hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "char list \<Rightarrow> com option"   and x2 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and x3 :: "char list \<Rightarrow> int \<Rightarrow> int"   and P :: "(char list \<Rightarrow> com option) \<Rightarrow> com \<times> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "x1: x2 \<Rightarrow> x3"   and "\<And>\<pi> s. P \<pi> (SKIP, s) s"   and "\<And>\<pi> x i a s. P \<pi> (x[i] ::= a, s) (s(x := (s x)(aval i s := aval a s)))"   and "\<And>\<pi> x y s. P \<pi> (x[] ::= y, s) (s(x := s y))"   and "\<And>\<pi> x s. P \<pi> (CLEAR x[], s) (s(x := \<lambda>_. 0))"   and "\<And>\<pi> l s. P \<pi> (Assign_Locals l, s) <l|s>"   and "\<And>\<pi> c\<^sub>1 s\<^sub>1 s\<^sub>2 c\<^sub>2 s\<^sub>3. \<lbrakk>\<pi>: (c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2; P \<pi> (c\<^sub>1, s\<^sub>1) s\<^sub>2; \<pi>: (c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3; P \<pi> (c\<^sub>2, s\<^sub>2) s\<^sub>3\<rbrakk> \<Longrightarrow> P \<pi> (c\<^sub>1;; c\<^sub>2, s\<^sub>1) s\<^sub>3"   and "\<And>b s \<pi> c\<^sub>1 t c\<^sub>2. \<lbrakk>bval b s; \<pi>: (c\<^sub>1, s) \<Rightarrow> t; P \<pi> (c\<^sub>1, s) t\<rbrakk> \<Longrightarrow> P \<pi> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) t"   and "\<And>b s \<pi> c\<^sub>2 t c\<^sub>1. \<lbrakk>\<not> bval b s; \<pi>: (c\<^sub>2, s) \<Rightarrow> t; P \<pi> (c\<^sub>2, s) t\<rbrakk> \<Longrightarrow> P \<pi> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) t"   and "\<And>\<pi> c s s'. \<lbrakk>\<pi>: (c, <<>|s>) \<Rightarrow> s'; P \<pi> (c, <<>|s>) s'\<rbrakk> \<Longrightarrow> P \<pi> (SCOPE c, s) <s|s'>"   and "\<And>b s \<pi> c. \<not> bval b s \<Longrightarrow> P \<pi> (WHILE b DO c, s) s"   and "\<And>b s\<^sub>1 \<pi> c s\<^sub>2 s\<^sub>3. \<lbrakk>bval b s\<^sub>1; \<pi>: (c, s\<^sub>1) \<Rightarrow> s\<^sub>2; P \<pi> (c, s\<^sub>1) s\<^sub>2; \<pi>: (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3; P \<pi> (WHILE b DO c, s\<^sub>2) s\<^sub>3\<rbrakk> \<Longrightarrow> P \<pi> (WHILE b DO c, s\<^sub>1) s\<^sub>3"   and "\<And>\<pi> p c s t. \<lbrakk>\<pi> p = Some c; \<pi>: (c, s) \<Rightarrow> t; P \<pi> (c, s) t\<rbrakk> \<Longrightarrow> P \<pi> (PCall p, s) t"   and "\<And>\<pi>' c s t \<pi>. \<lbrakk>\<pi>': (c, s) \<Rightarrow> t; P \<pi>' (c, s) t\<rbrakk> \<Longrightarrow> P \<pi> (PScope \<pi>' c, s) t" shows "P x1 x2 x3"
mat_inverse_var_def: fixes A :: "'a mat" shows "mat_inverse_var A = (if dim_row A = dim_col A then let one = 1\<^sub>m (dim_row A) in case gauss_jordan A one of (B, C) \<Rightarrow> if mat_equal B one then Some C else None else None)"
functor_axioms: shows "functor D C Adj.FGF.map"
BetaImpl: fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list"   and B :: "('a, 'b) form" assumes "\<stileturn> Neg A # G"   and "\<stileturn> B # G" shows "\<stileturn> Impl A B # G"
local_stepsE: fixes T :: "((('a \<Rightarrow> 'b) \<times> 'c) \<times> 'd \<times> ('a \<Rightarrow> 'e) \<times> 'f) set"   and J :: "'a set"   and \<sigma> :: "'a \<Rightarrow> 'b"   and s :: "'c"   and a :: "'d"   and \<sigma>' :: "'a \<Rightarrow> 'e"   and s' :: "'f"   and \<zeta> :: "'a \<Rightarrow> 'b" assumes "local_steps T J"   and "((\<sigma>, s), a, \<sigma>', s') \<in> T"   and "\<forall>j\<in>J. \<zeta> j = \<sigma> j" shows "\<exists>\<zeta>'. (\<forall>j\<in>J. \<zeta>' j = \<sigma>' j) \<and> ((\<zeta>, s), a, \<zeta>', s') \<in> T"
iT_cut_ge1_3: fixes t :: "nat"   and n :: "nat"   and d :: "nat" assumes "t \<in> [n\<dots>,d]" shows "[n\<dots>,d] \<down>\<ge> t = [t\<dots>,n + d - t]"
prime_elem_dvd_multD: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times p (times a b)" shows "dvd.dvd times p a \<or> dvd.dvd times p b"
simps_2: fixes nodes :: "'c set"   and edges :: "('c \<times> 'c) set"   and more :: "'d" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
geodesic_segment_distance: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and w :: "'b" assumes "geodesic_segment_between G x y" shows "dist w x + dist w y \<le> dist x y + 2 * infdist w G"
AE_T_reachable: fixes s :: "'a cfg" shows "almost_everywhere (MDP.MC.T s) (alw (HLD (MDP.MC.acc `` {s})))"
abs_eq: shows "0 = Interval (0::'b, 0::'b)"
Pause_parametric': fixes C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and A :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun C (rel_fun (rel_fun R (rel_gpv'' A C R)) (rel_gpv'' A C R)) Generative_Probabilistic_Value.Pause Generative_Probabilistic_Value.Pause"
init_opar_comp: fixes s :: "((nat \<Rightarrow> 'c) \<times> 'd, 'f seq_action) automaton"   and i :: "nat"   and t :: "('e, 'f seq_action) automaton" shows "init (s \<langle>\<langle>\<^bsub>i\<^esub> t) = extg ` (init s \<times> init t)"
B_eq: shows "B = Gauss_Jordan_in_ij A (0::'rows) (mod_type_class.from_nat k)"
ltl_FG_to_rabin_def: fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" shows "ltl_FG_to_rabin \<Sigma> \<G> w \<equiv> Only_G \<G> \<and> range w \<subseteq> \<Sigma> \<and> finite \<Sigma>"
cprank_max1_shallow_model': fixes y :: "nat"   and Z :: "nat"   and M :: "nat"   and N :: "nat"   and s :: "bool"   and w :: "nat \<Rightarrow> real" assumes "y < output_size (shallow_model' Z M N)" shows "cprank_max1 (tensors_from_net (insert_weights s (shallow_model' Z M N) w) $ y)"
position_cases: fixes x :: "'b sort_key extended" obtains y :: "'b" and z :: "'b sort_key" where "x = \<lbrakk>NonFinal (y__, sort_dir.Left) z__\<rbrakk>"   | y :: "'b" and z :: "'b sort_key" where "x = \<lbrakk>NonFinal (ya__, sort_dir.Right) z\<rbrakk>"   | y :: "'b" where "x = \<lbrakk>Final y\<rbrakk>"   | "x = \<turnstile>"   | "x = \<stileturn>"
reachable1_reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>+ v"   and "v \<rightarrow>\<^sup>* w" shows "u \<rightarrow>\<^sup>+ w"
of_int_eq_0_iff: fixes z :: "int" shows "(of_int z = (0::'a)) = (z = 0)"
cong_eq: shows "[fst (bezw e N) mod int N * int e = 1] (mod int N)"
spr_submodel_aux: fixes t :: "'s Trace"   and s :: "'s set \<times> 's" assumes "t \<in> SPR.jkbpC"   and "s \<in> worlds (spr_simMCt t)" shows "gen_model spr_simMC s = gen_model (spr_simMCt t) s"
pickE: fixes xl :: "Preliminaries.var list"   and al :: "univ list"   and i :: "nat" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> eintT (GE.tpOfV (xl ! i)) (al ! i)"   and "i < length xl" shows "pickE xl al (xl ! i) = al ! i"
finite_lin_dep: fixes S :: "'a vec set" assumes "finite S"   and "lin_dep S"   and "S \<subseteq> carrier_vec n" shows "\<exists>a v. True \<and> lincomb a S = 0\<^sub>v n \<and> v \<in> S \<and> a v \<noteq> (0::'a)"
maxl_fun_asc: shows "\<forall>x\<in>carrier D. x \<preceq> maxl_fun D x"
qSwap_preserves_qAFresh: fixes ys :: "'a"   and zs :: "'a"   and z1 :: "'b"   and z2 :: "'b"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" shows "qAFresh ys (y @ys[z1 \<and> z2]_zs) (X #[[z1 \<and> z2]]_zs) = qAFresh ys y X"
adm_all: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>y. adm (\<lambda>x. P x y)" shows "adm (\<lambda>x. \<forall>y. P x y)"
ovsmfidiesi: shows "(ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>) O (ov\<inverse> \<union> s\<inverse> \<union> m\<inverse> \<union> f \<union> d \<union> e \<union> s) \<subseteq> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> m \<union> m\<inverse>"
max_bag_card_def: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.max_bag_card T bag \<equiv> Max {card (bag t) |t. t \<in> V\<^bsub>T\<^esub>}"
envObs_rel_equiv: shows "equiv UNIV (rel_ext (envObs_rel (envObs agent)))"
obtains_dbm_le: fixes x :: "'a"   and m :: "int" assumes "x \<in> X"   and "m \<le> int (k x)" obtains M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" where "{u \<in> V. u x \<le> real_of_int m} = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n" and "normalized M"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> bankerb_gets" shows "\<exists>A. Says A B X \<in> set evs"
iprev_all_ge_fix: fixes I :: "nat set"   and n :: "nat" assumes "\<forall>x\<in>I. n \<le> x" shows "iprev n I = n"
hd_conv_nth: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "hd xs = xs ! 0"
composition_of_substs_eq: fixes e :: "'b equation"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list" shows "subst_equation (subst_equation e \<sigma>) \<eta> = subst_equation e (\<sigma> \<lozenge> \<eta>)"
convergent_imp_Bseq: fixes f :: "nat \<Rightarrow> 'a" assumes "convergent f" shows "Bseq f"
le_inf_iff: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(x \<le> inf y z) = (x \<le> y \<and> x \<le> z)"
induct: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and x1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and x2 :: "nat"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> bool" assumes "count_it \<pi> b c x1 x2"   and "\<And>s. \<not> bval b s \<Longrightarrow> P s 0"   and "\<And>s s' n. \<lbrakk>bval b s; \<pi>: (c, s) \<Rightarrow> s'; count_it \<pi> b c s' n; P s' n\<rbrakk> \<Longrightarrow> P s (Suc n)" shows "P x1 x2"
WT_plus_oracleD2: fixes \<I>l :: "('a, 'b) \<I>"   and \<I>r :: "('c, 'd) \<I>"   and s :: "'s" assumes "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>" shows "\<I>r \<turnstile>c right s \<surd>"
map_closed: fixes a :: "'b" assumes "a \<in> \<FF> (fst y)" shows "\<rho> (fst y) z' a \<in> \<FF> z'"
\<phi>_5: fixes i :: "nat" assumes "i \<in> {0, 1}" shows "csupport_on carrier (\<phi> i) \<inter> carrier \<subseteq> V i"
scale_bind_spmf: fixes r :: "real"   and p :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf" assumes "r \<le> 1" shows "scale_spmf r (p \<bind> f) = p \<bind> (\<lambda>x. scale_spmf r (f x))"
assms_2: fixes child :: "(_) object_ptr" assumes "\<And>parent child_node. \<lbrakk>child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node; h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent\<rbrakk> \<Longrightarrow> P parent"   and "\<And>host shadow_root. \<lbrakk>child = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root; h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host\<rbrakk> \<Longrightarrow> P (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)" shows "P child"
simps_1: fixes edge :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and weight :: "'a \<Rightarrow> ennreal"   and A :: "'a set"   and B :: "'a set"   and more :: "'b" shows "weight \<lparr>edge = edge, weight = weight, A = A, B = B, \<dots> = more\<rparr> = weight"
less_half_sum: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b" shows "less a (divide (plus a b) (plus one one))"
homeomorphism_image1: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "homeomorphism S T f g" shows "f ` S = T"
head_isnpolyh_Suc': fixes n :: "nat"   and p :: "'b poly" assumes "0 < n"   and "isnpolyh p n" shows "head p = p"
l1: shows "left r1 = left (Abs_real_int (left r1, right r3))"
truncate_up_nonpos: fixes x :: "real"   and prec :: "nat" assumes "x \<le> 0" shows "truncate_up prec x \<le> 0"
zero_less_iff_neq_zero_hmset: fixes M :: "hmultiset" shows "(0 < M) = (M \<noteq> 0)"
last_prog_hd_prog_append': fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'b"   and sb :: "'b memref list"   and sb' :: "'b memref list" assumes "program.history_consistent program_step \<theta> (hd_prog p (sb @ sb')) (sb @ sb')"   and "last_prog p (sb @ sb') = p" shows "last_prog (hd_prog p sb') sb = hd_prog p sb'"
has_integral_mult_right: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and i :: "'a set"   and c :: "'b" assumes "(f has_integral y) i" shows "((\<lambda>x. c * f x) has_integral c * y) i"
hmset_of_enat_le_\<omega>: fixes n :: "enat" shows "hmset_of_enat n \<le> \<omega>"
MatchNot_ternary_ternary_eval: fixes \<beta> :: "'b \<Rightarrow> 'c \<Rightarrow> ternaryvalue"   and p :: "'c"   and m' :: "'b match_expr"   and m :: "'b match_expr"   and \<alpha> :: "action \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "action" assumes "ternary_ternary_eval (map_match_tac \<beta> p m') = ternary_ternary_eval (map_match_tac \<beta> p m)" shows "matches (\<beta>, \<alpha>) (MatchNot m') a p = matches (\<beta>, \<alpha>) (MatchNot m) a p"
T_isort_length: fixes xs :: "'a list" shows "T_isort xs \<le> (length xs + 1)\<^sup>2"
compliant_stateful_ACS_only_state_violations: shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (flows_state \<T>)"
optimal_forest_def: fixes F :: "('c, 'd) graph"   and G :: "('c, 'd) graph" shows "optimal_forest F G \<equiv> \<forall>F'. spanning_forest F' G \<longrightarrow> edge_weight F \<le> edge_weight F'"
mat0_index: fixes i :: "nat"   and nc :: "nat"   and j :: "nat"   and nr :: "nat"   and ze :: "'c" assumes "i < nc"   and "j < nr" shows "mat0I ze nr nc ! i ! j = ze"
prod_induct3: fixes P :: "'a \<times> 'b \<times> 'c \<Rightarrow> bool"   and x :: "'a \<times> 'b \<times> 'c" assumes "\<And>a b c. P (a, b, c)" shows "P x"
m1_def: shows "m1 = Min {k. k < length xs \<and> xs ! k = i}"
comp_timpl_closure_is_timpl_closure_set: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set"   and TI :: "('sets set \<times> 'sets set) list" assumes "finite M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M" shows "comp_timpl_closure M (set TI) = timpl_closure_set M (set TI)"
hsubsetI: fixes A :: "hf"   and B :: "hf" assumes "\<And>x. x \<^bold>\<in> A \<Longrightarrow> x \<^bold>\<in> B" shows "A \<le> B"
exhaust: fixes y :: "ternaryvalue" obtains "y = TernaryTrue"   | "y = TernaryFalse"   | "y = TernaryUnknown"
eq: shows "(\<Sum>v\<in>T. u v *s v) = (0::'b)"
map_M_ok_I: fixes xs :: "'m list"   and h :: "'n"   and f :: "'m \<Rightarrow> ('n, 'o, 'p) prog" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<turnstile> ok f x"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "h \<turnstile> ok map_M f xs"
eq_sym: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a"   and y :: "'a" assumes "class.Eq_sym eq" shows "eq\<cdot>x\<cdot>y = eq\<cdot>y\<cdot>x"
map_of_alist_nil: fixes ys :: "('b \<times> 'c) list"   and x :: "'b" shows "has_key_alist ys x = (map_of_alist ys\<langle>x\<rangle> \<noteq> None)"
ray_def: fixes a :: "'a"   and b :: "'a" shows "ray a b \<equiv> insert b (segment a b \<union> prolongation a b)"
2: shows "map ((!) (as @ bs) \<circ> (+) (length as)) (fst ys) = map ((!) bs) (fst ys)"
assms_3: shows "x \<in>\<^sub>\<circ> A (vcard xs)"
simps_6: fixes f1 :: "fdecl \<Rightarrow> 'b"   and f2 :: "mdecl \<Rightarrow> 'b"   and x2 :: "mdecl" shows "(case mdecl x2 of fdecl x \<Rightarrow> f1 x | mdecl x \<Rightarrow> f2 x) = f2 x2"
fw_shortest: 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 n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "less_eq (linordered_ab_monoid_add.fw plus less_eq m n n i j) (plus (linordered_ab_monoid_add.fw plus less_eq m n n i k) (linordered_ab_monoid_add.fw plus less_eq m n n k j))"
poly_mod_type_simps_3: fixes m :: "int" shows "poly_mod_prime_type TYPE('b) m = (m = int CARD('b))"
ca_prod: fixes CA :: "'a literal multiset" assumes "CA \<in> set CAs" shows "gr.production M N CA \<noteq> {}"
pow_int_0: fixes x :: "'b" shows "pow_int x 0 = (1::'b)"
not_zero_le_neg_numeral: fixes n :: "num" shows "\<not> (0::'b) \<le> - numeral n"
simps_2: fixes xs :: "nat option list"   and zs :: "nat list"   and i :: "nat"   and T :: "bool list list" shows "mk_eqcl (None # xs) zs i T = (let (xs', y) = mk_eqcl (mk_eqcl' xs i (Suc i) (length zs) T) (zs @ [i]) (Suc i) T in (length zs # xs', y))"
lexord_asym: fixes R :: "('g \<times> 'g) set" assumes "asym R" shows "asym (lexord R)"
pmod_const_2: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "degree p < degree q" shows "UP.pmod p q = p"
memb: fixes a :: "'a"   and bs :: "'b" shows "memb a bs = (a \<in> set_of bs)"
ih3: fixes a :: "coname"   and M :: "trm" assumes "<a>:M \<in> \<parallel><B>\<parallel>" shows "SNa M"
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and R :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 R,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Seq c\<^sub>1 c\<^sub>2 Q,A"
mutual_information_nonneg': fixes S :: "'d measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> real"   and Pxy :: "'d \<times> 'e \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "finite_entropy S X Px"   and "finite_entropy T Y Py"   and "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "0 \<le> mutual_information b S T X Y"
bis_def: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and B1' :: "'h set"   and B2' :: "'i set"   and s1' :: "'h \<Rightarrow> ('f, 'h, 'i) F1"   and s2' :: "'i \<Rightarrow> ('g, 'h, 'i) F2"   and R1 :: "('d \<times> 'h) set"   and R2 :: "('e \<times> 'i) set" shows "bis B1 B2 s1 s2 B1' B2' s1' s2' R1 R2 = ((R1 \<subseteq> B1 \<times> B1' \<and> R2 \<subseteq> B2 \<times> B2') \<and> (\<forall>b1 b1'. (b1, b1') \<in> R1 \<longrightarrow> (\<exists>z\<in>F1in UNIV R1 R2. F1map id fst fst z = s1 b1 \<and> F1map id snd snd z = s1' b1')) \<and> (\<forall>b2 b2'. (b2, b2') \<in> R2 \<longrightarrow> (\<exists>z\<in>F2in UNIV R1 R2. F2map id fst fst z = s2 b2 \<and> F2map id snd snd z = s2' b2')))"
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 \<le> B" shows "thesis"
sup_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "sup f g x = sup (f x) (g x)"
emsapss_encode_help5_def: fixes DB :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_encode_help5 DB H emBits = emsapss_encode_help6 DB (MGF H (length DB)) H emBits"
Omega_fun_def: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'f" shows "Omega_fun f g = (\<sqinter>) g \<circ> (\<circ>) f"
aXb: shows "Derives1 \<delta> i r (a @ [X] @ b)"
AssertionStatImpTrans: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" assumes "AssertionStatImp \<Psi> \<Psi>'"   and "AssertionStatImp \<Psi>' \<Psi>''" shows "AssertionStatImp \<Psi> \<Psi>''"
round_4: fixes prf_key_gen :: "'a spmf"   and prf_fun :: "'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_domain :: "bool list set"   and prf_dlen :: "nat"   and prf_clen :: "nat"   and upf_key_gen :: "'b spmf"   and \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'c, (bool list \<times> bool list \<times> 'c) option + bool list option) gpv"   and upf_fun :: "'b \<Rightarrow> bool list \<Rightarrow> 'c" assumes "simple_cipher prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen"   and "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) \<A>" shows "map_spmf fst (simple_cipher.game3 prf_key_gen prf_domain prf_clen upf_key_gen upf_fun \<A>) = coin_spmf"
DIAndCut34Elim1_def: shows "DIAndCut34Elim1 = (([[$\<alpha> vid1]](Pc pid3 && Pc pid4) \<rightarrow> Pc pid3) \<rightarrow> ([[$\<alpha> vid1]]Pc pid3 && Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid3)"
c_rec: shows "c_rec \<le> (\<Sum>ii = Suc i..<m. 2 * n + (2 * n + 1 + 2 * ii) * ii)"
test_eq_conv: fixes x :: "'a" assumes "is_test x" shows "x\<^sup>T = x"
elims: fixes x :: "('a, 'b) Term.term"   and y :: "('a \<times> nat) option" assumes "root x = y" obtains xa :: "'b" where "x = Var xa" and "y = None"   | f :: "'a" and ts :: "('a, 'b) Term.term list" where "x = Fun f ts" and "y = Some (f, length ts)"
cringE_1: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "Group.comm_monoid R"
step: fixes r :: "('a, 'b, 'c, 'd) resumption"   and r' :: "('a, 'b, 'c, 'd) resumption" assumes "X r r'" shows "(\<exists>a. r = Terminal a \<and> r' = Terminal a) \<or> (\<exists>R R' b. r = Linear b R \<and> r' = Linear b R' \<and> (X R R' \<or> R = R')) \<or> (\<exists>rs rs' c. r = Branch c rs \<and> r' = Branch c rs' \<and> (\<forall>d. X (rs d) (rs' d) \<or> rs d = rs' d))"
Sup_subset_mono: fixes B :: "'al set"   and A :: "'al set" assumes "B \<subseteq> U"   and "A \<subseteq> B" shows "\<Squnion>\<^sub>o\<^sub>w A \<le>\<^sub>o\<^sub>w \<Squnion>\<^sub>o\<^sub>w B"
\<omega>_in_vhom: shows "\<guillemotleft>\<omega> : w \<star>\<^sub>B w \<Rightarrow> w\<guillemotright>"
refl_on_def': fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "refl_on A r = ((\<forall>(x, y)\<in>r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
holomorphic_on_compose: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f holomorphic_on s"   and "g holomorphic_on f ` s" shows "g \<circ> f holomorphic_on s"
op_bicategory_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b" shows "op_bicategory V H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<equiv> bicategory V H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"