hcju/isabelleps huggingface dataset:
finite_induct: fixes F :: "'e set"   and P :: "'e set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
HMA_M_def: shows "HMA_Connect.HMA_M = (\<lambda>a b. a = HMA_Connect.from_hma\<^sub>m b)"
one_sum_group: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" shows "\<one>\<^bsub>sum_group I G\<^esub> = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)"
distinct_card: fixes xs :: "'c list" assumes "cycle xs" shows "card (set xs) = length xs"
map_add_upd_left: fixes m :: "'b"   and e2 :: "'b \<Rightarrow> 'c option"   and e1 :: "'b \<Rightarrow> 'c option"   and u1 :: "'c" assumes "m \<notin> dom e2" shows "e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
lunit_in_hom: fixes a :: "'d" assumes "D.ide a" shows "\<guillemotleft>\<l>\<^sub>1[a] : \<I>\<^sub>1 \<otimes>\<^sub>D a \<rightarrow>\<^sub>D a\<guillemotright>"
child_ex_neighbour: fixes b :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat" shows "\<exists>b'. child b dir d = child b' (Grid_Point.inv dir) d"
maxR_rangeD_2: fixes x :: "'a"   and y :: "'a" assumes "maxR x y \<noteq> y" shows "maxR x y = x"
conuc_lax: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "conucleus f" shows "f x \<cdot> f y \<le> f (x \<cdot> y)"
valid_sharing: shows "valid_sharing \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
leq_mask_shift: fixes x :: "'b word"   and low_bits :: "nat"   and high_bits :: "nat" assumes "x \<le> mask (low_bits + high_bits)" shows "x >> low_bits \<le> mask high_bits"
combining_sys_satisfies_properties_helper_R: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list"   and subsets1 :: "(nat list \<times> nat list) list"   and signs1 :: "rat list list"   and matrix1 :: "rat mat"   and subsets2 :: "(nat list \<times> nat list) list"   and signs2 :: "rat list list"   and matrix2 :: "rat mat" assumes "p \<noteq> 0"   and "0 < length qs1"   and "0 < length qs2"   and "satisfies_properties_R p qs1 subsets1 signs1 matrix1"   and "satisfies_properties_R p qs2 subsets2 signs2 matrix2" shows "satisfies_properties_R p (qs1 @ qs2) (get_subsets_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_signs_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_matrix_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
hyps_1: assumes "list_all stateful_strand_step.is_Receive (unlabel A)" shows "list_all stateful_strand_step.is_Send (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
lookup_simps_3: fixes b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option"   and x :: "'h"   and xs :: "'h list" shows "lookup (Env b es) (x # xs) = (case es x of None \<Rightarrow> None | Some e \<Rightarrow> lookup e xs)"
complex_minus: fixes a :: "real"   and b :: "real" shows "- Complex a b = Complex (- a) (- b)"
kop_galois_var: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set" shows "(op\<^sub>K \<circ> f = g) = (op\<^sub>K \<circ> g = f)"
star_or_dist1: fixes A :: "assn"   and B :: "assn"   and C :: "assn" shows "(A \<or>\<^sub>A B) * C = A * C \<or>\<^sub>A B * C"
fst_gb_schema_dummy_dgrad_p_set_init: fixes d :: "'a \<Rightarrow> nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and m :: "nat"   and rc :: "nat"   and data :: "nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set" assumes "dickson_grading d"   and "struct_spec sel ap ab compl"   and "fst ` (set gs \<union> set bs) \<subseteq> dgrad_p_set d m" shows "fst ` set (fst (gb_schema_dummy (rc, data) D (ab gs [] bs data) (ap gs [] [] bs data))) \<subseteq> dgrad_p_set d m"
mst_reachable_inv: fixes w :: "'a"   and v :: "'a"   and e :: "'a"   and r :: "'a"   and t :: "'a" assumes "regular (prim_EP w v e)"   and "vector r"   and "e \<le> v * - v\<^sup>T"   and "vector v"   and "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"   and "t \<le> w"   and "t \<le> v * v\<^sup>T"   and "w * v \<le> v" shows "r\<^sup>T * w\<^sup>\<star> \<le> r\<^sup>T * prim_W w v e\<^sup>\<star>"
extensions_def: shows "M.extensions \<equiv> {L. field L \<and> M.indexed_const \<in> ring_hom M L \<and> Ball (carrier L) M.carrier_coeff \<and> (\<forall>\<P>\<in>carrier L. \<forall>P\<in>carrier (poly_ring M). \<forall>i. \<not> M.index_free \<P> (P, i) \<longrightarrow> M.indexed_var (P, i) \<in> carrier L \<and> ring.eval L (M.\<sigma> P) (M.indexed_var (P, i)) = \<zero>\<^bsub>L\<^esub>)}"
plus_assoc: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "semigroup_add_ow U plus"   and "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "plus (plus a b) c = plus a (plus b c)"
dist_decreases_closed_segment: fixes x :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "x \<in> closed_segment a b" shows "dist c x \<le> dist c a \<or> dist c x \<le> dist c b"
elims: fixes x :: "nat \<Rightarrow> nat"   and xa :: "thread"   and y :: "thread" assumes "map_ticket x xa = y" obtains f :: "nat \<Rightarrow> nat" where "x = f__" and "xa = INIT" and "y = INIT"   | f :: "nat \<Rightarrow> nat" and tk :: "nat" where "x = fa__" and "xa = WAIT tk__" and "y = WAIT (fa__ tk__)"   | f :: "nat \<Rightarrow> nat" and tk :: "nat" where "x = fb__" and "xa = HOLD tka__" and "y = HOLD (fb__ tka__)"   | f :: "nat \<Rightarrow> nat" and tk :: "nat" where "x = f" and "xa = REL tk" and "y = REL (f tk)"
intro: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e"   and alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "monad_fail return bind fail"   and "monad_alt return bind alt"   and "monad_fail_alt_axioms fail alt" shows "monad_fail_alt return bind fail alt"
plus_mwb_lens: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c" assumes "mwb_lens x"   and "mwb_lens y"   and "x \<bowtie> y" shows "mwb_lens (x +\<^sub>L y)"
snd_scaleC: fixes r :: "complex"   and A :: "'d \<times> 'c" shows "snd (r *\<^sub>C A) = r *\<^sub>C snd A"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ord.rbt_sorted less Empty = True"
assms: shows "f\<^sub>0 = \<tau> i(gap i t \<mapsto> 0)"
valid_dbm_def: fixes M :: "'c \<Rightarrow> 'c \<Rightarrow> 'd DBMEntry"   and n :: "'c" shows "valid_dbm M n \<equiv> dbm_int M n \<and> (\<forall>i\<le>n. M (0::'c) i \<le> \<one>)"
a_get_tag_name_def: fixes element_ptr :: "(_) element_ptr" shows "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name element_ptr = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name"
prim_self_root: fixes x :: "'b list" assumes "primitive x" shows "\<rho> x = x"
assms: shows "llist_all2 P (lappend xs xs') ys"
P3: shows "A1 B' C1 CongA A'' B'' C''"
induct: fixes x1 :: "('Var, 'AExp, 'BExp) Stmt"   and x2 :: "Mode \<Rightarrow> 'Var set"   and x3 :: "'Var \<Rightarrow> 'Val"   and x4 :: "('Var, 'AExp, 'BExp) Stmt"   and x5 :: "Mode \<Rightarrow> 'Var set"   and x6 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and P :: "('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var \<Rightarrow> 'Val) \<Rightarrow> ('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "\<langle>x1, x2, x3\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>x4, x5, x6\<rangle>"   and "\<And>\<Gamma> \<S> Pa c mds mem\<^sub>1 mem\<^sub>2. \<lbrakk>\<turnstile> \<Gamma>,\<S>,Pa {c} \<Gamma>',\<S>',P'; tyenv_wellformed mds \<Gamma> \<S> Pa; mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2; pred Pa mem\<^sub>1; pred Pa mem\<^sub>2; \<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x\<rbrakk> \<Longrightarrow> P c mds mem\<^sub>1 c mds mem\<^sub>2" shows "P x1 x2 x3 x4 x5 x6"
cart_basis'_def: shows "cart_basis' = (\<chi>i. axis i (1::'d))"
len_lexext_compat_cons: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "len_lexext gt ys xs" shows "len_lexext gt (x # ys) (x # xs)"
wellformed: fixes b :: "'a set" assumes "b \<in># add_block bl" shows "b \<subseteq> \<V> \<union> bl"
inj: shows "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
assms: shows "set_kdt kdt \<union> set ps - set (nearest_nbors n ps p kdt) \<noteq> {}"
is_bcv_kildall: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and n :: "nat" assumes "Semilat A r f"   and "acc r"   and "Semilat.top r T"   and "pres_type step n A"   and "bounded step n"   and "SemilatAlg.mono r step n A" shows "is_bcv r T step n A (kildall r f step)"
gWlsBinp_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) Recursion.model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'i option" shows "gWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (gWlsAbs MOD) (barOf delta) binp"
ex_fresh_gt: fixes A :: "'e set"   and i :: "'e"   and f :: "'e \<Rightarrow> 'd" assumes "\<exists>g. surj g"   and "finite A"   and "i \<notin> A" shows "\<exists>j. j \<notin> f ` A"
e_type_label: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and n :: "nat"   and es0 :: "e list"   and es :: "e list"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [Label n es0 es] : ts _> ts'" shows "\<exists>tls t2s. ts' = ts @ t2s \<and> length tls = n \<and> \<S>\<bullet>\<C> \<turnstile> es0 : tls _> t2s \<and> \<S>\<bullet>\<C>\<lparr>label := [tls] @ label \<C>\<rparr> \<turnstile> es : [] _> t2s"
order_def: fixes S :: "('e, 'f) monoid_scheme" shows "order S = card (carrier S)"
lt2: shows "Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z) < 2"
wt_err_imp_wt_app_eff: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ts :: "'a err list"   and app :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list" assumes "wt_err_step r (err_step (length ts) app step) ts"   and "bounded (err_step (length ts) app step) (length ts)" shows "wt_app_eff r app step (map ok_val ts)"
tagged_fun_relD_lhs: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'f"   and A :: "('c \<times> 'e) set"   and B :: "('d \<times> 'f) set"   and x :: "'c"   and x' :: "'e" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, f' x') \<in> B"
replacefacesAt2_nth: fixes k :: "nat"   and ns :: "nat list"   and F :: "face list list"   and oldf :: "face"   and newfs :: "face list" assumes "k \<in> set ns"   and "k < |F|"   and "oldf \<notin> set newfs"   and "distinct (F ! k)"   and "distinct newfs"   and "oldf \<in> set (F ! k) \<longrightarrow> set newfs \<inter> set (F ! k) \<subseteq> {oldf}" shows "replacefacesAt2 ns oldf newfs F ! k = replace oldf newfs (F ! k)"
inducts: fixes x1 :: "('a, 'b) Proc"   and x2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and P :: "('a, 'b) Proc \<Rightarrow> ('a, 'b) Proc \<Rightarrow> bool" assumes "(x1, x2) \<in> encoding.indRelR Enc"   and "\<And>S. P (SourceTerm S) (TargetTerm (Enc S))" shows "P x1 x2"
degree_minus_eq_right: fixes q :: "'b poly"   and p :: "'b poly" assumes "degree q < degree p" shows "degree (p - q) = degree p"
isomorphic_tuple_def: fixes isom :: "('a, 'b, 'c) tuple_isomorphism" shows "isomorphic_tuple isom \<equiv> (\<forall>x. abst isom (repr isom x) = x) \<and> (\<forall>y. repr isom (abst isom y) = y)"
path_append_elim: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and r :: "(('d \<times> 'e) \<times> 'f) list"   and s :: "(('d \<times> 'e) \<times> 'f) list"   and p :: "'f" assumes "path A (r @ s) p" obtains "path A r p" and "path A s (target r p)"
at_top_within_def: fixes s :: "'c set" shows "at_top_within s = (INF k\<in>s. principal ({k..} \<inter> s))"
\<R>\<^sub>GI: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and \<beta> :: "'e \<Rightarrow> 'e" assumes "bij \<alpha>"   and "\<And>r. s r = None \<Longrightarrow> s' (\<alpha> r) = None"   and "\<And>r \<sigma> \<tau> e. s r = Some (\<sigma>, \<tau>, e) \<Longrightarrow> s' (\<alpha> r) = Some (\<R>\<^sub>L \<alpha> \<beta> (\<sigma>, \<tau>, e))" shows "\<R>\<^sub>G \<alpha> \<beta> s = s'"
abd_acd_abcacb: fixes a :: "'a"   and b :: "'a"   and d :: "'a"   and c :: "'a" assumes "[[a b d]]"   and "[[a c d]]"   and "b \<noteq> c" shows "[[a b c]] \<or> [[a c b]]"
1: shows "|~ \<langle>Deq mid q2 out\<rangle>_vars = Deq mid q2 out"
typeof_unboxed_eq_const_1: fixes x :: "('a, 'b, 'c) unboxed" shows "(typeof x = None) = (\<exists>d. x = OpDyn d)"
sheaf_of_rings_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" shows "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> sheaf_of_rings_axioms S is_open \<FF> \<rho> zero_str"
iT_Mult_0_if: fixes I :: "nat set" shows "I \<otimes> 0 = (if I = {} then {} else [\<dots>0])"
ldrop_iterates: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "ldrop (enat n) (iterates f x) = iterates f ((f ^^ n) x)"
sfis_mon_conv_mono: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and y :: "real"   and r :: "real"   and s :: "'b \<Rightarrow> real" assumes "u\<up>f"   and "\<And>n. x n \<in> sfis (u n) M"   and "x\<up>y"   and "r \<in> sfis s M"   and "s \<le> f"   and "measure_space M" shows "r \<le> y"
take_vs: shows "take (Suc (es ! (n - 1))) vs = vs"
ord_Sab: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S_a"   and "b \<in> S_b" shows "b < a"
lubI: fixes S :: "'a set"   and L :: "'a" assumes "S \<subseteq> A"   and "L \<in> A"   and "\<forall>x\<in>S. (x, L) \<in> r"   and "\<forall>z\<in>A. (\<forall>y\<in>S. (y, z) \<in> r) \<longrightarrow> (L, z) \<in> r" shows "L = lub S cl"
that: assumes "arr_Par \<alpha> S"   and "S\<lparr>ArrDom\<rparr> = A"   and "S\<lparr>ArrCod\<rparr> = B" shows "thesis"
assoc'_in_hom: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "ide a"   and "ide b"   and "ide c" shows "\<guillemotleft>\<a>\<^sup>-\<^sup>1[a, b, c] : a \<otimes> b \<otimes> c \<rightarrow> (a \<otimes> b) \<otimes> c\<guillemotright>"
less_asym': fixes a :: "'t"   and b :: "'t"   and P :: "bool" assumes "a \<prec>\<^sub>t b"   and "b \<prec>\<^sub>t a" shows "P"
case_prod_eta: fixes f :: "'d \<times> 'e \<Rightarrow> 'f" shows "(\<lambda>(x, y). f (x, y)) = f"
floor_eq_iff: fixes x :: "'b"   and a :: "int" shows "(\<lfloor>x\<rfloor> = a) = (of_int a \<le> x \<and> x < of_int a + (1::'b))"
rel_intros: fixes R3 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "R3 x y" shows "rel_writerT R1 R2 R3 (WriterT x) (WriterT y)"
case3: assumes "0 \<le> x"   and "y \<le> 0" shows "x * y \<le> 4611686018427387904"
right_total_mod_ring_rel: shows "right_total mod_ring_rel"
trss_2empty_to_2return: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), w, [], c') \<in> trcl (trss fg)"   and "s \<noteq> []" shows "\<exists>w' p. w = w' @ [LRet] \<and> ((s, c), w', [return fg p], c') \<in> trcl (trss fg)"
map_add_find_right: fixes n :: "'c \<Rightarrow> 'b option"   and k :: "'c"   and xx :: "'b"   and m :: "'c \<Rightarrow> 'b option" assumes "n k = \<lfloor>xx\<rfloor>" shows "(n \<Oplus> m) k = \<lfloor>xx\<rfloor>"
subst_elimD'': fixes \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term"   and v :: "'e"   and w :: "'c" assumes "subst_elim \<sigma> v" shows "v \<notin> fv (\<sigma> w)"
that: assumes "h' = snd (f h)"   and "r = fst (f h)"   and "P h" shows "thesis"
mat_assoc_test_14: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "A + B + C = C + B + A"
simps: fixes a1 :: "'d"   and a2 :: "'d list" shows "ListMem a1 a2 = ((\<exists>x xs. a1 = x \<and> a2 = x # xs) \<or> (\<exists>x xs y. a1 = x \<and> a2 = y # xs \<and> ListMem x xs))"
decseq_f: shows "decseq (\<lambda>i. f (A i - \<Inter> (range A)))"
fn_eq: fixes x :: "'b"   and y :: "'b"   and B :: "'b trm"   and A :: "'b trm"   and T :: "PreSimplyTyped.type" assumes "x \<noteq> y"   and "x \<notin> fvs B"   and "A = [x \<leftrightarrow> y] \<cdot> B" shows "Fn x T A = Fn y T B"
preserves_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "S.src (S.UP \<mu>) = UP.map\<^sub>0 (src \<mu>)"
dot_matrix_product: fixes x :: "(real, 'a) vec"   and y :: "(real, 'a) vec" shows "x \<bullet> y = (rowvector x ** columnvector y) $ 1 $ 1"
subset_closed_def: fixes C :: "'b set set" shows "subset_closed C = (\<forall>S'\<in>C. \<forall>S\<subseteq>S'. S \<in> C)"
filtermap_at_right_shift: fixes d :: "real"   and a :: "real" shows "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d)"
total1I: fixes f :: "nat \<Rightarrow> nat option" assumes "\<And>x. f x \<down>" shows "total1 f"
Hoare_unhalt_def: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and p :: "(action \<times> nat) list" shows "{P} p \<up> \<equiv> \<forall>tp. P tp \<longrightarrow> (\<forall>n. \<not> is_final (steps0 (1, tp) p n))"
IH: fixes \<gamma> :: "V" assumes "\<gamma> \<in> elts (\<omega> \<up> ord_of_nat n)" shows "\<exists>ns. \<gamma> = omega_sum ns \<and> length ns = n"
that: fixes z :: "'a" assumes "z \<in> rel_frontier S"   and "y \<in> closed_segment x z"   and "open_segment x z \<subseteq> rel_interior S" shows "thesis"
assms_2: fixes x :: "'a"   and a :: "'a" assumes "x \<in> (\<Sum>X\<leftarrow>As. X)"   and "a \<in> A"   and "a + x = (0::'a)" shows "a = (0::'a)"
approx_merge: fixes t1 :: "'a \<Rightarrow> 'b option"   and s :: "'a \<Rightarrow> 'b"   and t2 :: "'a \<Rightarrow> 'b option" assumes "approx t1 s \<or> approx t2 s" shows "approx (merge t1 t2) s"
opeS: shows "openin (top_of_set (affine hull S)) S"
axioms_1: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "proper_sub_set_system \<U> \<A> \<V> \<B>" shows "incidence_system \<V> \<B>"
linear_imp_scale: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "vector_space scale"   and "Vector_Spaces.linear (*) scale D" obtains d :: "'b" where "D = (\<lambda>x. scale x d)"
subgroups_partial_order: shows "partial_order \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>"
prems_2: shows "\<forall>aa. ListMem aa (a # as) \<longrightarrow> fmdom' (fst aa) \<subseteq> fmdom' s1"
bz: shows "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
empty_correct'_2: shows "HashMap_Impl.invar (HashMap_Impl.empty ())"
cases: fixes x :: "(node \<times> node) list \<times> node list \<times> node list" obtains g :: "(node \<times> node) list" and ys :: "node list" where "x = (g__, [], ys__)"   | g :: "(node \<times> node) list" and xa :: "node" and xs :: "node list" and ys :: "node list" where "x = (g, xa # xs, ys)"
setFinal_def: fixes f :: "face" shows "setFinal f \<equiv> Face (vertices f) Final"
BackwardSlice_axioms: shows "BackwardSlice sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (WStandardControlDependence.PDG_BS_s prog)"
inject: fixes x1 :: "'a"   and y1 :: "'a" shows "(NormalAction x1 = NormalAction y1) = (x1 = y1)"
Scons_mono: fixes M :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and N' :: "('a, 'b) node set" assumes "M \<subseteq> M'"   and "N \<subseteq> N'" shows "Scons M N \<subseteq> Scons M' N'"
less_not_sym: fixes y :: "'d"   and x :: "'d" assumes "y < x" shows "\<not> x < y"
num_reachable_norefl_def: fixes G :: "'b list_graph"   and v :: "'b" shows "FiniteListGraph.num_reachable_norefl G v = length (filter (\<lambda>x. x \<noteq> v) (FiniteListGraph.succ_tran G v))"
integral: fixes a :: "'a"   and b :: "'a" assumes "a \<otimes> b = \<zero>"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a = \<zero> \<or> b = \<zero>"
lex_ord_raw_EqD: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and ko :: "'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> order option"   and k :: "'a" assumes "oalist_inv xs"   and "oalist_inv ys"   and "lex_ord_raw ko f xs ys = Some Eq"   and "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)" shows "f k (lookup_raw xs k) (lookup_raw ys k) = Some Eq"
object_more_extend_id: fixes x :: "(_) RObject_ext"   and y :: "'a" shows "RObject.more (RObject.extend x y) = y"
comp_ipresIGCons: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'k, 'm, 'p) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'l, 'n, 'q) model_scheme"   and h' :: "'l \<Rightarrow> 'r"   and hA' :: "'n \<Rightarrow> 's"   and MOD'' :: "('index, 'bindex, 'varSort, 't, 'opSym, 'o, 'r, 's, 'u) model_scheme" assumes "ipresIGWlsAll h hA MOD MOD'"   and "ipresIGCons h hA MOD MOD'"   and "ipresIGCons h' hA' MOD' MOD''" shows "ipresIGCons (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
app_subst_id_tel: shows "$ id_subst = (\<lambda>A. A)"
image_Union: fixes f :: "'c \<Rightarrow> 'a"   and S :: "'c set set" shows "f ` \<Union> S = \<Union> ((`) f ` S)"
Ord_succ: fixes x :: "V" assumes "Ord x" shows "Ord (ZFC_in_HOL.succ x)"
set_rw: shows "{0..<IArray.length (vec_to_iarray a)} = range mod_type_class.to_nat"
adv_P2_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool"   and D :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "OT_12.adv_P2 m1 m2 D = \<bar>spmf (R2 m1 m2 \<bind> D) True - spmf (funct_OT_12 m1 m2 \<bind> (\<lambda>(o1, o2). S2 m2 o2 \<bind> D)) True\<bar>"
SD_implies_BSD_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>' :: "'b V_rec" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "SD \<V> Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "BSD \<V>' Tr\<^bsub>ES\<^esub>"
f'g': shows "internal_equivalence f' g' \<phi>' \<psi>'"
assoc'_simps_1: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "arr \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' (\<cdot>) prj0 prj1 f g h, Dom = Cod (f \<star> g \<star> h), Cod = Dom ((f \<star> g) \<star> h)\<rparr>"