hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "(nat \<times> val) list"   and x2 :: "exp"   and x3 :: "val"   and P :: "(nat \<times> val) list \<Rightarrow> exp \<Rightarrow> val \<Rightarrow> bool" assumes "x1 \<turnstile> x2 \<Rightarrow> x3"   and "\<And>\<rho> n. P \<rho> (ENat n) (VNat n)"   and "\<And>\<rho> e1 n1 e2 n2 f. \<lbrakk>\<rho> \<turnstile> e1 \<Rightarrow> VNat n1; P \<rho> e1 (VNat n1); \<rho> \<turnstile> e2 \<Rightarrow> VNat n2; P \<rho> e2 (VNat n2)\<rbrakk> \<Longrightarrow> P \<rho> (EPrim f e1 e2) (VNat (f n1 n2))"   and "\<And>\<rho> x v' v. \<lbrakk>lookup \<rho> x = Some v'; v \<sqsubseteq> v'\<rbrakk> \<Longrightarrow> P \<rho> (EVar x) v"   and "\<And>t x \<rho> e. \<forall>v v'. (v, v') \<in> fset t \<longrightarrow> (x, v) # \<rho> \<turnstile> e \<Rightarrow> v' \<and> P ((x, v) # \<rho>) e v' \<Longrightarrow> P \<rho> (ELam x e) (VFun t)"   and "\<And>\<rho> e1 t e2 v2 v3 v3' v. \<lbrakk>\<rho> \<turnstile> e1 \<Rightarrow> VFun t; P \<rho> e1 (VFun t); \<rho> \<turnstile> e2 \<Rightarrow> v2; P \<rho> e2 v2; (v3, v3') \<in> fset t; v3 \<sqsubseteq> v2; v \<sqsubseteq> v3'\<rbrakk> \<Longrightarrow> P \<rho> (EApp e1 e2) v"   and "\<And>\<rho> e1 n e2 v e3. \<lbrakk>\<rho> \<turnstile> e1 \<Rightarrow> VNat n; P \<rho> e1 (VNat n); n \<noteq> 0; \<rho> \<turnstile> e2 \<Rightarrow> v; P \<rho> e2 v\<rbrakk> \<Longrightarrow> P \<rho> (EIf e1 e2 e3) v"   and "\<And>\<rho> e1 n e3 v e2. \<lbrakk>\<rho> \<turnstile> e1 \<Rightarrow> VNat n; P \<rho> e1 (VNat n); n = 0; \<rho> \<turnstile> e3 \<Rightarrow> v; P \<rho> e3 v\<rbrakk> \<Longrightarrow> P \<rho> (EIf e1 e2 e3) v" shows "P x1 x2 x3"
cases: fixes a1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and a2 :: "'e"   and a3 :: "'f \<Rightarrow> 'g"   and a4 :: "'g"   and a5 :: "'e"   and a6 :: "'f \<Rightarrow> 'g" assumes "a1 \<turnstile> \<langle>a2, a3\<rangle> \<rightarrow>\<^bsup>a4\<^esup> \<langle>a5, a6\<rangle>" obtains u :: "'f \<Rightarrow> 'g" and A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)" and l :: "'e" and d :: "'g" where "a1 = A" and "a2 = l" and "a3 = u" and "a4 = d" and "a5 = l" and "a6 = u \<oplus> d" and "u \<turnstile> inv_of A l" and "u \<oplus> d \<turnstile> inv_of A l" and "(0::'g) \<le> d"
is_full: fixes a :: "'c"   and a' :: "'c"   and g :: "'d" assumes "C.ide a"   and "C.ide a'"   and "\<guillemotleft>g : F a' \<rightarrow>\<^sub>D F a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow>\<^sub>C a\<guillemotright> \<and> F f = g"
inject_3: fixes x31 :: "Label \<Rightarrow> sterm option"   and x32 :: "Sigma.type"   and y31 :: "Label \<Rightarrow> sterm option"   and y32 :: "Sigma.type" shows "(Obj x31 x32 = Obj y31 y32) = (x31 = y31 \<and> x32 = y32)"
is_strongly_dom_component_safe_def: shows "Core_DOM_Components.is_strongly_dom_component_safe \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe Core_DOM_Functions.get_root_node Core_DOM_Functions.to_tree_order"
inf_left_commute: fixes x :: "'e multiset"   and y :: "'e multiset"   and z :: "'e multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
assms_1: shows "(strict {(x, y). x \<sqsubseteq> y})\<inverse> \<subseteq> measure m"
intersection_number_def: fixes b1 :: "'a set"   and b2 :: "'a set" shows "b1 |\<inter>| b2 \<equiv> int (card (b1 \<inter> b2))"
l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs ShadowRootClass.type_wf"
events_in_local_order: fixes pre :: "('a \<times> ('a, 'b) operation) event list"   and e2 :: "('a \<times> ('a, 'b) operation) event"   and i :: "nat"   and e1 :: "('a \<times> ('a, 'b) operation) event" assumes "pre @ [e2] prefix of i"   and "e1 \<in> set pre" shows "e1 \<sqsubset>\<^sup>i e2"
keys_hom_componentD_2: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" assumes "t \<in> keys (hom_component p n)" shows "deg_pm t = n"
assms: fixes x :: "hf" assumes "\<And>y. y \<^bold>\<in> x \<Longrightarrow> P y" shows "P x"
positive_imp_inverse_positive: 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less zero (inverse a)"
base: assumes "4 / 3 < \<alpha>" shows "1 < base"
complex_rotations_inverses: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> complex_rotations" shows "\<exists>y\<in>complex_rotations. y \<circ> r = id"
Un_absorb1: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> B" shows "A \<union> B = B"
join_mmsaux_abs_eq: fixes args :: "Monitor.args"   and cur :: "nat"   and nt :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'a option list set) queue"   and data_in :: "(nat \<times> 'a option list set) queue"   and tuple_in :: "('a option list, nat) mapping"   and tuple_since :: "('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list"   and X :: "'a option list set" assumes "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"   and "table (args_n args) (args_L args) X" shows "join_mmsaux args X (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) = join_mmsaux_abs args X (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)"
composable_implies_arr_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.arr V \<nu>"
adopt_node_locs_def: shows "adopt_node_locs = adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
Rep_cltn2_inverse: fixes x :: "cltn2" shows "Abs_cltn2 (Rep_cltn2 x) = x"
ite_impl_lu_R: fixes s :: "bdd"   and ii :: "nat"   and i :: "nat ifex"   and ti :: "nat"   and t :: "nat ifex"   and ei :: "nat"   and e :: "nat ifex" assumes "bdd_sane s"   and "(ii, i) \<in> Rmi s"   and "(ti, t) \<in> Rmi s"   and "(ei, e) \<in> Rmi s" shows "ospec (mi.ite_impl_lu ii ti ei s) (\<lambda>(r, s'). (r, ifex_ite_opt i t e) \<in> Rmi s' \<and> bdd_sane s' \<and> mi.les s s')"
castAbs_def: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" shows "castAbs xs s A \<equiv> SOME x_X. wls s (snd x_X) \<and> A = Abs xs (fst x_X) (snd x_X)"
local_lipschitz: shows "local_lipschitz UNIV X (\<lambda>_. f)"
Fvars_Pf: shows "Fvars Pf = {yy, xx}"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>pD_1: fixes p :: "'a \<times> ('b, 'c) Term.term \<times> ('b, 'c) Term.term"   and a :: "'a \<times> ('b, 'c) stateful_strand_step" assumes "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a" shows "fst p = fst a"
map_filter_M_pure: fixes xs :: "'x list"   and f :: "'x \<Rightarrow> ('y, 'z, 'aa option) prog"   and h :: "'y" assumes "\<And>x h. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_filter_M f xs) h"
hyps_1: shows "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle>"
hom_root_unity: fixes n :: "nat" shows "map_poly hom (root_unity n) = root_unity n"
wf_ts_def: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and j :: "nat"   and \<phi> :: "'b MFOTL.formula"   and \<psi> :: "'b MFOTL.formula"   and ts :: "nat list" shows "wf_ts \<sigma> j \<phi> \<psi> ts = list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)..<j] ts"
of_int_poly_pCons: fixes c :: "int"   and p :: "int poly" shows "of_int_poly (pCons c p) = pCons (of_int c) (of_int_poly p)"
subsetCE: fixes A :: "'f set"   and B :: "'f set"   and c :: "'f" assumes "A \<subseteq> B" obtains "c \<notin> A"   | "c \<in> B"
pmat_extension_carrier: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and dims :: "nat list"   and m :: "'a mat" assumes "partial_state2 vars1 vars2" shows "partial_state2.pmat_extension dims vars1 vars2 m \<in> carrier_mat (partial_state2.d0 dims vars1 vars2) (partial_state2.d0 dims vars1 vars2)"
elims_2: fixes x :: "('a, 'b) rbt" assumes "cl_inv' x" obtains "x = Leaf"   | l :: "('a, 'b) rbt" and c :: "color" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l c k v r" and "cl_inv l \<and> cl_inv r"
sum_indicator_eq_card2: fixes I :: "'b set"   and P :: "'b \<Rightarrow> 'c set"   and x :: "'c" assumes "finite I" shows "(\<Sum>i\<in>I. indicator (P i) x) = card {i \<in> I. x \<in> P i}"
ithE: fixes x :: "'b"   and B :: "('c \<Rightarrow> 'b) set"   and i :: "'c" assumes "x \<in> ith B i" obtains f :: "'c \<Rightarrow> 'b" where "f \<in> B" and "f i = x"
extendContain: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and ps :: "('a, 'b) sequent list"   and c :: "('a, 'b) sequent"   and Ps :: "('a, 'b) sequent list"   and C :: "('a, 'b) sequent"   and S :: "('a, 'b) sequent"   and p :: "('a, 'b) sequent" assumes "r = (ps, c)"   and "(Ps, C) = extendRule S r"   and "p \<in> set ps" shows "extend S p \<in> set Ps"
IH: fixes B :: "'a set"   and Ba :: "'a set" assumes "B \<subset> A"   and "finite Ba" shows "random_bst B \<bind> (\<lambda>t1. random_bst Ba \<bind> mrbst_inter_diff b t1) = random_bst (B \<diamondop> Ba)"
asym_trans_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sqsubset> x"   and "asympartp (\<sqsubset>) z y"   and "x \<in> P"   and "y \<in> P"   and "z \<in> P" shows "asympartp (\<sqsubset>) z x"
prems: fixes a :: "'a" assumes "a \<in> insert x F" shows "f a \<noteq> (0::'b)"
isTrans_def: fixes event :: "('a, 'b, 'c) event" shows "isTrans event = (\<exists>x11 x12 x13. event = Trans x11 x12 x13)"
image_eqI: fixes b :: "'k"   and f :: "'l \<Rightarrow> 'k"   and x :: "'l"   and A :: "'l set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
pow_Suc_list: fixes u :: "'b list"   and n :: "nat" shows "u \<^sup>@ Suc n = u \<cdot> u \<^sup>@ n"
F_inj_composition_permutation: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and \<phi> :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n"   and "\<phi> permutes {0..<n}"   and "g \<in> Cauchy_Binet.F_inj n m" shows "g \<circ> \<phi> \<in> Cauchy_Binet.F_inj n m"
mult_scalar_pprod_def: shows "mult_scalar_pprod = pprod.mult_scalar"
sin_30: shows "sin (pi / 6) = 1 / 2"
defNode_1: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and v :: "'d"   and g :: "'a" assumes "CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"   and "v \<in> CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g" shows "CFG_SSA_wf_base.defNode \<alpha>n defs phis g v \<in> set (\<alpha>n g)"
conv_has_start_end_points_path: fixes x :: "'a" shows "has_start_end_points_path x = has_start_end_points_path (x\<^sup>T)"
lmap_iterates: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "lmap f (iterates f x) = iterates f (f x)"
discrete_def: fixes S :: "'b set" shows "discrete S = (\<forall>x\<in>S. x isolate S)"
hurwitz_formula_bound1: fixes r :: "real" assumes "0 < r" obtains C :: "real" where "0 \<le> C" and "\<And>z. z \<notin> (\<Union>n. ball (complex_of_real (real_of_int (2 * n) * pi) * \<i>) r) \<Longrightarrow> cmod (exp (complex_of_real a * z) / (1 - exp z)) \<le> C"
simps: fixes A :: "('b \<times> nat) tree"   and a :: "'b"   and uu :: "nat"   and B :: "('b \<times> nat) tree"   and b :: "'b"   and C :: "('b \<times> nat) tree" shows "rotateR \<langle>A, (a, uu), B\<rangle> b C = (if single B A then rot1R A a B b C else rot2 A a B b C)"
frac_add: fixes x :: "'d"   and y :: "'d" shows "frac (x + y) = (if frac x + frac y < (1::'d) then frac x + frac y else frac x + frac y - (1::'d))"
cinner_commute: fixes x :: "'a"   and y :: "'a" shows "cinner x y = cnj (cinner y x)"
assms: shows "\<forall>x y. x \<rhd> y = (x \<rhd> (1::'a)) \<cdot> y"
tensor_mat_trace: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)" shows "trace (tensor_mat dims vars m1 m2) = trace m1 * trace m2"
emeasure_pos_unionE: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>N. A N \<in> sets M"   and "0 < emeasure M (\<Union> (range A))" shows "\<exists>N. 0 < emeasure M (A N)"
prems_2: shows "g \<in> insert a A \<rightarrow> carrier G"
CD_on_range: fixes ds :: "'d set"   and A :: "'x set" shows "CD_on ds A \<subseteq> A \<inter> (\<Union>d\<in>ds. Field (Pd d))"
assms_1: shows "\<forall>t1\<le>t. a t1 = []"
map_sel_2: fixes a :: "('g, 'h, 'i) generat"   and f1 :: "'g \<Rightarrow> 'k"   and f2 :: "'h \<Rightarrow> 'j"   and f3 :: "'i \<Rightarrow> 'l" assumes "\<not> is_Pure a" shows "generat.output (map_generat f1 f2 f3 a) = f2 (generat.output a)"
balance_right_rbt_sorted: fixes l :: "('a, 'c) rbt"   and r :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c" assumes "rbt_sorted l"   and "rbt_sorted r"   and "l |\<guillemotleft> k"   and "k \<guillemotleft>| r" shows "rbt_sorted (balance_right l k v r)"
1: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(mu * q' g) \<odot> g \<noteq> 0" shows "lt ((mu * q' g) \<odot> g) \<prec>\<^sub>t v"
WbisT_coinduct: fixes \<phi> :: "(test, atom) com \<Rightarrow> (test, atom) com \<Rightarrow> bool"   and c :: "(test, atom) com"   and d :: "(test, atom) com" assumes "\<phi> c d"   and "\<And>c d. \<phi> c d \<Longrightarrow> \<phi> d c"   and "\<And>c s d t c' s'. \<lbrakk>\<phi> c d; Example_PL.indisAbbrev s t; Example_PL.transC (c, s) (c', s')\<rbrakk> \<Longrightarrow> \<exists>d' t'. Example_PL.MtransC (d, t) (d', t') \<and> Example_PL.indisAbbrev s' t' \<and> (\<phi> c' d' \<or> c' \<approx>wT d')"   and "\<And>c s d t s'. \<lbrakk>\<phi> c d; Example_PL.indisAbbrev s t; Example_PL.transT (c, s) s'\<rbrakk> \<Longrightarrow> \<exists>t'. Example_PL.MtransT (d, t) t' \<and> Example_PL.indisAbbrev s' t'" shows "c \<approx>wT d"
pdevs_eqI: fixes x :: "'a pdevs"   and y :: "'a pdevs" assumes "\<And>i. pdevs_apply x i = pdevs_apply y i" shows "x = y"
munionM_comp_fun_commute: shows "comp_fun_commute munionM"
bin_sign_def: fixes k :: "int" shows "bin_sign k = (if 0 \<le> k then 0 else - 1)"
gcd_add_dvd_left1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b" shows "gcd (b + c) a = gcd c a"
wf_equality: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and ac :: "poscheckvariant"   and t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term"   and S' :: "('b, 'a) strand_step list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>t V (S @ \<langle>ac: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"   and "mgu t t' = Some \<delta>"   and "bvars\<^sub>s\<^sub>t (S @ \<langle>ac: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S') \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "wf\<^sub>s\<^sub>t (fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)) (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>)"
h: shows "sorted_wrt (\<lambda>(uu_, i) (uu_, y). i < y) (chop (to_list v p))"
Un_Diff_cancel: fixes A :: "'k set"   and B :: "'k set" shows "A \<union> (B - A) = A \<union> B"
cmem_def: fixes M :: "heap" shows "state_dp_consistency.cmem M \<equiv> \<forall>param\<in>dom (state_dp_consistency.map_of M). state_dp_consistency.map_of M param = Some (dp param)"
intros_7: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> \<sigma>" shows "OrderedSet \<tau> \<sqsubset> OrderedSet \<sigma>"
ik_init_def: shows "ik_init \<equiv> {priK C |C. C \<in> bad_init} \<union> {pubK A |A. True} \<union> {shrK A B |A B. A \<in> bad_init \<or> B \<in> bad_init} \<union> Tags"
Phi_conv: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" shows "(\<Phi> w x y z)\<^sup>\<smile> = y ; w\<^sup>\<smile> \<cdot> z ; x\<^sup>\<smile>"
rtranclp_trans: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
A2: fixes i :: "'b"   and p :: "'b fm"   and q :: "'b fm" shows "\<turnstile> (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i q)"
lm066: fixes R :: "('d \<times> 'c) set"   and X :: "'d set" shows "Range (Outside' X R) = R `` (Domain R - X)"
insert: fixes A :: "nat set"   and x :: "nat" assumes "finite A" shows "F (insert x A) = lcm x (F A)"
linear_linear: fixes f :: "'d \<Rightarrow> 'e" shows "bounded_linear f = linear f"
par_anycast_msg_zhops: fixes i :: "nat" shows "opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. rreq_rrep_sn) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(uu_, a, uu_). anycast msg_zhops a)"
dim_submatrix_2: fixes A :: "'a mat"   and I :: "nat set"   and J :: "nat set" shows "dim_col (submatrix A I J) = card {j. j < dim_col A \<and> j \<in> J}"
assms_3: shows "val_Zp y \<le> val_Zp x"
intro: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "f \<sim>[at_top] (\<lambda>x. x)"   and "\<And>F. \<forall>\<^sub>F x in F. g x = exp (f x)" shows "exp_asymp_equiv_linear f g"
assms_3: shows "B \<in>\<^sub>\<circ> Vset \<alpha>"
history_consistent_access_last_read': fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'p"   and sb :: "'p memref list"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" assumes "history_consistent \<theta> p (sb @ [Read\<^sub>s\<^sub>b volatile a t v])" shows "\<theta> t = Some v"
del_invalid_point: fixes p :: "'a" assumes "p \<notin> \<V>" shows "del_point p = \<V>"
disc_2: fixes x21 :: "'h"   and x22 :: "'i" shows "\<not> is_Pure (IO x21 x22)"
step_hext_incr: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "hp s \<unlhd> hp s'"
welltyped_commute: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and \<sigma>' :: "'addr option \<times> 'heap \<times> 'addr frame list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "\<Phi> \<turnstile> t:\<sigma> \<surd>" shows "P,t \<turnstile> Normal \<sigma> -ta-jvmd\<rightarrow> Normal \<sigma>' = P,t \<turnstile> \<sigma> -ta-jvm\<rightarrow> \<sigma>'"
image_mset_fun_upd: fixes x :: "'c"   and M :: "'c multiset"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "x \<notin># M" shows "image_mset (f(x := y)) M = image_mset f M"
V_full_prop: shows "nodes (generate_valid_topology M (fully_connected G)) = V"
preserves_ide: fixes a :: "'c local.term set" assumes "ide a" shows "\<F>\<^sub>SC.ide (D a)"
assms_3: shows "monom P (p (deg R p)) (deg R p) = monom P (q (deg R q)) (deg R q)"
denot_lam_implies_ts: fixes e :: "exp"   and f :: "(val \<times> val) list"   and x :: "nat"   and \<rho> :: "(nat \<times> val) list" assumes "\<forall>v \<rho>. v \<in> E e \<rho> \<longrightarrow> Tenv \<rho> \<turnstile> e : T v"   and "\<forall>v1 v2. (v1, v2) \<in> set f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)" shows "Tenv \<rho> \<turnstile> ELam x e : TFun (Tf f)"
zero_hom_0_axioms: shows "zero_hom_0 of_real"
base: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list" assumes "execution trans sends start exec trace" shows "asynchronousSystem.initial start (hd exec)"
simps: fixes x :: "nat"   and n :: "nat" shows "senior x n = the (oldest_token (token_run x n) n)"
echelon_form_condition1: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "echelon_form A" shows "\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)"
E: shows "t2 \<in> {t. \<exists>t1. x = t1 @ t} \<and> infinite (f t2)"
runit_tensor': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<r>[a \<otimes> b] \<cdot> \<a>\<^sup>-\<^sup>1[a, b, \<I>] = a \<otimes> \<r>[b]"
assms: shows "f $ 0 \<noteq> (0::'a)"
p'_5: fixes x1 :: "'n"   and K1 :: "'n set"   and x2 :: "'n"   and K2 :: "'n set" assumes "(x1, K1) \<in> p"   and "(x2, K2) \<in> p"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
simps_1: fixes p2 :: "'a list list" shows "UP.poly_mult [] p2 = []"
D: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list" assumes "trans r"   and "irrefl r" shows "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>' = ((r|\<tau>'| -s r \<down>l \<sigma>, r|\<tau>|) \<in> mul_eq r \<and> (r|\<sigma>'| -s r \<down>l \<tau>, r|\<sigma>|) \<in> mul_eq r)"
indep_var_distribution_eq: fixes S :: "'c measure"   and X :: "'b \<times> 'b \<Rightarrow> 'c"   and T :: "'c measure"   and Y :: "'b \<times> 'b \<Rightarrow> 'c" shows "P.indep_var S X T Y = (P.random_variable S X \<and> P.random_variable T Y \<and> distr P S X \<Otimes>\<^sub>M distr P T Y = distr P (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
seteqI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>a. a \<in> A \<Longrightarrow> a \<in> B"   and "\<And>b. b \<in> B \<Longrightarrow> b \<in> A" shows "A = B"
assms: fixes x :: "'a" assumes "a \<le> x"   and "x \<le> b" shows "\<exists>y. (f has_field_derivative y) (at x)"
subseq_good: fixes f :: "nat \<Rightarrow> 'b list" shows "good subseq f"