hcju/isabelleps huggingface dataset:
inject: fixes x1 :: "(nat \<times> linear_poly) list"   and x2 :: "(nat, 'a \<times> 'b) mapping"   and x3 :: "(nat, 'a \<times> 'b) mapping"   and x4 :: "(nat, 'b) mapping"   and x5 :: "bool"   and x6 :: "'a list option"   and y1 :: "(nat \<times> linear_poly) list"   and y2 :: "(nat, 'a \<times> 'b) mapping"   and y3 :: "(nat, 'a \<times> 'b) mapping"   and y4 :: "(nat, 'b) mapping"   and y5 :: "bool"   and y6 :: "'a list option" shows "(State x1 x2 x3 x4 x5 x6 = State y1 y2 y3 y4 y5 y6) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4 \<and> x5 = y5 \<and> x6 = y6)"
diff_le_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a - b \<le> c) = (a \<le> c + b)"
v1: shows "range ((*\<^sub>V) P) \<subseteq> (\<lambda>x. x - P *\<^sub>V x) -` {0::'a}"
4: shows "m \<in> {m. 0 < m \<and> prime_factors m \<subseteq> {..n}}"
set_of_sub_inc_left: fixes A :: "'a interval"   and A' :: "'a interval"   and B :: "'a interval" assumes "set_of A \<subseteq> set_of A'" shows "set_of (A - B) \<subseteq> set_of (A' - B)"
nat_in_between_eq_2: fixes a :: "nat"   and b :: "nat" shows "(a \<le> b \<and> b < Suc a) = (b = a)"
shows_sep_map: fixes f :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and sep :: "char list \<Rightarrow> char list"   and g :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "shows_sep f sep (map g xs) = shows_sep (f \<circ> g) sep xs"
monomials_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "monomials (MPoly_Type.monom m a) = (if a = (0::'a) then {} else {m})"
suma_left_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I" shows "Tarski_neutral_dimensionless.SumA Bet Cong C B A D E F G H I"
CMP_def: fixes r :: "'a"   and s :: "'a" shows "CMP r s \<equiv> \<lparr>Chn = \<lbrakk>\<lbrakk>two_composable_identities_in_bicategory_of_spans.cmp (\<cdot>) (\<star>) \<a> \<i> src trg r s\<rbrakk>\<rbrakk>, Dom = Dom (SPN r \<circ> SPN s), Cod = Dom (SPN (r \<star> s))\<rparr>"
d_IN_h_source: fixes i :: "nat" shows "d_IN (h_diff i) (source \<Delta>) = 0"
rev_nonempty_induct: fixes xs :: "'b list"   and P :: "'b list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "\<And>x. P [x]"   and "\<And>x xs. \<lbrakk>xs \<noteq> []; P xs\<rbrakk> \<Longrightarrow> P (xs @ [x])" shows "P xs"
has_derivative_zero_connected_constant: fixes S :: "'a set"   and k :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "connected S"   and "open S"   and "finite k"   and "continuous_on S f"   and "\<forall>x\<in>S - k. (f has_derivative (\<lambda>h. 0::'b)) (at x within S)" obtains c :: "'b" where "\<And>x. x \<in> S \<Longrightarrow> f x = c"
simps_5: fixes b :: "bool"   and val :: "val" shows "assigns_if b (Lit val) = (if val = Bool b then {} else UNIV)"
rbt_union_with_key_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('b, 'c) rbt"   and t2 :: "('b, 'c) rbt" shows "ord.rbt_union_with_key less f t1 t2 = (case RBT_Impl.compare_height t1 t1 t2 t2 of compare.LT \<Rightarrow> RBT_Impl.fold (ord.rbt_insert_with_key less (\<lambda>k v w. f k w v)) t1 t2 | compare.GT \<Rightarrow> RBT_Impl.fold (ord.rbt_insert_with_key less f) t2 t1 | compare.EQ \<Rightarrow> rbtreeify (ord.sunion_with less f (RBT_Impl.entries t1) (RBT_Impl.entries t2)))"
borel_measurable_iff_halfspace_le: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i \<le> a} \<in> sets M)"
safe_combined_def: fixes c :: "('b, 't, 'loc) Combined.configuration" shows "safe_combined c \<equiv> \<forall>loc1 loc2 t s p. 0 < zcount (cri.records (exchange_config c)) (loc1, t) \<and> s \<in>\<^sub>A path_summary loc1 loc2 \<and> init c p \<longrightarrow> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp (prop_config c p) loc2) \<and> t' \<le> results_in t s)"
cinner_complex_def: fixes x :: "complex"   and y :: "complex" shows "cinner x y = cnj x * y"
simps: fixes x :: "int"   and n :: "nat" shows "incr2 (x, n) = (x, n + 1)"
convex_connected: fixes S :: "'c set" assumes "convex S" shows "connected S"
mult_sign_intros_1: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(0::'a) \<le> a * b"
neq_s1_s2: shows "s1 \<noteq> s2"
nop: fixes loc1 :: "'c"   and loc2 :: "'c"   and t1 :: "'t"   and t2 :: "'t"   and c :: "('c, 't, 'd) Propagate.configuration_scheme"   and \<Delta> :: "('c \<times> 't) zmultiset" shows "loc1 \<noteq> loc2 \<or> t1 \<noteq> t2 \<longrightarrow> zcount (c_pts (apply_cm c loc2 t2 (zcount \<Delta> (loc2, t2))) loc1) t1 = zcount (c_pts c loc1) t1"
generator_sub_carrier: fixes H :: "'a set" assumes "generator R M H" shows "H \<subseteq> carrier M"
prj_simps_11: shows "C.seq trg_f.prj\<^sub>1 \<p>\<^sub>0[trg.leg0, trg.leg1 \<cdot> trg_f.prj\<^sub>1]"
If: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>t {\<lambda>l s. P l s \<and> bval b s} c\<^sub>1 {Q}"   and "\<turnstile>\<^sub>t {\<lambda>l s. P l s \<and> \<not> bval b s} c\<^sub>2 {Q}" shows "\<turnstile>\<^sub>t {P} IF b THEN c\<^sub>1 ELSE c\<^sub>2 {Q}"
alpha_Tree_eqvt': fixes t1 :: "('a, 'b, 'c, 'd) Tree"   and t2 :: "('a, 'b, 'c, 'd) Tree"   and p :: "perm" shows "(t1 =\<^sub>\<alpha> t2) = (p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2)"
hyps_2: shows "sourcenode a -a # as'\<rightarrow>\<^sub>c\<^sub>d n''"
Par_Arr_norm: fixes t :: "'c local.term" assumes "Arr t" shows "Par \<^bold>\<parallel>t\<^bold>\<parallel> t"
degree_polyneg: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p n" shows "Reflected_Multivariate_Polynomial.degree (~\<^sub>p p) = Reflected_Multivariate_Polynomial.degree p"
gcp_append: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "gcp (xs @ ys) (xs @ zs) = xs @ gcp ys zs"
dom_f: shows "\<D>\<^sub>\<circ> f = n"
cong_conga_perp: fixes B :: "'p"   and P :: "'p"   and A :: "'p"   and C :: "'p" assumes "B P TS A C"   and "Cong A B C B"   and "A B P CongA C B P" shows "A C Perp B P"
worst_chain_bad: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "wf {(x, y). gt y x}"   and "bad p x" shows "inf_chain p (worst_chain p gt)"
simps_1: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and c :: "('c \<times> (Mode \<Rightarrow> 'a set)) list \<times> ('a \<Rightarrow> 'b)"   and c' :: "('c \<times> (Mode \<Rightarrow> 'a set)) list \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.meval_sched eval [] c c' = (c = c')"
simps_2: fixes A :: "'b set" assumes "infinite A" shows "esize A = \<infinity>"
synth_payload2: fixes Y :: "msg set"   and X :: "msg set" assumes "Y \<inter> payload = {}" shows "synth (Y \<union> X) \<subseteq> synth X \<union> - payload"
extend_map_affine_to_sphere_cofinite_gen: fixes S :: "'a set"   and U :: "'b set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and L :: "'a set" assumes "compact S"   and "convex U"   and "bounded U"   and "affine T"   and "S \<subseteq> T"   and "aff_dim T \<le> aff_dim U"   and "continuous_on S f"   and "f ` S \<subseteq> rel_frontier U"   and "\<And>C. \<lbrakk>C \<in> components (T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}" obtains K :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite K" and "K \<subseteq> L" and "K \<subseteq> T" and "disjnt K S" and "continuous_on (T - K) g" and "g ` (T - K) \<subseteq> rel_frontier U" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
GroupHom_idhom: shows "GroupHom M (id \<down> M)"
sup_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "\<Squnion>\<^bsub>L\<^esub>A \<in> carrier L"
PDG_path_ddep: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and control_dependence :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b"   and V :: "'d"   and n' :: "'b" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "PDG.PDG_edge sourcenode targetnode valid_edge Def Use control_dependence n (Some V) n'" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n'"
IH: fixes e :: "'v \<times> 'v" shows "h_plus i e \<le> g e"
pmf_of_multiset_pmf_of_mapping: fixes A :: "'b multiset"   and xs :: "'b list" assumes "A \<noteq> {#}"   and "set xs = set_mset A"   and "distinct xs" shows "mapping_of_pmf (pmf_of_multiset A) = Mapping.tabulate xs (\<lambda>x. real (count A x) / real (size A))"
with_proj_simps_2: fixes G :: "'b pair_pre_digraph" shows "arcs (with_proj G) = parcs G"
right_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
inject: fixes x1 :: "'e option"   and x2 :: "('d \<times> ('d, 'e) trie) list"   and y1 :: "'e option"   and y2 :: "('d \<times> ('d, 'e) trie) list" shows "(Trie x1 x2 = Trie y1 y2) = (x1 = y1 \<and> x2 = y2)"
mC: shows "mC_ZOC ZObis (fst cf1) (fst cf2) (snd cf1) (snd cf2) I0 P F"
b: shows "abc_steps_l (0, lm) ap (Suc stp) = (as, am)"
taus: shows "{\<tau>1, \<tau>2, \<tau>3', \<delta>1'''} \<subseteq> seq ars"
inject_1: fixes a :: "'d"   and a' :: "'d" shows "(Inl a = Inl a') = (a = a')"
prj_simps_3: shows "C.arr \<mu>\<nu>\<pi>.Prj\<^sub>0"
assms: shows "get_S w1 = 0 \<and> w2 AND 128 = 0 \<and> w3 AND 128 = 0"
triangle_num_def: fixes n :: "nat" shows "triangle_num n = n * (n + 1) div 2"
emeasure_real: fixes A :: "'a set" shows "\<exists>r\<ge>0. emeasure M A = ennreal r"
rat_to_normalized_int_poly_code: fixes p :: "rat poly" shows "rat_to_normalized_int_poly p = (if p = 0 then (1, 0) else case rat_to_int_poly p of (s, q) \<Rightarrow> let c = content q in (rat_of_int c / rat_of_int s, sdiv_poly q c))"
aff_dim_def: fixes V :: "'c set" shows "aff_dim V = (SOME d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = d + 1)"
prefixeq_vertices: fixes c :: "'form"   and "is" :: "nat list"   and is' :: "nat list" assumes "(c, is) |\<in>| vertices"   and "prefix is' is" shows "(c, is') |\<in>| vertices"
map_list_eq_cong: fixes b :: "'b list"   and ba :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "b \<approx> ba" shows "map f b \<approx> map f ba"
him_hv: fixes x :: "(real, 3) vec" shows "HIm (Hv x) = x"
evaln_elim_cases_14: fixes G :: "prog"   and s :: "st"   and va :: "Term.var"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (Acc va)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains vb :: "val" and f :: "val \<Rightarrow> state \<Rightarrow> state" where "v = In1 vb" and "G\<turnstile>Norm s \<midarrow>va=\<succ>(vb, f)\<midarrow>n\<rightarrow> s'"
prems: shows "position_of t ((a # p) @ q @ [dq])"
supp_subst: fixes e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var" shows "supp e[y::=x] \<subseteq> supp e - {atom y} \<union> {atom x}"
results_gpv_mono: fixes \<I>' :: "('h, 'i) \<I>"   and \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv" assumes "\<I>' \<le> \<I>"   and "\<I>' \<turnstile>g gpv \<surd>" shows "results_gpv \<I> gpv \<subseteq> results_gpv \<I>' gpv"
Sup_measure_def: shows "Sup = Sup_lexord space (Sup_lexord sets local.Sup_measure' (\<lambda>U. sigma (\<Union> (space ` U)) (\<Union> (sets ` U)))) (\<lambda>U. sigma (\<Union> (space ` U)) {})"
reflFreshAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'e, 'c, 'f) Recursion.model_scheme" shows "reflFreshAbs hA MOD \<equiv> \<forall>ys y us s A. wlsAbs (us, s) A \<longrightarrow> gFreshAbs MOD ys y A (hA A) \<longrightarrow> freshAbs ys y A"
wf_tuple_length: fixes n :: "nat"   and V :: "nat set"   and x :: "'b option list" assumes "wf_tuple n V x" shows "length x = n"
def_gfp_unfold: fixes A :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "A \<equiv> gfp f"   and "mono f" shows "A = f A"
assms_5: shows "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"
properL_update: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'e) cmd list"   and c' :: "('c, 'd, 'e) cmd"   and n :: "nat" assumes "PL cval"   and "properL cl"   and "proper c'" shows "properL (cl[n := c'])"
active_threadsp_active_threads_eq: fixes final :: "'b \<Rightarrow> bool"   and r :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and s :: "('d, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" shows "multithreaded_base.active_threadsp final r s = (\<lambda>x. x \<in> multithreaded_base.active_threads final r s)"
quotE: fixes R :: "'d set" assumes "R \<in> quot" obtains a :: "'d" where "R = {x. a \<sim> x}"
producesD: fixes C :: "'a literal multiset"   and A :: "'a" assumes "production C = {A}" shows "C \<in> N \<and> C \<noteq> {#} \<and> Pos A = Max_mset C \<and> \<not> interp C \<Turnstile> C \<and> S C = {#}"
hcomp_ide_cod: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd" assumes "natural_transformation A B F G \<tau>" shows "identity_functor.map B \<circ> \<tau> = \<tau>"
poly_y_x_def: fixes p :: "'c poly poly" shows "poly_y_x p \<equiv> \<Sum>i\<le>degree p. \<Sum>j\<le>degree (coeff p i). monom (monom (coeff (coeff p i) j) i) j"
thm_relation_negation_1_3: fixes v :: "i"   and F :: "\<Pi>\<^sub>3"   and x :: "\<nu>"   and y :: "\<nu>"   and z :: "\<nu>" shows "[\<lparr>F\<^sup>-,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr> in v]"
get_component_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_dom_component ptr"
bij_is_inj: fixes f :: "'d \<Rightarrow> 'e" assumes "bij f" shows "inj f"
mult_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<cdot> b \<in> carrier G"
mat_oner: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "f \<otimes> \<epsilon> = f"
eq_fin_prefix_fininf_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
assms_3: shows "e1 \<in> in_arcs G v"
lipschitz_grid: fixes j :: "nat" assumes "t (Suc j) \<le> t1"   and "dist (x (t j)) (grid_function (discrete_evolution incr) (x (t 0)) t j) \<le> \<bar>r\<bar>" shows "dist (incr (stepsize j) (t j) (x (t j))) (incr (stepsize j) (t j) (grid_function (discrete_evolution incr) (x (t 0)) t j)) \<le> L * dist (x (t j)) (grid_function (discrete_evolution incr) (x (t 0)) t j)"
language_eq_right_lang: shows "Accessible.language = Accessible.right_lang (init Accessible_dfa)"
comp_inv_arr: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c" assumes "C.VV.inverse_arrows f g" shows "C.VV.comp g f = C.VV.dom f"
v2: shows "atom v2 \<sharp> (t, X1, X3, X4)"
plus_Nat_def: fixes a :: "'a"   and b :: "'a" shows "a + b = AbsNat (RepNat a + RepNat b)"
pred_map: fixes Q :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'c" shows "pred_fun (\<lambda>_. True) Q (f \<circ> x) = pred_fun (\<lambda>_. True) (Q \<circ> f) x"
simps_6: fixes I :: "('b, 'c, 'a) interp"   and \<alpha> :: "('b, 'c, 'a) hp"   and \<phi> :: "('b, 'c, 'a) formula" shows "fml_sem I (\<langle> \<alpha> \<rangle> \<phi>) = {uu_. \<exists>\<nu> \<omega>. uu_ = \<nu> \<and> (\<nu>, \<omega>) \<in> prog_sem I \<alpha> \<and> \<omega> \<in> fml_sem I \<phi>}"
eventually_at_bot_dense: fixes P :: "'a \<Rightarrow> bool" shows "eventually P at_bot = (\<exists>N. \<forall>n<N. P n)"
invar_cons_up: fixes t :: "('c, 'd) BinomialTree"   and bq :: "('c, 'd) BinomialTree list"   and t' :: "('c, 'd) BinomialTree" assumes "invar (t # bq)"   and "rank t' < rank t"   and "tree_invar t'" shows "invar (t' # t # bq)"
XD_XH_lfp_F_1: shows "fst (lfp_F ds) \<subseteq> fst XD_XH"
abs_summable_on_union: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "f abs_summable_on A"   and "f abs_summable_on B" shows "f abs_summable_on A \<union> B"
enat_ord_code_1: fixes m :: "nat"   and n :: "nat" shows "(enat m \<le> enat n) = (m \<le> n)"
card_of_UNION_ordLeq_infinite: fixes B :: "'d set"   and I :: "'e set"   and A :: "'e \<Rightarrow> 'f set" assumes "infinite B"   and "ordLeq3 (card_of I) (card_of B)"   and "\<forall>i\<in>I. ordLeq3 (card_of (A i)) (card_of B)" shows "ordLeq3 (card_of (\<Union> (A ` I))) (card_of B)"
eval_subst_1: fixes \<phi> :: "'a ltln"   and m :: "'a ltln \<Rightarrow> 'a ltln option" assumes "eval \<phi> = Yes" shows "eval (subst \<phi> m) = Yes"
respects_labels_adjacent: fixes B :: "'c set"   and \<phi> :: "'a \<Rightarrow> 'c"   and C :: "'a set"   and D :: "'a set" assumes "label_wrt B \<phi>"   and "chamber C"   and "chamber D"   and "C \<sim> D"   and "\<forall>v\<in>C. \<phi> (f v) = \<phi> v" shows "\<forall>v\<in>D. \<phi> (f v) = \<phi> v"
supp_sum_empty: fixes f :: "'b \<Rightarrow> 'a" shows "supp_sum f {} = (0::'a)"
mon_pl_cons: fixes e :: "'b set \<times> 'b set"   and w :: "('b set \<times> 'b set) list" shows "mon_pl (e # w) = fst e \<union> snd e \<union> mon_pl w"
zero_order_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
\<U>_def: shows "\<U> = (\<lambda>a b. uniform_measure lborel {a..b})"
tendsto_add_ereal_nonneg: fixes x :: "ereal"   and y :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal" assumes "x \<noteq> - \<infinity>"   and "y \<noteq> - \<infinity>"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
\<Sigma>_concat: fixes f :: "'d \<Rightarrow> 'c"   and j :: "int"   and k :: "int"   and l :: "int" shows "\<Sigma> f j k + \<Sigma> f k l = \<Sigma> f j l"
inner_quad_gt: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "2 * (a \<bullet> b) < a \<bullet> a + b \<bullet> b"
simps: fixes x :: "'a extB" shows "x\<^sup>T = x"
poly_subst_cong: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and f :: "'b \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and g :: "'b \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "p = q"   and "\<And>x. x \<in> indets q \<Longrightarrow> f x = g x" shows "poly_subst f p = poly_subst g q"
root_rat: fixes p :: "nat"   and x :: "rat" assumes "p \<noteq> 0 \<or> x \<noteq> 1" shows "set (root_rat p x) = {y. y ^ p = x}"
distinct_6: fixes x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree"   and x1 :: "('e, 'f, 'g) Tree set['e]" shows "tAct x41 x42 \<noteq> tConj x1"
ord_iso_ord_embed: fixes f :: "'b \<Rightarrow> 'c" assumes "ord_iso f" shows "ord_embed f"
presWls_wlsBinp: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'c, 'i) Recursion.model_scheme" assumes "wlsBinp delta binp"   and "presWlsAbs hA MOD" shows "gWlsBinp MOD delta (lift hA binp)"
simps_1: fixes f :: "'c" shows "\<lbrace>\<^bold>\<langle>f\<^bold>\<rangle>\<rbrace> = V f"
concat_last_suf: fixes ws :: "'b list list" assumes "ws \<noteq> \<epsilon>" shows "last ws \<le>s concat ws"
intros: fixes A :: "'b set"   and R :: "'d set"   and conv :: "('b, 'a, 'c, 'd) converter"   and B :: "'a \<Rightarrow> bool"   and C :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>results_converter (\<I>_uniform A UNIV) (\<I>_uniform UNIV R) conv. B x"   and "\<forall>out\<in>outs_converter (\<I>_uniform A UNIV) (\<I>_uniform UNIV R) conv. C out" shows "pred_converter A B C R conv"
assms_1: shows "T\<^sub>p init qs Strat = T\<^sub>p_opt init qs"
assms_15: fixes \<phi> :: "'a formula"   and \<psi> :: "'a formula"   and I :: "\<I>" assumes "fv \<phi> \<subseteq> fv \<psi>"   and "safe_formula \<phi>"   and "P \<phi>"   and "P \<psi>" shows "P (Until \<phi> I \<psi>)"
less_le_not_le: fixes x :: "'c word"   and y :: "'c word" shows "(x <s y) = (x \<le>s y \<and> \<not> y \<le>s x)"
nAct_active: fixes i :: "nat"   and t :: "cnf llist"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "enat i < llength t"   and "active c (lnth t i)" shows "dynamic_component.nAct tCMP active c (enat (Suc i)) t = eSuc (dynamic_component.nAct tCMP active c (enat i) t)"
simps_2: fixes x :: "'a"   and v :: "'b"   and l :: "('a, 'b) rbt"   and c :: "color"   and y :: "'a"   and w :: "'b"   and r :: "('a, 'b) rbt" shows "ins x v (Node l c y w r) = (if c = B then if x = y then Node l B x v r else if x < y then balance (ins x v l) y w r else balance l y w (ins x v r) else if x = y then Node l R x v r else if x < y then Node (ins x v l) R y w r else Node l R y w (ins x v r))"
lookup_tensor_prod: fixes is1 :: "nat list"   and A :: "'a tensor"   and is2 :: "nat list"   and B :: "'a tensor" assumes "is1 \<lhd> dims A"   and "is2 \<lhd> dims B" shows "lookup (local.tensor_prod_otimes A B) (is1 @ is2) = lookup A is1 * lookup B is2"
leadsTo_Un_distrib: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "(F \<in> A \<union> B \<longmapsto> C) = (F \<in> A \<longmapsto> C \<and> F \<in> B \<longmapsto> C)"
upRulesp_upRules_eq: shows "upRulesp = (\<lambda>x xa. (x, xa) \<in> upRules)"
stabilizer_contains_P: fixes P :: "'a set"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := conj.stabilizer P\<rparr>))"   and "P \<in> stabGroup.subgroups_of_size p" shows "P \<subseteq> group_action.stabilizer (G\<lparr>carrier := conj.stabilizer P\<rparr>) (stabGroup.conjugation_action p) P"
diff_eq_0_ennreal: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < top"   and "a \<le> b" shows "a - b = 0"
state_set_card: fixes X :: "'a list" shows "card (state_set X) = length X"
nested_propos_bounded: fixes \<phi> :: "'a ltl"   and \<nu> :: "'b" shows "nested_propos (f \<phi> \<nu>) \<subseteq> nested_propos \<phi>"
conj_ac_1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a AND b) AND c = a AND b AND c"
mu_exact_triangle: fixes v :: "'a"   and s :: "'a"   and c :: "'b \<Rightarrow> real" assumes "v \<noteq> s"   and "s \<rightarrow>\<^sup>* v"   and "\<And>e. e \<in> arcs G \<Longrightarrow> 0 \<le> c e" obtains u :: "'a" and e :: "'b" where "\<mu> c s v = \<mu> c s u + ereal (c e)" and "arc e (u, v)"
st2pvsttv: shows "st2.ptensor_vec = st.tensor_vec"