And>u. z \<longrightarrow>\<^sub>a u \<Longrightarrow> SNa u"   and "\<And>u. z \<longrightarrow>\<^sub>a u \<Longrightarrow> P x u" shows "P x z"
last_append: fixes xs :: "'f list"   and ys :: "'f list" shows "last (xs @ ys) = (if ys = [] then last xs else last ys)"
Retr_incl: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set" assumes "PL_Indis cval indis" shows "PL_Indis.Sretr aval tval cval indis theta \<subseteq> PL_Indis.ZOretr aval tval cval indis theta"
matrix_vector_mult_add_rdistrib: fixes A :: "(('a, 'c) vec, 'b) vec"   and B :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "(A + B) *v x = A *v x + B *v x"
path_component_subset_connected_component: fixes S :: "'a set"   and x :: "'a" shows "path_component_set S x \<subseteq> connected_component_set S x"
divide_uexpr_def: fixes u :: "('c, 'd) uexpr"   and v :: "('c, 'd) uexpr" shows "u div v = bop (div) u v"
inf_greatest: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" assumes "x \<subseteq># y"   and "x \<subseteq># z" shows "x \<subseteq># y \<inter># z"
rank_transpose: shows "vec_space.rank (length signs) (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) = vec_space.rank (length (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))\<^sup>T"
l_set_child_nodes_get_shadow_root_def: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_shadow_root set_child_nodes_locs get_shadow_root_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
elims_2: fixes x :: "'a list" assumes "Lyndon x" obtains w :: "'a list" where "x = w" and "w \<noteq> \<epsilon> \<and> (\<forall>n. 0 < n \<and> n < \<^bold>|w\<^bold>| \<longrightarrow> w <lex rotate n w)"
inv_solve_right: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<in> carrier (G i)"   and "b \<in> carrier (G i)"   and "c \<in> carrier (G i)" shows "(a = b \<otimes>\<^bsub>G i\<^esub> inv\<^bsub>G i\<^esub> c) = (b = a \<otimes>\<^bsub>G i\<^esub> c)"
assms: shows "timpls_transformable_to_pred A (Var x) t"
split_face_empty_ram2_ram1_in_f12': fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat" assumes "pre_split_face oldF ram1 ram2 []" shows "(ram2, ram1) \<in> \<E> (fst (split_face oldF ram1 ram2 []))"
eqD_1: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and x :: "'b"   and y :: "'b" assumes "class.Eq_eq eq"   and "eq\<cdot>x\<cdot>y = TT" shows "x = y"
s0_trans_def: shows "s0_trans \<equiv> (\<Union> (range s0_add_secret) \<union> \<Union> (range s0_learn))\<^sup>="
call_intra_edge_False: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "intra_kind (kind a')" shows "kind a' = (\<lambda>cf. False)\<^sub>\<surd>"
bind_lunit: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a option" shows "Option.bind (Some x) f = f x"
bij: shows "bij_betw (ordermap U r) U (elts \<beta>)"
set_take_if_index: fixes xs :: "'b list"   and x :: "'b"   and i :: "nat" assumes "index xs x < i"   and "i \<le> length xs" shows "x \<in> set (take i xs)"
sub_top_def: shows "top = (THE x. \<forall>y. x = - y \<squnion> - - y)"
qSwapBinp_preserves_qGoodBinp: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGoodBinp qbinp" shows "qGoodBinp (qSwapBinp xs x1 x2 qbinp)"
rreqs_increase: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). rreqs \<xi> \<subseteq> rreqs \<xi>')"
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"   and "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf" shows "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
Ig_nonzero: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I"   and "I \<noteq> {\<zero>\<^bsub>Vr K v\<^esub>}" shows "Ig K v I \<noteq> \<zero>"
cinner_zero_right: fixes x :: "'a" shows "\<langle>x, 0::'a\<rangle> = 0"
depth_prm: fixes \<pi> :: "'b prm"   and A :: "'b trm" shows "depth (\<pi> \<cdot> A) = depth A"
Quotient_rep_abs_eq: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c" assumes "Quotient R Abs Rep T"   and "R t t"   and "R \<le> (=)" shows "Rep (Abs t) = t"
I_def: shows "I \<equiv> \<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3\<^sub>' c (wp\<^sub>3\<^sub>' (WHILE b DO c) Q) (ps, n) else Q (ps, n))"
assms: fixes x :: "'b" shows "(0::'a) \<le> f x"
cone_zero: fixes U :: "'x set" shows "cone (0, U) = {0}"
ltln_weak_strong_stable_words_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<phi> U\<^sub>n (\<psi> or\<^sub>n G\<^sub>n \<phi>[\<G>\<F> \<phi> w]\<^sub>\<Pi>\<^sub>1)"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<tau>' f) = Cop.arr f"
bisim1AAcc1: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr expr1"   and a' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and i :: "'addr expr1" assumes "P,a,h \<turnstile> (a', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,a\<lfloor>i\<rceil>,h \<turnstile> (a'\<lfloor>i\<rceil>, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
universal_affirmation_def: fixes A :: "'b set"   and B :: "'b set" shows "A Q B \<equiv> \<forall>b\<in>B. b \<in> A"
assms_4: shows "set qs \<subseteq> {x, y}"
atLeast_iff: fixes i :: "'a Word.word"   and k :: "'a Word.word" shows "(i \<in> signed.atLeast k) = (k \<le>s i)"
linorder_ow_axioms: shows "linorder_ow U (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w)"
r: shows "path_connected S \<and> (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))"
algebraic_int_altdef_ipoly: fixes x :: "'b" shows "algebraic_int x = (\<exists>p. poly (map_poly of_int p) x = (0::'b) \<and> lead_coeff p = 1)"
wt_kil_correct: fixes P :: "jvm_prog"   and Ts :: "ty list"   and p :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and b :: "staticb"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "bytecode"   and xt :: "ex_table" assumes "start_context P Ts p C"   and "wt_kildall P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt" shows "\<exists>\<tau>s. wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s"
map_comp: fixes g :: "'n \<Rightarrow> 'm"   and f :: "'o \<Rightarrow> 'n"   and v :: "'o fset" shows "g |`| f |`| v = (g \<circ> f) |`| v"
sum_list'_def: fixes p :: "nat"   and xs :: "'a list" shows "sum_list' p xs = fold (\<lambda>a b. eucl_truncate_up p (a + b)) xs (0::'a)"
length_nths': fixes xs :: "'a list"   and I :: "nat set" shows "length (nths xs I) = card {i \<in> I. i < length xs}"
synWhileSC_def: fixes e :: "('c, 'd) Expr" shows "synWhileSC e \<equiv> \<exists>d. {} \<turnstile>\<^bsub>\<E>\<^esub> e : d \<and> (\<forall>d'. d \<le> d')"
assms_1: shows "{x \<in> space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
upred_lattice_inf: fixes A :: "'a upred set" shows "Lattice.inf \<P> A = \<Or>A"
is_unit_gauss_int_iff': fixes x :: "gauss_int" shows "is_unit x = (gauss_int_norm x = 1)"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vint_uminus" shows "-\<^sub>\<int> a = b"
r_consistent_converg: fixes r_psi :: "recf"   and e :: "nat"   and i :: "nat" assumes "recfn 2 r_psi"   and "\<forall>k<e_length e. eval r_psi [i, k] \<down>" shows "eval (r_consistent r_psi) [i, e] \<down>= (if \<forall>k<e_length e. eval r_psi [i, k] \<down>= e_nth e k then 0 else 1)"
transpos_id: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and x :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j"   and "x \<in> {k. k \<le> n} - {i, j}" shows "(\<tau>\<^bsub>i j\<^esub>) x = x"
simps_21: fixes v :: "'a rexp"   and vc :: "'a rexp"   and vb :: "'a" shows "local.le_rexp (Times v vc) (Atom vb) = False"
keys_tabulate: fixes ks :: "'c list"   and f :: "'c \<Rightarrow> 'd" shows "Mapping.keys (Mapping.tabulate ks f) = set ks"
simps_1: fixes k :: "nat"   and sub :: "'b btree" shows "root_order_up\<^sub>i k (T\<^sub>i sub) = root_order k sub"
path_cons_conv: fixes E :: "('e \<times> 'e) set"   and u :: "'e"   and u' :: "'e"   and l :: "'e list"   and w :: "'e" shows "path E u (u' # l) w = (u' = u \<and> (\<exists>v. (u, v) \<in> E \<and> path E v l w))"
advantage_def: fixes \<A> :: "(bool, '\<alpha>, '\<gamma>) gpv" shows "local.prf.advantage \<A> = \<bar>spmf (local.prf.game_0 \<A>) True - spmf (local.prf.game_1 \<A>) True\<bar>"
cases: fixes x :: "'b list" obtains "x = []"   | y :: "'b" and ys :: "'b list" where "x = y # ys"
distinct_11: fixes x3 :: "nat"   and x41 :: "dbtm"   and x42 :: "dbtm" shows "DBInd x3 \<noteq> DBEats x41 x42"
sym: fixes x :: "'a"   and y :: "'a" assumes "x \<bowtie> y"   and "x \<in> A"   and "y \<in> A" shows "y \<bowtie> x"
is_hom_GB_boundD: fixes F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and b :: "nat"   and g :: "('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "extended_ord.is_hom_GB_bound F b"   and "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "g \<in> extended_ord.punit.reduced_GB F" shows "poly_deg g \<le> b"
det_wa_wa_def: fixes DWA :: "('a, 'b) det_while_algo_scheme" shows "det_wa_wa DWA \<equiv> \<lparr>wa_cond = {s. dwa_cond DWA s}, wa_step = {(s, dwa_step DWA s) |s. True}, wa_initial = {dwa_initial DWA}, wa_invar = dwa_invar DWA\<rparr>"
intro: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d" assumes "subdivision_step G rev_G H rev_H (u, v, w) (uv, uw, vw)" shows "subdiv_step G rev_G H rev_H u v w uv uw vw"
rev_filter: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "rev (filter P xs) = filter P (rev xs)"
g_def: shows "g \<equiv> \<lambda>z. if z \<in> closure S then f z else z"
CHECK_monadic_rule_iff: fixes c :: "('c + bool) nres"   and e :: "'c"   and E :: "'c \<Rightarrow> bool"   and P :: "unit \<Rightarrow> bool" shows "(CHECK_monadic c e \<le> ESPEC E P) = (c \<le> ESPEC E (\<lambda>r. (r \<longrightarrow> P ()) \<and> (\<not> r \<longrightarrow> E e)))"
radj_def: fixes f :: "'c \<Rightarrow> 'b" shows "radj f = (\<lambda>y. \<Squnion> {x. f x \<le> y})"
app_def: fixes f :: "V"   and x :: "V" shows "f\<lparr>x\<rparr> \<equiv> THE y. \<langle>x, y\<rangle> \<in>\<^sub>\<circ> f"
uncurry_def: fixes f :: "'e \<Rightarrow> 'g \<Rightarrow> 'h" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
Gauss_Jordan_det_P_iarrays_def: fixes A :: "'d iarray iarray" shows "Gauss_Jordan_det_P_iarrays A = Gauss_Jordan_upt_k_det_P_iarrays A (ncols_iarray A - 1)"
inj_inj_on: fixes F :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj F" shows "inj_on F A"
finite_Sup: fixes X :: "'a set set" assumes "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<subseteq> z)" shows "finite (\<Union> X)"
Pair_inject: fixes a :: "'b"   and b :: "'c"   and a' :: "'b"   and b' :: "'c" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
D: fixes x :: "complex" assumes "x \<in> T" shows "cmod (f x) \<le> D"
bi_unique_alt_def2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "bi_unique R = rel_fun R (rel_fun R (=)) (=) (=)"
overlap_eq: fixes x :: "i"   and y :: "i" assumes "MM3" shows "O x y = (\<exists>z. P z x \<and> P z y)"
swapEnvDom_def: fixes xs :: "'f"   and x :: "'g"   and y :: "'g"   and rho :: "'f \<Rightarrow> 'g \<Rightarrow> 'h" shows "swapEnvDom xs x y rho \<equiv> \<lambda>zs z. rho zs (z @zs[x \<and> y]_xs)"
square_part_square_dvd: fixes n :: "'b" shows "(square_part n)\<^sup>2 dvd n"
to_nat_less_card: fixes j :: "'a" shows "to_nat j < CARD('a)"
functors_preserve_homsets: fixes A :: "'o"   and B :: "'o"   and f :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "f \<in> Hom A B" shows "F \<^bsub>\<a>\<^esub> f \<in> Hom\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A F \<^bsub>\<o>\<^esub> B"
implInsec_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and I :: "msg set" assumes "H \<subseteq> payload \<union> asym.valid \<union> range LtK \<union> Tags"   and "asym.implInsec A B M \<in> synth (analz H)" shows "asym.implInsec A B M \<in> I \<or> M \<in> synth (analz H)"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length fs_init" shows "rat_of_int (gs.Gramian_determinant fs_init k) = (\<Prod>j<k. \<parallel>gs.gso j\<parallel>\<^sup>2)"
i_take_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Down> n \<equiv> map f [0..<n]"
lincomb_insert: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier V"   and "a \<in> S \<union> {v} \<rightarrow> carrier K"   and "v \<notin> S"   and "v \<in> carrier V" shows "lincomb a (S \<union> {v}) = a v \<odot>\<^bsub>V\<^esub> v \<oplus>\<^bsub>V\<^esub> lincomb a S"
take_all_iff: fixes n :: "nat"   and xs :: "'b list" shows "(take n xs = xs) = (length xs \<le> n)"
simps_4: fixes x2 :: "sig"   and x1 :: "vname" shows "mid x2 \<noteq> fid x1"
isomorphic_implies_ide_2: fixes f :: "'a"   and g :: "'a" assumes "f \<cong>\<^sub>B g" shows "B.ide g"
DERIV_isconst_end: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "continuous_on {a..b} f"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_real_derivative 0) (at x)" shows "f b = f a"
index_mat_four_block_1: fixes i :: "nat"   and A :: "'b mat"   and D :: "'b mat"   and j :: "nat"   and B :: "'b mat"   and C :: "'b mat" assumes "i < dim_row A + dim_row D"   and "j < dim_col A + dim_col D" shows "four_block_mat A B C D $$ (i, j) = (if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A) else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))"
OclANY_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)" shows "x->any\<^sub>S\<^sub>e\<^sub>t() = (\<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> then if (\<delta> x and x->notEmpty\<^sub>S\<^sub>e\<^sub>t()) \<tau> = true \<tau> then SOME y. y \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> else null \<tau> else \<bottom>)"
know_composition1: fixes m :: "KS"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "m \<notin> specKeysSecrets P"   and "m \<notin> specKeysSecrets Q"   and "know P m"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ" shows "know PQ m"
module_hom_on_def: fixes S1 :: "'c set"   and S2 :: "'d set"   and s1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" shows "module_hom_on S1 S2 s1 s2 f \<equiv> module_on S1 s1 \<and> module_on S2 s2 \<and> module_hom_on_axioms S1 s1 s2 f"
fsubset_finsert: fixes x :: "'b"   and A :: "'b fset"   and B :: "'b fset" assumes "x |\<notin>| A" shows "(A |\<subseteq>| finsert x B) = (A |\<subseteq>| B)"
unat_arith_simps_5: fixes a :: "'b word"   and b :: "'b word" shows "unat (a + b) = (if unat a + unat b < 2 ^ LENGTH('b) then unat a + unat b else unat a + unat b - 2 ^ LENGTH('b))"
goodEnv_def: fixes rho :: "'f \<Rightarrow> 'g \<Rightarrow> ('h, 'i, 'varSort, 'var, 'j) qTerm set option" shows "goodEnv rho \<equiv> (\<forall>ys. liftAll good (rho ys)) \<and> (\<forall>ys. |{y. rho ys y \<noteq> None}| <o |UNIV|)"
path_contains_arc: fixes p :: "real \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "path p"   and "pathstart p = a"   and "pathfinish p = b"   and "a \<noteq> b" obtains q :: "real \<Rightarrow> 'b" where "arc q" and "path_image q \<subseteq> path_image p" and "pathstart q = a" and "pathfinish q = b"
set_sel_1: fixes a :: "'e list" assumes "a \<noteq> []" shows "hd a \<in> set a"
less_eq_gexp_def: fixes e1 :: "'a gexp"   and e2 :: "'a gexp" shows "e1 \<le> e2 \<equiv> e1 < e2 \<or> e1 = e2"
rtrancl3p_trans: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and bs :: "'c list"   and a' :: "'b"   and bs' :: "'c list"   and a'' :: "'b" assumes "rtrancl3p r a bs a'"   and "rtrancl3p r a' bs' a''" shows "rtrancl3p r a (bs @ bs') a''"
unstream_def: shows "unstream \<equiv> \<mu> unstream. \<Lambda> v. Fixrec.run\<cdot>(match_Stream\<cdot>v\<cdot>(\<Lambda> h s. Fixrec.succeed\<cdot>(unfold\<cdot>h\<cdot>s)))"
facet_of_imp_face_of: fixes F :: "'a set"   and S :: "'a set" assumes "F facet_of S" shows "F face_of S"
size_split_min: fixes t :: "('b \<times> nat) tree" assumes "t \<noteq> \<langle>\<rangle>" shows "size t = Suc (size (snd (split_min t)))"
symrun_interp_commute: fixes \<Gamma>\<^sub>1 :: "'a constr list"   and \<Gamma>\<^sub>2 :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>2 @ \<Gamma>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
wf_finalize: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" assumes "local.wf idx \<phi>" shows "local.wf idx (finalize idx \<phi>)"
iFROM_D: fixes x :: "nat"   and n :: "nat" assumes "x \<in> [n\<dots>]" shows "n \<le> x"
execute_parallel_operator_def: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list" shows "execute_parallel_operator s ops \<equiv> foldl (++) s (map (map_of \<circ> effect_to_assignments) ops)"
tail_add_vert: fixes u :: "'a" shows "tail (add_vert u) = tail G"
set_6: fixes x31 :: "'f"   and x32 :: "'e" shows "set2_step (Yield x31 x32) = {x32}"
orthogonal_transformation_def: fixes f :: "'c \<Rightarrow> 'd" shows "orthogonal_transformation f = (linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w))"
UN_ball_bex_simps_3: fixes A :: "'a set set"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>\<Union> A. P x) = (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
valuation_def: fixes M :: "object atom formula set" shows "valuation M \<equiv> \<lambda>x. case x of predAtm p xs \<Rightarrow> Atom (predAtm p xs) \<in> M | Eq a b \<Rightarrow> a = b"
XOR_lower: fixes x :: "int"   and y :: "int" assumes "0 \<le> x"   and "0 \<le> y" shows "0 \<le> x XOR y"
positive_weight: fixes x :: "'a" shows "0 < \<integral>\<^sup>+ x. ennreal (R x x) \<partial>count_space UNIV"
assms_8: shows "poincare_collinear {0\<^sub>h, a, i}"