hcju/isabelleps huggingface dataset:
distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A"   and "x \<in>\<^sub>\<circ> B" shows "False"
in_set_dropD: fixes x :: "'a"   and n :: "nat"   and xs :: "'a list" assumes "x \<in> set (drop n xs)" shows "x \<in> set xs"
max_fun_diff: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> rst.FINFUNC"   and "g \<in> rst.FINFUNC" shows "\<exists>a b. a \<noteq> b \<and> a \<in> Field rs.oexp \<and> b \<in> Field rs.oexp \<and> f (t.max_fun_diff f g) = a \<and> g (t.max_fun_diff f g) = b"
fusion_gfp_geq: fixes H :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and G :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" assumes "mono (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "dist_over_inf (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "\<And>x::'b::refinement_lattice. ((H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) x \<sqsubseteq> (F \<circ> (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) x" shows "gfp (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<sqsubseteq> (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (gfp (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice))"
Nil_tl: fixes xs :: "'b list" shows "(\<epsilon> = tl xs) = (xs = \<epsilon> \<or> (\<exists>x. xs = [x]))"
leI: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
merge_def: fixes t1 :: "'a \<Rightarrow> 'b option"   and t2 :: "'a \<Rightarrow> 'b option" shows "merge t1 t2 = (\<lambda>m. if t1 m = t2 m then t1 m else None)"
transrec_def: fixes H :: "(V \<Rightarrow> V) \<Rightarrow> V \<Rightarrow> V"   and a :: "V" shows "transrec H a \<equiv> wfrec {(x, y). x \<in> elts y} H a"
assms_17: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
uminus_mult_left_mat: shows "- A * B = - (A * B)"
add_strict_increasing: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "(0::'a) < a"   and "b \<le> c" shows "b < a + c"
size_single: fixes b :: "'c" shows "size {#b#} = 1"
distinct_hd_tl: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs"   and "x = hd xs" shows "x \<notin> set (tl xs)"
intro: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option" assumes "Port_Graph nodes inPorts outPorts vertices nodeOf edges"   and "Port_Graph_Signature_Scoped inPorts outPorts hyps" shows "Scoped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"
that: fixes A :: "'a multiset"   and C :: "'a zmultiset"   and B :: "'a multiset" assumes "M = zmset_of A + C"   and "N = zmset_of B + C"   and "A \<subset># B" shows "thesis"
psubst_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set" assumes "goodEnv rho"   and "good X" shows "good X #[rho]"
ub_ln_2: fixes y :: "float"   and prec :: "nat"   and x :: "float" assumes "Some y = ub_ln prec x" shows "0 < real_of_float x"
tabs_1: shows "table (args_n args) (args_L args) rel1"
to_total_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" shows "to_total \<Gamma> \<equiv> \<lambda>v. if v \<in> dom \<Gamma> then the (\<Gamma> v) else dma_type v"
sa_def: fixes G :: "('h, 'i, 'j) sa_rec_scheme" shows "sa G \<equiv> graph G"
WT_elim_cases_12: fixes 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"   and e :: "expr"   and T :: "ty" assumes "P,E \<turnstile> throw e :: T" obtains C :: "char list" where "T = Void" and "P,E \<turnstile> e :: Class C"
insert_iff: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" shows "(a \<in> Set.insert b A) = (a = b \<or> a \<in> A)"
pCons_0_as_mult: fixes p :: "'a poly" shows "pCons (0::'a) p = [:0::'a, 1::'a:] * p"
R: fixes A :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and B :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and n :: "nat"   and s :: "(char list \<Rightarrow> int option) \<times> nat" assumes "\<And>s. A s \<longrightarrow> B s" shows "(A \<and>* $ n) s \<longrightarrow> (B \<and>* $ n) s"
real_non_neg_vec_def: fixes v :: "(complex, 'a) Finite_Cartesian_Product.vec" shows "real_non_neg_vec v \<equiv> \<forall>a\<in>vec_elements_h v. a \<in> \<real> \<and> 0 \<le> Re a"
abs_eq: fixes xa :: "'b ltln"   and x :: "'b ltln" shows "ltln\<^sub>P_eq_test (abs_ltln\<^sub>P xa) (abs_ltln\<^sub>P x) = xa \<sim>\<^sub>P x"
runit_char_eqn: fixes a :: "'a" assumes "ide a" shows "local.prod (runit a) \<one> = local.prod a \<iota> \<cdot> assoc a \<one> \<one>"
compare_expansions_EQ: fixes F :: "'a"   and G :: "'a"   and c :: "real"   and c' :: "real"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "compare_expansions F G = (EQ, c, c')"   and "trimmed F"   and "trimmed G"   and "(f expands_to F) basis"   and "(g expands_to G) basis"   and "basis_wf basis" shows "(\<lambda>x. c' * f x) \<sim>[at_top] (\<lambda>x. c * g x)"
vt_id_1: fixes v :: "val" shows "V (T v) = v"
comp_swap: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'c" shows "f \<circ> Fun.swap a b g = Fun.swap a b (f \<circ> g)"
wordinterval_UNIV_def: shows "wordinterval_UNIV \<equiv> WordInterval 0 (- 1)"
integral_distr: fixes g :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integral\<^sup>L (distr M N g) f = LINT x|M. f (g x)"
ab_1: shows "0 \<le> a \<and> a \<le> 2 \<and> cos a = 0"
BijGroup_def: fixes S :: "'d set" shows "BijGroup S = \<lparr>carrier = Bij S, mult = \<lambda>g\<in>Bij S. restrict (compose S g) (Bij S), one = \<lambda>x\<in>S. x\<rparr>"
simps: fixes args :: "Monitor.args"   and cur :: "nat"   and dt :: "nat"   and tp :: "nat"   and tss :: "nat queue"   and len :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and a1_map :: "('a option list, nat) mapping"   and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping"   and "done" :: "'a option list set list"   and done_length :: "nat"   and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list" shows "valid_mmuaux' args cur dt (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) auxlist = (args_L args \<subseteq> args_R args \<and> maskL = join_mask (args_n args) (args_L args) \<and> maskR = join_mask (args_n args) (args_R args) \<and> len \<le> tp \<and> length (linearize tss) = len \<and> sorted (linearize tss) \<and> (\<forall>t\<in>set (linearize tss). t \<le> cur \<and> enat cur \<le> enat t + right (args_ivl args)) \<and> table (args_n args) (args_L args) (Mapping.keys a1_map) \<and> Mapping.keys a2_map = {tp - len..tp} \<and> (\<forall>xs\<in>Mapping.keys a1_map. case Mapping.lookup a1_map xs of Some tp' \<Rightarrow> tp' < tp) \<and> (\<forall>tp'\<in>Mapping.keys a2_map. case Mapping.lookup a2_map tp' of Some m \<Rightarrow> table (args_n args) (args_R args) (Mapping.keys m) \<and> (\<forall>xs\<in>Mapping.keys m. case Mapping.lookup m xs of Some tstp \<Rightarrow> tstp_lt tstp (cur - (left (args_ivl args) - 1)) tp \<and> isl tstp = (0 < left (args_ivl args)))) \<and> length done = done_length \<and> length done + len = length auxlist \<and> rev done = map proj_thd (take (length done) auxlist) \<and> Ball (set (take (length done) auxlist)) (check_before (args_ivl args) dt) \<and> sorted (map fst auxlist) \<and> list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map (args_pos args) tp' a1_map) (\<lambda>ts' tp'. filter_a2_map (args_ivl args) ts' tp' a2_map)) (drop (length done) auxlist) (zip (linearize tss) [tp - len..<tp]))"
simps: fixes p :: "real poly" shows "pders p = (if p = 0 then [] else p # pders (pderiv p))"
convex_hull_finite_union: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}" shows "convex hull \<Union> (S ` I) = {\<Sum>i\<in>I. c i *\<^sub>R s i |c s. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
steps_append: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and s :: "'c list"   and qs' :: "'c list"   and as' :: "(nat \<times> nat list) list" assumes "length qs = length as" shows "Partial_Cost_Model.steps s (qs @ qs') (as @ as') = Partial_Cost_Model.steps (Partial_Cost_Model.steps s qs as) qs' as'"
spanning_forest_def: fixes F :: "('a, 'b) graph"   and G :: "('a, 'b) graph" shows "spanning_forest F G \<equiv> forest F \<and> maximally_connected F G \<and> subgraph F G"
parts_partsD: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> parts (parts H)" shows "X \<in> parts H"
finals_def: fixes es :: "'c exp list" shows "finals es \<equiv> (\<exists>vs. es = map Val vs) \<or> (\<exists>vs a es'. es = map Val vs @ Throw a # es')"
len_eq': shows "length (fst gc\<^sub>C) = length cms"
assms_5: shows "4 \<le> rk {A, B, C, A', B', C', P, Q}"
finite_max_next_pcs: fixes bp :: "instr list" shows "finite (max_next_pcs bp)"
BS1: fixes r :: "'a" assumes "C.ide r" shows "\<exists>f g. C.is_left_adjoint f \<and> C.is_left_adjoint g \<and> r \<cong>\<^sub>C g \<star>\<^sub>C f\<^sup>*\<^sup>C"
vsv_is_vconst_onI: fixes r :: "V"   and A :: "V"   and a :: "V" assumes "vsv r"   and "\<D>\<^sub>\<circ> r = A"   and "\<R>\<^sub>\<circ> r = ZFC_in_HOL.set {a}" shows "r = vconst_on A a"
r_phi_def: shows "r_phi \<equiv> r_universal 1"
dagger_unfoldl_distr: fixes y :: "('a \<times> ('b \<times> 'a) list) set"   and x :: "('a \<times> ('b \<times> 'a) list) set" shows "y \<union> t_prod (t_prod x (t_star x)) y = t_prod (t_star x) y"
Diag_implies_Arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.Arr C t"
interleave_assoc_1: fixes tu :: "'b event list"   and t :: "'b event list"   and A :: "'b event set"   and u :: "'b event list"   and tuv :: "'b event list"   and v :: "'b event list" assumes "tu setinterleaves ((t, u), A)"   and "tuv setinterleaves ((tu, v), A)" shows "\<exists>uv. uv setinterleaves ((u, v), A) \<and> tuv setinterleaves ((t, uv), A)"
monom_in_carrier: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "set (local.monom a n) \<subseteq> carrier R"
Suc: shows "fps_of_poly (eulerian_poly k) = Abs_fps (\<lambda>n. of_nat (n + 1) ^ k) * (1 - fps_X) ^ (k + 1)"
prod_dividef: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.field divide inverse times one plus zero minus uminus" shows "comm_monoid_mult.prod times one (\<lambda>x. divide (f x) (g x)) A = divide (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
that: fixes x :: "'b" assumes "fmlookup m x = Some y" shows "thesis"
Fvars_lcnj: fixes \<phi>s :: "'fmla list"   and F :: "'a set" assumes "set \<phi>s \<subseteq> fmla"   and "finite F" shows "Fvars (lcnj \<phi>s) = \<Union> (set (map Fvars \<phi>s))"
wtE_pickE: fixes xl :: "Preliminaries.var list"   and al :: "univ list" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> eintT (TE.tpOfV (xl ! i)) (al ! i)" shows "wtE (pickE xl al)"
id: shows "step_3_a (Suc i) j A = step_3_a i j B"
AB_def: shows "AB = (\<lambda>l. \<beta> l / \<alpha> l)"
sub2_preserves_triangle: fixes d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub2 d l (r, A) = (r', A')"   and "local.triangular_to l A"   and "l < dim_col A"   and "l < d"   and "d \<le> dim_row A" shows "local.triangular_to l A'"
rep_eq: fixes x :: "'b antichain"   and xa :: "'b antichain" shows "set_antichain (x - xa) = set_antichain x - set_antichain xa"
not_model_mono_imp_addedge_mono: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and a1 :: "'c"   and a2 :: "'c"   and X :: "('c \<times> 'c) set"   and Y :: "('c \<times> 'c) set"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph G"   and "(a1, a2) \<in> edges G"   and "X \<subseteq> Y"   and "\<not> sinvar (add_edge a1 a2 (delete_edges G Y)) nP" shows "\<not> sinvar (add_edge a1 a2 (delete_edges G X)) nP"
bij_betw_inter_subsets: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and a1 :: "'c set"   and a2 :: "'c set" assumes "bij_betw f A B"   and "a1 \<subseteq> A"   and "a2 \<subseteq> A" shows "f ` (a1 \<inter> a2) = f ` a1 \<inter> f ` a2"
f_Exec_N_eq_last_f_Exec_Stream_Init_take: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and n :: "nat"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_N trans_fun n xs c = last (f_Exec_Comp_Stream_Init trans_fun (xs \<down> n) c)"
assms_2: shows "\<not> is_sig_red (\<prec>\<^sub>t) (=) G (monom_mult (1::'b) (pp_of_term u - lp g) g)"
compress_normalize_input_interfaces_def: fixes m :: "'b common_primitive match_expr" shows "compress_normalize_input_interfaces m \<equiv> compress_normalize_primitive (is_Iiface, iiface_sel) IIface compress_interfaces m"
h2_def: fixes x :: "'a" shows "h2 x = (if g x = (0::'b) then if f x = (0::'b) then h x else h x / f x else g x)"
Number_EphK: fixes X :: "nat"   and X' :: "ephkey" shows "Number X \<noteq> EphK X'"
convolution_of_aezfun_is_aezfun: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "convolution f g \<in> aezfun_set"
nat_mult_distrib: fixes z :: "int"   and z' :: "int" assumes "0 \<le> z" shows "nat (z * z') = nat z * nat z'"
unat_word_ariths_2: fixes a :: "'b word"   and b :: "'b word" shows "unat (a * b) = unat a * unat b mod 2 ^ LENGTH('b)"
vsv_vrange_vempty: fixes r :: "V" assumes "vsv r"   and "\<R>\<^sub>\<circ> r = []\<^sub>\<circ>" shows "r = []\<^sub>\<circ>"
normalize_power_normalize: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and n :: "nat" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize (power.power one times (normalize x) n) = normalize (power.power one times x n)"
distinct_49: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "Stmt.If x51 x52 x53 \<noteq> Stop"
match_simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t\<^sub>1 :: "term"   and t\<^sub>2 :: "term"   and u\<^sub>1 :: "'c"   and u\<^sub>2 :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.match unapp unconst (t\<^sub>1 $ t\<^sub>2) (app u\<^sub>1 u\<^sub>2) = term.match unapp unconst t\<^sub>1 u\<^sub>1 \<bind> (\<lambda>env\<^sub>1. term.match unapp unconst t\<^sub>2 u\<^sub>2 \<bind> (\<lambda>env\<^sub>2. Some (env\<^sub>1 ++\<^sub>f env\<^sub>2)))"
subcls_C_Object: fixes P :: "'a prog"   and C :: "String.literal"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "is_class P C"   and "wf_prog wf_md P" shows "P \<turnstile> C \<preceq>\<^sup>* Object"
linord_of_listP_linord_of_list: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "linord_of_listP x y xs = ((x, y) \<in> linord_of_list xs)"
pw_ref_iff: fixes S :: "'c nres"   and R :: "('c \<times> 'd) set"   and S' :: "'d nres" shows "(S \<le> \<Down> R S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> (\<exists>s'. (x, s') \<in> R \<and> inres S' s')))"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Can \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Can t \<and> Can u \<and> Can v)"
simps_10: shows "reg\<^sub>C \<noteq> control_var\<^sub>C"
PO_a0i_obs_agreement: shows "oreach a0i \<subseteq> a0i_agreement"
Inf_ow_axioms: shows "Inf_ow U\<^sub>1 Inf\<^sub>1"
star_rot: fixes q :: "assertion"   and p :: "assertion"   and r :: "assertion" shows "q \<star> p \<star> r = p \<star> q \<star> r"
Un_empty_left: fixes B :: "'a set" shows "{} \<union> B = B"
d_inter_r: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" shows "mrel_c_lattice.d x \<cdot> (y \<parallel> z) = mrel_c_lattice.d x \<cdot> y \<parallel> (mrel_c_lattice.d x \<cdot> z)"
box_equals: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a = b"   and "a = c"   and "b = d" shows "c = d"
hom_0: fixes x :: "int" assumes "of_int x = (0::'c)" shows "x = 0"
simps: fixes Enc :: "'a \<Rightarrow> 'b"   and a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc" shows "encoding.indRelRp Enc a1 a2 = (\<exists>S. a1 = SourceTerm S \<and> a2 = TargetTerm (Enc S))"
ans4: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_ad (x \<union> y) = rel_ad x ; rel_ad y"
CondBool: fixes Env :: "env"   and c :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and B :: "lname set"   and C :: "assigned"   and E1 :: "assigned"   and E2 :: "assigned"   and A :: "assigned" assumes "Env\<turnstile>c ? e1 : e2\<Colon>-PrimT Boolean"   and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C"   and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1"   and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2"   and "nrm A = B \<union> assigns_if True (c ? e1 : e2) \<inter> assigns_if False (c ? e1 : e2)"   and "brk A = (\<lambda>l. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A"
isom_gch_units_transpTr6: fixes n :: "nat"   and g :: "nat \<Rightarrow> ('e set, 'f) Group_scheme"   and i :: "nat" assumes "Gchain n g"   and "i \<le> n" shows "Group (g i)"
naturality: fixes f :: "'c" assumes "C.arr f" shows "E (C.cod f) \<cdot> E f = E f \<cdot> E (C.dom f)"
"2_2": shows "RETURN xs \<le> it_to_sorted_list (\<lambda>_ _. True) g"
p3_def: shows "p3 = p \<circ>\<^sub>p [:ub, Complex (Re lb - Re ub) 0:]"
assms: fixes \<xi> :: "'a"   and p :: "'b"   and a :: "'c"   and \<xi>' :: "'a"   and p' :: "'b" assumes "(\<xi>, p) \<in> reachable A I"   and "((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A"   and "I a" shows "P ((\<xi>, p), a, \<xi>', p')"
strategyproofness_1: shows "(pmf (sds R37) a + pmf (sds R37) c < pmf (sds R46) d + pmf (sds R46) b \<or> pmf (sds R37) a + (pmf (sds R37) c + pmf (sds R37) d) < pmf (sds R46) d + (pmf (sds R46) b + pmf (sds R46) a)) \<or> pmf (sds R37) a + pmf (sds R37) c = pmf (sds R46) d + pmf (sds R46) b \<and> pmf (sds R37) a + (pmf (sds R37) c + pmf (sds R37) d) = pmf (sds R46) d + (pmf (sds R46) b + pmf (sds R46) a)"
distinct_77: fixes x31 :: "char list"   and x32 :: "expr"   and x15 :: "expr" shows "\<lparr>x31\<rparr>x32 \<noteq> throw x15"
lossless_restrict_gpvI: fixes \<I> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "lossless_gpv \<I> (restrict_gpv \<I> gpv)"
mkAutoEC_implements: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr" shows "IncrEnvironment.implements jkbp envInit envAction envTrans envVal jview envObs (IncrEnvironment.mkAutoEC jkbp envInit envAction envTrans envVal jview jviewInit jviewIncr)"
ToTC_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "H.ToTC f \<equiv> if VxVxV.arr f then fst (fst (fst f, fst (snd f)) \<star> snd (fst f, fst (snd f)), snd (snd f)) \<star> snd (fst (fst f, fst (snd f)) \<star> snd (fst f, fst (snd f)), snd (snd f)) else null"
PiE_I: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and B :: "'e \<Rightarrow> 'f set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x"   and "\<And>x. x \<notin> A \<Longrightarrow> f x = undefined" shows "f \<in> Pi\<^sub>E A B"
R: fixes x :: "'a"   and e :: "real" assumes "x \<in> closure S \<and> 0 < e" shows "\<exists>d>0. \<forall>y. y \<in> S \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e"
norm1_ge_0: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "0 \<le> norm1 v"
sel_13: fixes x1 :: "('d, 'c) Term.term" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>x1\<rangle>) = []"
mht_eq_min_height: fixes t :: "('b \<times> nat) tree" assumes "ltree t" shows "mht t = min_height t"
prems_2: shows "\<forall>rs. set rs \<subseteq> R \<and> length rs = length (m # ms) \<and> rs \<bullet>\<cdot> (m # ms) = (0::'m) \<longrightarrow> set rs \<subseteq> 0"
rtrancl_on_induct: fixes a :: "'a"   and b :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "(a, b) \<in> rtrancl_on F r"   and "a \<in> F \<Longrightarrow> P a"   and "\<And>y z. \<lbrakk>(a, y) \<in> rtrancl_on F r; (y, z) \<in> r; y \<in> F; z \<in> F; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
zmset_of_inject: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(zmset_of M = zmset_of N) = (M = N)"
\<alpha>_def: fixes G :: "nat set \<times> nat set set" shows "\<alpha> G \<equiv> \<Squnion>vs\<in>independent_sets G. enat (card vs)"
AllQ: fixes xs :: "real list" shows "eval (clearQuantifiers \<phi>) xs = eval \<phi> xs"
distinct_1: fixes x1 :: "nat"   and x2 :: "(val \<times> val) fset" shows "VNat x1 \<noteq> VFun x2"
vlen_def: fixes v :: "(nat \<Rightarrow> real) \<times> nat" shows "vlen v = snd v"