hcju/isabelleps huggingface dataset:
Suc_innermost_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "fw m n (Suc k) i j i j \<le> fw m n k i j i j"
empty_plan_is_valid: fixes PROB :: "'d set" shows "[] \<in> valid_plans PROB"
lossless_iter: fixes n :: "nat"   and s :: "'a" assumes "\<And>s. guard s \<Longrightarrow> lossless_spmf (iter (N s) s)" shows "lossless_spmf (fuse.iter n s)"
rep_eq: fixes x :: "real_alg" shows "to_rat_real_alg_main x = to_rat_3 (rep_real_alg x)"
inject_2: fixes x41 :: "'b rexp"   and x42 :: "'b rexp"   and y41 :: "'b rexp"   and y42 :: "'b rexp" shows "(Plus x41 x42 = Plus y41 y42) = (x41 = y41 \<and> x42 = y42)"
h5: shows "invertible_mat (get_matrix (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
comb_planar_def: fixes G :: "('b, 'c) pre_digraph" shows "comb_planar G \<equiv> \<exists>M. digraph_map G M \<and> pre_digraph_map.euler_genus G M = 0"
foldl_rule_aux_P: fixes I :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and l0 :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "I \<sigma>0 l0"   and "\<And>l1 l2 x \<sigma>. \<lbrakk>l0 = l1 @ x # l2; I \<sigma> (x # l2)\<rbrakk> \<Longrightarrow> I (f \<sigma> x) l2"   and "\<And>\<sigma>. I \<sigma> [] \<Longrightarrow> P \<sigma>" shows "P (foldl f \<sigma>0 l0)"
Diff_subset: fixes A :: "'k set"   and B :: "'k set" shows "A - B \<subseteq> A"
lm016: fixes bids :: "'d \<times> 'e set \<Rightarrow> 'f"   and N :: "'d set"   and G :: "'e set" shows "summedBidVectorRel bids N G = {summedBid bids pair |pair. pair \<in> N \<times> (Pow G - {{}})}"
subgroup_imp_normal: fixes A :: "'a set" assumes "subgroup A G" shows "A \<lhd> G"
dvd_power_minus_1_conv2: fixes x :: "nat"   and a :: "nat"   and b :: "nat" assumes "1 < x"   and "0 < a"   and "a dvd b"   and "0 < b" shows "x ^ a - 1 dvd x ^ b - 1"
fun_rel_comp_dist: fixes R1 :: "('a \<times> 'g) set"   and R2 :: "('b \<times> 'h) set"   and R3 :: "('g \<times> 'c) set"   and R4 :: "('h \<times> 'f) set" shows "(R1 \<rightarrow> R2) O (R3 \<rightarrow> R4) \<subseteq> R1 O R3 \<rightarrow> R2 O R4"
asymp_equiv_imp_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "f \<in> \<Theta>[F](g)"
CCfix_unroll: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "CCfix \<Gamma>\<cdot>(ae, G) = ccBindsExtra \<Gamma>\<cdot>(ae, CCfix \<Gamma>\<cdot>(ae, G)) \<squnion> G"
df_sin: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x :: "real"   and g' :: "'b" assumes "(f has_vector_derivative f') (at x)"   and "(sin has_field_derivative g') (at (f x))" shows "((\<lambda>x. sin (f x)) has_vector_derivative f' * g') (at x)"
list_update_id: fixes xs :: "'b list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
simps_1: fixes x :: "'a"   and y :: "'a" shows "linord_of_listP x y [] = False"
homotopic_paths_subset: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a"   and t :: "'a set" assumes "homotopic_paths s p q"   and "s \<subseteq> t" shows "homotopic_paths t p q"
Ord_mono_imp_increasing: fixes h :: "V \<Rightarrow> V"   and D :: "V set"   and \<nu> :: "V" assumes "h \<in> D \<rightarrow> D"   and "strict_mono_on h D"   and "D \<subseteq> ON"   and "\<nu> \<in> D" shows "\<nu> \<le> h \<nu>"
sb_compose2: fixes ys :: "'a"   and xs :: "'a"   and x2 :: "'b"   and y1 :: "'b"   and y2 :: "'b"   and zs :: "'a"   and x1 :: "'b"   and z :: "'b" assumes "ys \<noteq> xs \<or> x2 \<notin> {y1, y2}" shows "z @zs[x1 / x2]_xs @zs[y1 / y2]_ys = z @zs[y1 / y2]_ys @zs[x1 @xs[y1 / y2]_ys / x2]_xs"
hyps: fixes yss :: "'a list list" assumes "distincts xss"   and "distincts yss"   and "set xss = set yss" shows "lists_succ xss = lists_succ yss"
l2_inv4D: fixes x :: "l2_state"   and Rb :: "fid_t" assumes "test_ended x"   and "guessed_frame test xsk = Some (NonceF (Rb $ sk))"   and "x \<in> l2_inv4" shows "progress x Rb = Some {xpkE, xsk}"
homologous_rel_diff: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and a' :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b' :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S a a'"   and "homologous_rel p X S b b'" shows "homologous_rel p X S (a - b) (a' - b')"
isotone_def: fixes A :: "('a, 'b) gorder_scheme"   and B :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "isotone A B f \<equiv> weak_partial_order A \<and> weak_partial_order B \<and> (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. x \<sqsubseteq>\<^bsub>A\<^esub> y \<longrightarrow> f x \<sqsubseteq>\<^bsub>B\<^esub> f y)"
std_lookup_action: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "abs_ast_prob problem"   and "is_standard_operator' ast_op"   and "ast_op \<in> set (ast_problem.ast\<delta> problem)" shows "\<exists>ast_op'. ast_problem.lookup_action problem (abs_ast_operator ast_op) = Some ast_op'"
finite_proj_diag: fixes d :: "'g \<Rightarrow> 'h \<Rightarrow> 'e"   and P :: "'f \<Rightarrow>\<^sub>F 'g"   and Q :: "'f \<Rightarrow>\<^sub>F 'h"   and S :: "'f set" shows "finite ((\<lambda>i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S)"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i \<sqsubset> n - m" shows "map f [m..<n] ! i = f (m + i)"
category_with_pullbacks_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "category_with_pullbacks C \<equiv> category C \<and> category_with_pullbacks_axioms C"
prod_2_id_def: fixes p :: "'a \<Rightarrow> 'b decision option"   and q :: "'a \<Rightarrow> 'c decision option" shows "p \<Otimes>\<^sub>2\<^sub>I q = p \<Otimes>\<^sub>2 q \<circ> (\<lambda>x. (x, x))"
gb_schema_aux_term2_def: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "gb_schema_aux_term2 d gs = {(a, b). dgrad_p_set_le d (args_to_set (gs, a)) (args_to_set (gs, b)) \<and> component_of_term ` Keys (args_to_set (gs, a)) \<subseteq> component_of_term ` Keys (args_to_set (gs, b))}"
filter_append: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "filter P (xs @ ys) = filter P xs @ filter P ys"
subst_apply_fv_unfold': fixes t :: "('d, 'e) Term.term"   and \<delta> :: "'e \<Rightarrow> ('d, 'c) Term.term" shows "fv (t \<cdot> \<delta>) = (\<Union>v\<in>fv t. fv (\<delta> v))"
neutral_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set" shows "neutral scf A Is \<equiv> \<forall>P P' x y z w. profile A Is P \<and> profile A Is P' \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> w \<in> A \<and> (\<forall>i\<in>Is. (x \<^bsub>(P i)\<^esub>\<preceq> y) = (z \<^bsub>(P' i)\<^esub>\<preceq> w)) \<and> (\<forall>i\<in>Is. (y \<^bsub>(P i)\<^esub>\<preceq> x) = (w \<^bsub>(P' i)\<^esub>\<preceq> z)) \<longrightarrow> (x \<^bsub>(scf P)\<^esub>\<preceq> y) = (z \<^bsub>(scf P')\<^esub>\<preceq> w) \<and> (y \<^bsub>(scf P)\<^esub>\<preceq> x) = (w \<^bsub>(scf P')\<^esub>\<preceq> z)"
atLeastAtMost_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "ord.atLeastAtMost less_eq l u \<equiv> ord.atLeast less_eq l \<inter> ord.atMost less_eq u"
simps_1: fixes x1 :: "real"   and y1 :: "real" shows "(ereal x1 = ereal y1) = (x1 = y1)"
RepFun_hinsert: fixes a :: "hf"   and b :: "hf"   and f :: "hf \<Rightarrow> hf" shows "RepFun (b \<triangleleft> a) f = RepFun b f \<triangleleft> f a"
comp_CLS: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "is_left_adjoint f"   and "is_left_adjoint g"   and "f \<star> g \<cong> h" shows "\<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>g\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>h\<rbrakk>\<rbrakk>"
irreducible_mono: fixes b :: "'a"   and a :: "'a" assumes "irreducible b"   and "a dvd b"   and "\<not> a dvd (1::'a)" shows "irreducible a"
assms_4: assumes "\<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)" shows "eval c t t' n' \<gamma>"
append_set_rem_last: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list"   and zs :: "'c list" assumes "set (xs @ [x]) = set (ys @ [x] @ zs)"   and "distinct (xs @ [x])"   and "distinct (ys @ [x] @ zs)" shows "set xs = set (ys @ zs)"
new_element_is_l_new_element: shows "l_new_element ShadowRootClass.type_wf"
fresh_psubst_E1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho"   and "rho ys y = None"   and "fresh zs z X #[rho]" shows "fresh ys y X \<or> ys \<noteq> zs \<or> y \<noteq> z"
group_commutes_pow: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<oplus> y = y \<oplus> x"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "[n] \<cdot> x \<oplus> y = y \<oplus> [n] \<cdot> x"
assms: fixes i :: "'a" assumes "i \<in> A" shows "prob_space (M i)"
permutations_of_set_lists: fixes A :: "'b set" shows "permutations_of_set A \<subseteq> lists A"
distinct_17: fixes x21 :: "floatarith"   and x22 :: "floatarith"   and x23 :: "form"   and x51 :: "floatarith"   and x52 :: "floatarith"   and x53 :: "floatarith" shows "Assign x21 x22 x23 \<noteq> AtLeastAtMost x51 x52 x53"
simps_2: fixes x :: "octo"   and y :: "octo" shows "Im1 (x + y) = Im1 x + Im1 y"
simps_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "vwellformed (Vabs cs \<Gamma>) = (pred_fmap id (fmmap vwellformed \<Gamma>) \<and> wellformed_clauses cs)"
ce_set_to_rel_def: shows "ce_set_to_rel = (\<lambda>A. {(c_fst x, c_snd x) |x. x \<in> A})"
simps_2: fixes u :: "'r"   and us :: "[:'r:]"   and pat :: "[:'r:]"   and vs :: "[:'r:]" assumes "u \<noteq> \<bottom>"   and "us \<noteq> \<bottom>" shows "left2\<cdot>pat\<cdot>(u :# us, vs) = next\<cdot>vs\<cdot>(sfoldl\<cdot>(op2\<cdot>pat)\<cdot>(root2\<cdot>pat)\<cdot>us)"
approx_PInf_emeasure_with_finite: fixes M :: "'b measure"   and W :: "'b set"   and C :: "real" assumes "sigma_finite_measure M"   and "W \<in> sets M"   and "emeasure M W = \<infinity>" obtains Z :: "'b set" where "Z \<in> sets M" and "Z \<subseteq> W" and "emeasure M Z < \<infinity>" and "ennreal C < emeasure M Z"
indexlt_trans: fixes k'' :: "nat"   and u'' :: "nat"   and k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" assumes "indexlt k'' u'' k' u'"   and "indexlt k' u' k u" shows "indexlt k'' u'' k u"
bounded_err_stepD: fixes n :: "nat"   and app :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and p :: "nat"   and a :: "'b"   and q :: "nat"   and b :: "'b" assumes "bounded (err_step n app step) n"   and "p < n"   and "app p a"   and "(q, b) \<in> set (step p a)" shows "q < n"
f_shrink_eq_f_last_message_hold_shrink_last: fixes xs :: "'d message_af list"   and k :: "nat" shows "xs \<div> k = xs \<longmapsto> k \<div>\<^sub>l k"
"1_2": fixes u :: "nat list"   and v :: "nat list" assumes "u @ v \<le>\<^sub>v x @ y"   and "(u, v) \<in> Minimal_Solutions a b" shows "thesis"
simps_2: fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree" shows "braun \<langle>l, x, r\<rangle> = ((size l = size r \<or> size l = size r + 1) \<and> braun l \<and> braun r)"
less_multiset\<^sub>D\<^sub>M: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(M < N) = (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"
borel_open: fixes A :: "'d set" assumes "open A" shows "A \<in> sets borel"
prime_factorization_nat_def: fixes n :: "nat" shows "prime_factorization_nat n \<equiv> rev (if n < 2 then [] else case next_candidates 0 of (j, is) \<Rightarrow> prime_factorization_nat_main n j is [])"
that: fixes g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>Y. f Y = f X + frechet_derivative f (at X) (Y - X) + (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. (Y - X) \<bullet> j * ((Y - X) \<bullet> i) * g i j Y)"   and "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk> \<Longrightarrow> smooth_on UNIV (g i j)" shows "thesis"
b: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and e :: "('a \<times> 'a) set"   and m :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and ov :: "('a \<times> 'a) set"   and d :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set"   and p :: "'a"   and q :: "'a" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> b) = (\<exists>t. meets p t \<and> meets t q)"
finite_ReZ_segment_cases: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "finite_ReZ_segments g z" obtains s :: "real" where "sa__ \<in> {0..<1}" and "sa__ = 0 \<or> Re (g sa__) = Re z" and "\<forall>t\<in>{sa__<..<1}. Re (g t) = Re z" and "finite_ReZ_segments (subpath 0 sa__ g) z"   | s :: "real" where "s \<in> {0..<1}" and "s = 0 \<or> Re (g s) = Re z" and "\<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z" and "finite_ReZ_segments (subpath 0 s g) z"
assms: shows "(ys, y, zs) \<in> set (list_splits xs)"
cf_list_length: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" shows "length (cf_list R p) = Suc (deg R p)"
healthy_sound: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "sound (t P)"
N: fixes n :: "nat" assumes "N \<le> n" shows "ereal (u n) < C"
not_outside_1: fixes x :: "'a"   and y :: "'a" assumes "of_weak_ranking As x y" shows "x \<in> \<Union> (set As)"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "B.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f)"
aezfun_setD: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set" shows "finite (supp f)"
lens_plus_eq_left: fixes X :: "'a \<Longrightarrow> 'b"   and Z :: "'c \<Longrightarrow> 'b"   and Y :: "'d \<Longrightarrow> 'b" assumes "X \<bowtie> Z"   and "X \<approx>\<^sub>L Y" shows "X +\<^sub>L Z \<approx>\<^sub>L Y +\<^sub>L Z"
mod_mult2_eq: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "m mod (n *n q) = n *n m div n mod q + m mod n"
4: shows "homeomorphism (cbox a d) (cbox a v) f4 g4"
iwlsFSbSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "iwlsFSbSw MOD \<equiv> iwlsFSb MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igSwapCls MOD"
no_call_in_path: fixes p :: "'a"   and x :: "'a list"   and \<pi> :: "perm" assumes "no_call_in_path p x" shows "no_call_in_path (\<pi> \<bullet> p) (\<pi> \<bullet> x)"
sum: shows "\<forall>x. c 2 * x\<^sup>2 + c (Suc 0) * x + c 0 = (\<Sum>i\<le>2. c i * x ^ i)"
primeness_condition: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c" assumes "principal_domain R"   and "p \<in> carrier R" shows "ring_irreducible\<^bsub>R\<^esub> p = ring_prime\<^bsub>R\<^esub> p"
elims_2: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and to_ns :: "('a \<times> constraint) list \<Rightarrow> ('a \<times> 'b ns_constraint) list"   and from_ns :: "(nat, 'b) mapping \<Rightarrow> 'b ns_constraint list \<Rightarrow> (nat, rat) mapping"   and x :: "('a \<times> constraint) list"   and xa :: "'a set"   and xb :: "'b ns_constraint list \<times> 'c" assumes "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns"   and "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex.checked_cs checked_nsc to_ns x xa xb" obtains cs :: "('a \<times> constraint) list" and J :: "'a set" and ds :: "'b ns_constraint list" and s :: "'c" where "x = cs" and "xa = J" and "xb = (ds, s)" and "ds = map snd (to_ns cs) \<and> checked_nsc (to_ns cs) J s"
bi_unique: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Ta :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "bi_total T"   and "bi_unique Ta" shows "bi_unique (pcr_mapping T Ta)"
simps_2: fixes h :: "nat"   and n_ops :: "nat"   and t :: "nat"   and k :: "nat" shows "cnf_to_dimacs.valid_state_var h n_ops (State t k) = (t < h)"
wt_jvm_prog_lbv_def: fixes P :: "jvm_prog"   and cert :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option err list" shows "wt_jvm_prog_lbv P cert \<equiv> wf_prog (\<lambda>P C (mn, b, Ts, T\<^sub>r, mxs, mxl\<^sub>0, ins, et). wt_lbv P C b Ts T\<^sub>r mxs mxl\<^sub>0 et (cert C mn) ins) P"
simps_18: fixes Vs :: "char list list"   and e\<^sub>1 :: "char list exp"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (try e\<^sub>1 catch(C V) e\<^sub>2) = try compE\<^sub>1 Vs e\<^sub>1 catch(C length Vs) compE\<^sub>1 (Vs @ [V]) e\<^sub>2"
cUN_cinsert: fixes B :: "'e \<Rightarrow> 'd cset"   and a :: "'e"   and A :: "'e cset" shows "cUnion (cimage B (cinsert a A)) = cUn (B a) (cUnion (cimage B A))"
less_imp_le: fixes x :: "'a word"   and y :: "'a word" assumes "x <s y" shows "x \<le>s y"
summable_comparison_test_ev: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. norm (f n) \<le> g n"   and "summable g" shows "summable f"
simps_11: fixes V :: "char list"   and T :: "ty"   and e :: "expr" shows "fv {V:T; e} = fv e - {V}"
div_by_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a zero = zero"
minus_real_def: fixes x :: "real"   and y :: "real" shows "x - y = x + - y"
leg0_simps_5: shows "cod r\<^sub>0 = r\<^sub>0"
islimpt_finite: fixes s :: "'c set"   and x :: "'c" assumes "finite s" shows "\<not> x islimpt s"
HPhase0Read_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv4c s q" shows "HInv4c s' q"
T_Tn_T_compose_1: fixes n :: "nat"   and x :: "'a" shows "T ((T ^^ n) x) = (T ^^ Suc n) x"
refl: fixes cmp :: "'b comparator"   and a :: "'b" shows "compare cmp a a = Equiv"
simps_6: fixes a :: "nat rexp"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "subst (Star a) \<sigma> = Star (subst a \<sigma>)"
UN_extend_simps_9: fixes C :: "'d \<Rightarrow> 'b set"   and B :: "'c \<Rightarrow> 'd set"   and A :: "'c set" shows "(\<Union>x\<in>A. \<Union> (C ` B x)) = \<Union> (C ` \<Union> (B ` A))"
simulation2: fixes s1 :: "'s1"   and s2 :: "'s2"   and tl2 :: "'tl2"   and s2' :: "'s2" assumes "s1 \<approx> s2"   and "s2 -2-tl2\<rightarrow> s2'" shows "\<exists>s1' tl1. s1 -1-tl1\<rightarrow> s1' \<and> s1' \<approx> s2' \<and> tl1 \<sim> tl2"
related: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'c set"   and h :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "R (0::'a) (0::'a)"   and "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 + y1) (x2 + y2)"   and "finite S"   and "\<forall>x\<in>S. R (h x) (g x)" shows "R (sum_0 h S) (sum_0 g S)"
exhaust: fixes y :: "'l \<times> 'm" obtains a :: "'l" and b :: "'m" where "y = (a, b)"
univ_poly_is_abelian_monoid: shows "abelian_monoid (K [X])"
summable_gbinomial: fixes z :: "real"   and a :: "real" assumes "norm z < 1" shows "summable (\<lambda>n. (a gchoose n) * z ^ n)"
LIM_offset: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and a :: "'a"   and k :: "'a" assumes "f \<midarrow>a\<rightarrow> L" shows "(\<lambda>x. f (x + k)) \<midarrow>a - k\<rightarrow> L"
list_induct_ssnoc: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>xs x y. P (xs @ [x]) \<Longrightarrow> P (xs @ [x, y])" shows "P xs"
division: fixes a :: "nat"   and b :: "nat" shows "\<exists>r q. a = Suc b * q + r \<and> r \<le> b"
poly_const_def: fixes a :: "'d" shows "poly_const a = (if a = (0::'d) then [] else [(1, a)])"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "strict_bicategory V H \<a> \<i> src trg"   and "adjunction_in_normal_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_strict_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
exact_decompD_2: fixes X :: "'x set"   and m :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "finite X"   and "exact_decomp X m ps"   and "(h, U) \<in> set ps" shows "U \<subseteq> X"
index_smult_mat_3: fixes a :: "'a"   and A :: "'a Matrix.mat" shows "dim_col (a \<cdot>\<^sub>m A) = dim_col A"
hcomp_arr_obj: fixes f :: "'a"   and a :: "'a" assumes "arr f"   and "obj a"   and "src f = a" shows "f \<star> a = f"
Un_subset_iff: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "order.greater_eq C (A \<union> B) = (order.greater_eq C A \<and> order.greater_eq C B)"
eq_Nil_null: fixes xs :: "'c list" shows "(xs = []) = List.null xs"
assms: shows "Y \<subseteq> f -` {0::'b}"
Ide_HcompNml: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Ide t"   and "Ide u"   and "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Ide (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
prod_ord_pair_alt: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys" shows "prod_ord_pair_tc P xs ys = (\<forall>k\<in>fst ` set xs \<union> fst ` set ys. P k (lookup_pair_tc xs k) (lookup_pair_tc ys k))"
zip_induct: fixes P :: "'e list \<Rightarrow> 'f list \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'f list" assumes "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>ys. P [] ys"   and "\<And>xs. P xs []" shows "P xs ys"
1: fixes f :: "nat \<Rightarrow> real" shows "valid_net' (insert_weights s m f)"
Rmi_sv_1: fixes s :: "bdd"   and ni :: "nat"   and n :: "nat ifex"   and ni' :: "nat"   and n' :: "nat ifex" assumes "bdd_sane s"   and "(ni, n) \<in> Rmi s"   and "(ni', n') \<in> Rmi s"   and "ni = ni'" shows "n = n'"