hcju/isabelleps huggingface dataset:
f: shows "(\<Sum>x\<in>UNIV. f x *s X $ x) = X $ j"
permutation_similar_mat: fixes A :: "'a mat"   and n :: "nat"   and p :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat n n"   and "p permutes {..<n}" shows "similar_mat A (Matrix.mat n n (\<lambda>(i, j). A $$ (p i, p j)))"
cont_discrete_cpo: fixes f :: "'a \<Rightarrow> 'b" shows "cont f"
minimum_spanning_tree_def: fixes t :: "'a"   and g :: "'a"   and r :: "'a" shows "minimum_spanning_tree t g r \<equiv> spanning_tree t g r \<and> (\<forall>u. spanning_tree u g r \<longrightarrow> sum_class.sum (t \<sqinter> g) \<le> sum_class.sum (u \<sqinter> g))"
atms_of_add_mset: fixes a :: "'b literal"   and A :: "'b literal multiset" shows "atms_of (add_mset a A) = insert (atm_of a) (atms_of A)"
complete_iff_conditionally_complete_extremed_pointed: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "UNIV-complete A less_eq = (conditionally_complete A less_eq \<and> extremed A less_eq \<and> extremed A (\<lambda>x y. less_eq y x))"
conj_disj_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<^bold>\<sqinter> z \<^bold>\<squnion> x = (y \<^bold>\<squnion> x) \<^bold>\<sqinter> (z \<^bold>\<squnion> x)"
404: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> y) \<sqinter> (- x \<sqinter> z) = - x \<sqinter> z"
distinct_27: fixes x61 :: "'b ltln"   and x62 :: "'b ltln" shows "false\<^sub>n \<noteq> x61 or\<^sub>n x62"
pure_tree_unfold: fixes x :: "'b" shows "pure x = Node x (pure x) (pure x)"
is_am_def: fixes am :: "nat \<Rightarrow> 'a list"   and psi :: "'a list Heap.array" shows "is_am am psi \<equiv> \<exists>\<^sub>Al. psi \<mapsto>\<^sub>a l * \<up> (length l = N \<and> (\<forall>i<N. l ! i = am i) \<and> (\<forall>i\<ge>N. am i = []))"
csrp_mem: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R" shows "csrp_fn R I (a \<uplus>\<^bsub>R\<^esub> I) \<in> a \<uplus>\<^bsub>R\<^esub> I"
list_of_R1: fixes x :: "'c list_R1.list_R1" shows "list_R1.list_to_list_R1 (list_R1.list_R1_to_list x) = x"
minus_swap: fixes a :: "atom"   and b :: "atom" shows "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
hom_mult: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) neg (x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y) = hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) neg x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) neg y"
sub1_preserves_dimensions_2: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub1 q k l (r, A) = (r', A')" shows "dim_col A' = dim_col A"
adjacent_to_adjacent_int: fixes C :: "'b set"   and D :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "C \<sim> D"   and "f ` C \<sim> f ` D"   and "f ` C \<noteq> f ` D" shows "f ` (C \<inter> D) = f ` C \<inter> f ` D"
normal_subgroup_axioms: shows "normal_subgroup image.induced.Ker kernel.Partition ([\<cdot>]) (kernel.Class \<one>)"
continuous_on_cong: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "s = t"   and "\<And>x. x \<in> t \<Longrightarrow> f x = g x" shows "continuous_on s f = continuous_on t g"
edges_of_graph_from_list: fixes l :: "('a \<times> 'a) list" assumes "valid_graph_repr l" shows "edges (graph_from_list l) = set l \<union> (set l)\<inverse>"
Bnd: shows "Bnd = Max (abs ` (elements_mat A \<union> set\<^sub>v b))"
counit_simps_2: shows "local.dom \<xi> = f \<star> g"
forall_sem: fixes I :: "('b, 'c, 'a) interp"   and x :: "'a"   and \<phi> :: "('b, 'c, 'a) formula" shows "fml_sem I (Forall x \<phi>) = {v. \<forall>r. repv v x r \<in> fml_sem I \<phi>}"
take_bit_eq_mod: fixes n :: "nat"   and a :: "'a" shows "take_bit n a = a mod (2::'a) ^ n"
rel_optionI: fixes x :: "'c option"   and y :: "'d option"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "Option.is_none x = Option.is_none y"   and "\<lbrakk>\<not> Option.is_none x; \<not> Option.is_none y\<rbrakk> \<Longrightarrow> P (the x) (the y)" shows "rel_option P x y"
prems_4: shows "p \<in> set (comp_red_basis_aux (a # xs) ys)"
prefix_length_le_length: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "prefix_length P xs \<le> length xs"
implicit_span_with: shows "span_with (+) (0::'a) (*\<^sub>R) = subspace.span"
inorder_del: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (del x t) = del_list x (Tree2.inorder t)"
induct: fixes x :: "'b fm"   and P :: "'b fm \<Rightarrow> bool" assumes "\<turnstile> x"   and "\<And>p. tautology p \<Longrightarrow> P p"   and "\<And>i p q. P (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i q)"   and "\<And>p q. \<lbrakk>\<turnstile> p; P p; \<turnstile> (p \<^bold>\<longrightarrow> q); P (p \<^bold>\<longrightarrow> q)\<rbrakk> \<Longrightarrow> P q"   and "\<And>p i. \<lbrakk>\<turnstile> p; P p\<rbrakk> \<Longrightarrow> P (K i p)" shows "P x"
induces_unit_counit_adjunction: shows "unit_counit_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<phi>\<psi>.\<eta> \<phi>\<psi>.\<epsilon>"
that: fixes node_ptr :: "(_) node_ptr" assumes "cast ptr = Some node_ptr" shows "thesis"
key_order_of_ord_def: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "key_order_of_ord ord = Abs_key_order (comp_of_ord ord)"
build_trace: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "SestoftCorrect.steps c c'" shows "\<exists>T. c \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> c'"
lprefix_lfilterI: fixes xs :: "'a llist"   and ys :: "'a llist"   and P :: "'a \<Rightarrow> bool" assumes "lprefix xs ys" shows "lprefix (lfilter P xs) (lfilter P ys)"
gcd_mono: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b d" shows "dvd.dvd times (gcd a b) (gcd c d)"
transitive_closurep_trans'_2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
dvd_mult_cancel: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k * m dvd k * n"   and "0 < k" shows "m dvd n"
extendLevel_step: fixes belowN :: "fmla stream"   and n :: "nat"   and N :: "fmla stream"   and st :: "fmla stream \<times> fmla stream" assumes "i.R belowN = {\<phi>. max_depth \<phi> < n}"   and "i.R N = {\<phi>. max_depth \<phi> = n}"   and "st = (belowN, N)" shows "\<exists>belowNext Next. extendLevel st = (belowNext, Next) \<and> i.R belowNext = {\<phi>. max_depth \<phi> < Suc n} \<and> i.R Next = {\<phi>. max_depth \<phi> = Suc n}"
int_sym: fixes a :: "'a"   and b :: "'a" shows "interval a b = interval b a"
simps_4: fixes xs :: "'a list"   and ys :: "'a list" shows "proper_interval (Some xs) (Some ys) = local.proper_interval_list_aux xs ys"
pr_in_hom_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "partial_magma.in_hom C (pr1 a b) (elementary_category_with_binary_products.prod C pr0 pr1 a b) a"
fresh_ctxt_set_eq: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list" shows "x \<sharp> \<Gamma> = (x \<notin> fst ` set \<Gamma>)"
sup_ge2: fixes y :: "'k multiset"   and x :: "'k multiset" shows "y \<subseteq># x \<union># y"
toCard_pred_toCard: fixes A :: "'c set"   and r :: "('d \<times> 'd) set" assumes "|A| \<le>o r"   and "Card_order r" shows "toCard_pred A r (toCard A r)"
r: shows "generalized_sfw (map simple_rule_dtor fw) p = Some (r, Accept)"
reduced_GB_D1: fixes G :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "extended_ord.punit.is_reduced_GB G" shows "extended_ord.punit.is_Groebner_basis G"
20: shows "irreflexive (prim_W w v e * (w \<sqinter> - prim_EP w v e)\<^sup>\<star> * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
is_red_singletonD: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_red {p} q"   and "p \<in> F" shows "is_red F q"
lim_last: shows "Liminf_llist (lmap (\<lambda>St. grounding_of_clss (wrp.N_of_wstate St \<union> wrp.P_of_wstate St \<union> wrp.Q_of_wstate St)) sswSts) = grounding_of_clss (wrp.N_of_wstate (llast sswSts) \<union> wrp.P_of_wstate (llast sswSts) \<union> wrp.Q_of_wstate (llast sswSts))"
rreq_def: shows "rreq \<equiv> \<lambda>(hops, dip, dsn, dsk, oip, osn, sip, handled). Rreq hops dip dsn dsk oip osn sip handled"
simps_2: fixes nothing :: "unit"   and more :: "'m" shows "RObject.more \<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = more"
prime_power_dvd_pderiv: fixes p :: "'b poly"   and f :: "'b poly" assumes "prime_elem p" shows "p ^ (multiplicity p f - 1) dvd pderiv f"
decseq_SucD: fixes A :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "decseq A" shows "A (Suc i) \<le> A i"
no_interleaving_ordered: fixes ops :: "('a \<times> 'a option) list"   and start :: "'a option"   and xs :: "('a \<times> 'a option) list"   and ys :: "('a \<times> 'a option) list" assumes "insert_ops ops"   and "insert_seq start xs"   and "insert_ops xs"   and "insert_seq start ys"   and "insert_ops ys"   and "set xs \<subseteq> set ops"   and "set ys \<subseteq> set ops"   and "distinct (map fst xs @ map fst ys)"   and "fst (hd xs) < fst (hd ys)"   and "\<And>r. start = Some r \<Longrightarrow> r \<in> set (interp_ins ops)" shows "(\<forall>x\<in>set (map fst xs). \<forall>y\<in>set (map fst ys). list_order ops y x) \<and> (\<forall>r. start = Some r \<longrightarrow> (\<forall>x\<in>set (map fst xs). list_order ops r x) \<and> (\<forall>y\<in>set (map fst ys). list_order ops r y))"
asympartp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "asympartp r x y \<equiv> r x y \<and> \<not> r y x"
simps: fixes m :: "'b simple_match"   and p :: "('b, 'c) simple_packet_scheme" shows "simple_matches m p = (match_iface (iiface m) (p_iiface p) \<and> match_iface (oiface m) (p_oiface p) \<and> simple_match_ip (src m) (p_src p) \<and> simple_match_ip (dst m) (p_dst p) \<and> match_proto (proto m) (p_proto p) \<and> simple_match_port (sports m) (p_sport p) \<and> simple_match_port (dports m) (p_dport p))"
composite_simps_1: shows "src (g \<star> f \<star> g) = src g"
sum_dcharacter_lessThan_le: fixes m :: "nat" assumes "principal_dchar n \<noteq> principal_dchar n" shows "cmod (sum (principal_dchar n) {..<m}) \<le> real (totient n)"
ego_other_ivt: fixes t :: "real" assumes "other.s t \<le> ego.s t" shows "collision {0..t}"
betweenE: fixes a :: "'b"   and b :: "'b"   and x :: "'b" assumes "between (a, b) x" obtains u :: "real" where "0 \<le> u" and "u \<le> 1" and "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
preserves_iso: fixes f :: "'a" assumes "local.iso f" shows "local.iso (L f)"
simps_3: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and \<phi> :: "'b MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Neg \<phi>) j = Monitor.progress \<sigma> \<phi> j"
partn_lst_def: fixes \<beta> :: "'a set"   and \<alpha> :: "nat list"   and \<gamma> :: "nat" shows "Ramsey.partn_lst \<beta> \<alpha> \<gamma> \<equiv> \<forall>f\<in>[\<beta>]\<^bsup>\<gamma>\<^esup> \<rightarrow> {..<length \<alpha>}. \<exists>i<length \<alpha>. \<exists>H\<in>[\<beta>]\<^bsup>(\<alpha> ! i)\<^esup>. f ` [H]\<^bsup>\<gamma>\<^esup> \<subseteq> {i}"
int_of_integer_plus: fixes x :: "integer"   and y :: "integer" shows "int_of_integer (x + y) = int_of_integer x + int_of_integer y"
solve_depressed_cubic_complex_def: fixes e :: "complex"   and f :: "complex" shows "solve_depressed_cubic_complex e f = Let (if e = 0 then all_croots 3 (- f) else let u = hd (croots2 [:- (e ^ 3 / 27), f, 1:]) in Let (all_croots 3 u) (map (\<lambda>z. z - e / (3 * z)))) remdups"
foldl1_append: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ys :: "'b list" assumes "xs \<noteq> []" shows "foldl1 f (xs @ ys) = foldl1 f (foldl1 f xs # ys)"
obtain_trail_in_fcs': fixes a :: "'b"   and u :: "'a"   and v :: "'a" assumes "a \<in> arcs G"   and "u \<in> tail G ` G.face_cycle_set a"   and "v \<in> tail G ` G.face_cycle_set a" obtains p :: "'b list" where "G.trail u p v" and "set p \<subseteq> G.face_cycle_set a"
simps_2: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
bot_less1: fixes X :: "'a process" shows "\<bottom> \<le> X"
0: shows "{a \<in> {a. a < i \<and> a \<in> S}. a < i} = {a. a < i \<and> a \<in> S}"
HOMSetIsSet: fixes X :: "'a"   and Y :: "'a" assumes "X \<in> obj"   and "Y \<in> obj" shows "HOMCollection C X Y \<in> range explode"
surjective_sum: fixes f :: "'a + 'b \<Rightarrow> 'c" shows "(\<lambda>x. f (Inl x)) (+) (\<lambda>y. f (Inr y)) = f"
split_list: fixes x :: "'a"   and A :: "'a list" assumes "x \<in> set A" shows "set (x # removeAll x A) = set A \<and> x \<notin> set (removeAll x A)"
vec_eq_iff: fixes x :: "('d, 'e) vec"   and y :: "('d, 'e) vec" shows "(x = y) = (\<forall>i. x $ i = y $ i)"
assms_2: shows "InvariantVarsF F F0 Vbl"
is_unit_field_poly: fixes p :: "'b poly" shows "is_unit p = (p \<noteq> 0 \<and> degree p = 0)"
elims_3: fixes x :: "'e \<times> ('f, 'g, 'h, 'i) SLin_state_scheme" assumes "\<not> P4 x" obtains s1 :: "'e" and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" where "x = (s1, s2)" and "(\<forall>p. status s2 p = Sleep) \<noteq> (initVals s2 = {})"
f_def: shows "f \<equiv> \<lambda>xs. SOME v. v \<in> S \<and> v \<in> set xs"
eAbs_nchotomy: fixes eX :: "'c withERR"   and xs :: "'varSort"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and x :: "'var" shows "(\<exists>s X. eX = OK X \<and> isInBar (xs, s) \<and> igWls MOD s X) \<or> eAbs MOD xs x eX = ERR"
dims_smult: fixes \<alpha> :: "'a"   and A :: "'a tensor" shows "dims (\<alpha> \<cdot> A) = dims A"
convergent_def: fixes "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a" assumes "class.topological_space open" shows "topological_space.convergent open X = (\<exists>L. filterlim X (topological_space.nhds open L) sequentially)"
union: shows "{- r / 2--r / 2} = {0--r / 2} \<union> {0--- r / 2}"
rep_eq: fixes x :: "('a, 'b) uexpr"   and xa :: "('a, 'b) uexpr" shows "(x \<le> xa) = (\<forall>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<le> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
equality: fixes r :: "'a obj_scheme"   and r' :: "'a obj_scheme" assumes "tag r = tag r'"   and "values r = values r'"   and "obj.more r = obj.more r'" shows "r = r'"
Cons_2: shows "\<forall>a\<in>set L'. case a of (p, i) \<Rightarrow> p = 0"
l_child_lower_bound_alt: fixes l :: "int"   and x :: "int"   and p :: "int" assumes "l \<le> x"   and "x \<le> p" shows "x < l_child l p"
elims: fixes x :: "nat"   and xa :: "('ekey \<times> 'dkey \<times> 'cipher list) option"   and xb :: "unit"   and y :: "('ekey \<times> ('ekey \<times> 'dkey \<times> 'cipher list) option) spmf" assumes "ekey_oracle x xa xb = y" obtains \<eta> :: "nat" and uu_ :: "unit" where "x = \<eta>'__" and "xa = None" and "xb = uu_" and "y = key_gen \<eta>'__ \<bind> (\<lambda>(ekey, dkey). return_spmf (ekey, Some (ekey, dkey, [])))"   | \<eta> :: "nat" and ekey :: "'ekey" and rest :: "'dkey \<times> 'cipher list" and uv_ :: "unit" where "x = \<eta>" and "xa = Some (ekey, rest)" and "xb = uv_" and "y = return_spmf (ekey, Some (ekey, rest))"
SUP_upper: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "i \<in> A" shows "f i \<le> \<Squnion> (f ` A)"
P_def: shows "P = {p. p \<in># prime_factorization n \<and> odd p}"
coincides_val_process: fixes Mkt :: "('c, 'd) discrete_market"   and Mkt2 :: "('c, 'd) discrete_market"   and A :: "'d set"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "coincides_on Mkt Mkt2 A"   and "support_set pf \<subseteq> A" shows "\<forall>n w. val_process Mkt pf n w = val_process Mkt2 pf n w"
diff_ge_0_iff_ge: fixes a :: "'b"   and b :: "'b" shows "((0::'b) \<le> a - b) = (b \<le> a)"
LPDs_Plus: fixes r :: "mregex"   and s :: "mregex" shows "LPDs (MPlus r s) \<subseteq> {MPlus r s} \<union> LPDs r \<union> LPDs s"
has_integral_integral_lebesgue_on: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "integrable (lebesgue_on S) f"   and "S \<in> sets lebesgue" shows "(f has_integral integral\<^sup>L (lebesgue_on S) f) S"
leftmost_Derivation: fixes i :: "nat"   and a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and j :: "nat" assumes "leftmost i a"   and "Derivation a D b"   and "j \<le> i" shows "derivation_ge D j"
predicate1I: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "P \<le> Q"
bounded_linear_def: fixes f :: "'c \<Rightarrow> 'd" shows "bounded_linear f \<equiv> linear f \<and> bounded_linear_axioms f"
assms_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R x y"   and "\<And>x y. R x y \<Longrightarrow> Q R (s1 x) (s2 y)" shows "P x y"
T: fixes t' :: "trm" assumes "t \<mapsto> t'"   and "r = t' \<star> k" shows "P"
simps_1: fixes x :: "nat"   and y :: "nat" shows "eval_AddNum [DNum x, DNum y] = DNum (x + y)"
qelim_ci: fixes qe :: "fm \<Rightarrow> fm"   and p :: "fm"   and bs :: "real list" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bs (qe p) = Ifm bs (E p)" shows "qfree (qelim p qe) \<and> Ifm bs (qelim p qe) = Ifm bs p"
h1: shows "minimal A (\<lambda>S. finite S \<and> S \<subseteq> carrier V \<and> gen_set S)"
facts_7: fixes n :: "nat" assumes "2 * card (uverts H) \<le> n" shows "1 / prob_space_with_indicators.\<mu> (MGn p n) {S. S \<subseteq> {1..n} \<and> card S = card (uverts H)} (\<lambda>S. {es \<in> space (MGn p n). subgraph (f S) (induced_subgraph S (edge_space.edge_ugraph n es))}) \<le> real (card (uverts H) ^ card (uverts H)) * (1 / (real n ^ card (uverts H) * p n ^ card (uedges H)))"
separatedin_empty_2: fixes X :: "'b topology"   and S :: "'b set" shows "separatedin X {} S = (S \<subseteq> topspace X)"
rev_subsetD: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "(c::'b) \<in> (A::'b set)"   and "(A::'b set) \<subseteq> (B::'b set)" shows "(c::'b) \<in> (B::'b set)"
fdr_step_induct: fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "nat"   and c :: "'a" assumes "\<And>v c. (\<And>b. \<lbrakk>v \<noteq> 0; v < n\<rbrakk> \<Longrightarrow> P (2 * v) ((2::'a) * c + (if b then 1::'a else (0::'a)))) \<Longrightarrow> P v c" shows "P v c"
sel_2: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<not> lnull xs"   and "P (lhd xs)" shows "ltl (ltakeWhile P xs) = ltakeWhile P (ltl xs)"
prems_3: shows "bvars\<^sub>s\<^sub>t (S @ \<langle>ac: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S' @ [x]) \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
intros_9: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "wf_term (sig \<Theta>) t"   and "\<turnstile>\<^sub>\<tau> t : \<tau> \<rightarrow> \<tau>'" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau> (t $ Bv 0)) t"
K: shows "{p. prime p \<and> [p = h] (mod n)} \<subseteq> {..<K}"
Re_Ln: fixes z :: "complex" assumes "z \<noteq> 0" shows "Re (Ln z) = ln (cmod z)"
convergent_eq_Cauchy: fixes S :: "nat \<Rightarrow> 'c" shows "(\<exists>l. S \<longlonglongrightarrow> l) = Cauchy S"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c llist"   and b :: "'d llist" shows "llist_all2 R\<inverse>\<inverse> a b = llist_all2 R b a"
times_matrix_I_2: fixes x :: "(nat \<times> nat) \<times> nat \<times> nat" shows "x \<otimes> I = x"
surj_on_multisets_mset: fixes A :: "'b set" shows "\<forall>M\<in>multisets A. \<exists>xs\<in>lists A. M = mset xs"
x_coord_smooth: fixes a :: "real"   and b :: "real" shows "x_coord C1_differentiable_on {a..b}"
intro: fixes R :: "('e, 'f) ring_scheme"   and H :: "'e set" assumes "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<And>h1 h2. \<lbrakk>h1 \<in> H; h2 \<in> H; h1 \<otimes>\<^bsub>R\<^esub> h2 = \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> h1 = \<zero>\<^bsub>R\<^esub> \<or> h2 = \<zero>\<^bsub>R\<^esub>" shows "subdomain_axioms H R"
path_components_of_subset: fixes C :: "'a set"   and X :: "'a topology" assumes "C \<in> path_components_of X" shows "C \<subseteq> topspace X"
finite_ReZ_segments_poly_linepath: fixes p :: "complex poly"   and a :: "complex"   and b :: "complex"   and z :: "complex" shows "finite_ReZ_segments (poly p \<circ> linepath a b) z"
attacker: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND"   and bc' :: "'nid list" assumes "\<not> honest nid" shows "eval nid t t' 0 (\<box>\<^sub>b[\<lambda>nd. bc' = (SOME b. b \<in> pin nd \<union> {bc nd})]\<^sub>b \<longrightarrow>\<^sup>b \<circle>\<^sub>b[\<lambda>nd. \<not> mining nd \<and> prefix (bc nd) bc' \<or> mining nd \<and> (\<exists>b. bc nd = bc' @ [b])]\<^sub>b)"
stuff_8: shows "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem \<le> dma mem x"
update': fixes a :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b"   and b :: "'b" assumes "a \<notin> keys f"   and "b \<noteq> (0::'b)"   and "P f" shows "P (update a b f)"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. f (x + y) = f x + f y" shows "additive f"
subst_cls_id_subst: fixes C :: "'a literal multiset" shows "C \<cdot> id_subst = C"
assms_3: shows "C = resolvent_upon P1 P2 A"
fcs_not_triv: shows "G.face_cycle_set a \<noteq> {a} \<or> G.face_cycle_set a' \<noteq> {a'}"
assms_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "p divides a \<otimes> b" shows "p divides a \<or> p divides b"
count: fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "consider x"   and "x \<in> outs_\<I> \<I>" shows "count s' \<le> Suc (count s)"