hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "'b formula set"   and x2 :: "'b formula"   and P :: "'b formula set \<Rightarrow> 'b formula \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>F \<Gamma>. F \<in> \<Gamma> \<Longrightarrow> P \<Gamma> F"   and "\<And>\<Gamma> F. \<lbrakk>\<Gamma> \<turnstile> \<^bold>\<not> F; P \<Gamma> (\<^bold>\<not> F); \<Gamma> \<turnstile> F; P \<Gamma> F\<rbrakk> \<Longrightarrow> P \<Gamma> \<bottom>"   and "\<And>F \<Gamma>. \<lbrakk>F \<triangleright> \<Gamma> \<turnstile> \<bottom>; P (F \<triangleright> \<Gamma>) \<bottom>\<rbrakk> \<Longrightarrow> P \<Gamma> (\<^bold>\<not> F)"   and "\<And>F \<Gamma>. \<lbrakk>\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> \<bottom>; P (\<^bold>\<not> F \<triangleright> \<Gamma>) \<bottom>\<rbrakk> \<Longrightarrow> P \<Gamma> F"   and "\<And>\<Gamma> F G. \<lbrakk>\<Gamma> \<turnstile> F \<^bold>\<and> G; P \<Gamma> (F \<^bold>\<and> G)\<rbrakk> \<Longrightarrow> P \<Gamma> F"   and "\<And>\<Gamma> F G. \<lbrakk>\<Gamma> \<turnstile> F \<^bold>\<and> G; P \<Gamma> (F \<^bold>\<and> G)\<rbrakk> \<Longrightarrow> P \<Gamma> G"   and "\<And>\<Gamma> F G. \<lbrakk>\<Gamma> \<turnstile> F; P \<Gamma> F; \<Gamma> \<turnstile> G; P \<Gamma> G\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<and> G)"   and "\<And>\<Gamma> F G. \<lbrakk>\<Gamma> \<turnstile> F; P \<Gamma> F\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<or> G)"   and "\<And>\<Gamma> G F. \<lbrakk>\<Gamma> \<turnstile> G; P \<Gamma> G\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<or> G)"   and "\<And>\<Gamma> F G H. \<lbrakk>\<Gamma> \<turnstile> F \<^bold>\<or> G; P \<Gamma> (F \<^bold>\<or> G); F \<triangleright> \<Gamma> \<turnstile> H; P (F \<triangleright> \<Gamma>) H; G \<triangleright> \<Gamma> \<turnstile> H; P (G \<triangleright> \<Gamma>) H\<rbrakk> \<Longrightarrow> P \<Gamma> H"   and "\<And>F \<Gamma> G. \<lbrakk>F \<triangleright> \<Gamma> \<turnstile> G; P (F \<triangleright> \<Gamma>) G\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<rightarrow> G)"   and "\<And>\<Gamma> F G. \<lbrakk>\<Gamma> \<turnstile> F \<^bold>\<rightarrow> G; P \<Gamma> (F \<^bold>\<rightarrow> G); \<Gamma> \<turnstile> F; P \<Gamma> F\<rbrakk> \<Longrightarrow> P \<Gamma> G" shows "P x1 x2"
happens_before_new_not_new: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat"   and a' :: "nat"   and P :: "'c prog" assumes "thread_start_actions_ok E"   and "a \<in> actions E"   and "a' \<in> actions E"   and "is_new_action (action_obs E a)"   and "\<not> is_new_action (action_obs E a')" shows "P,E \<turnstile> a \<le>hb a'"
connected_outside: fixes S :: "'a set" assumes "bounded S"   and "2 \<le> DIM('a)" shows "connected (outside S)"
sinit_0: fixes xs :: "'b seq" shows "sinit 0 xs = []"
starfun_star_of: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(*f* f) (star_of x) = star_of (f x)"
openin_subtopology: fixes U :: "'c topology"   and V :: "'c set"   and S :: "'c set" shows "openin (subtopology U V) S = (\<exists>T. openin U T \<and> S = T \<inter> V)"
allocationDisjoint: fixes a :: "('c \<times> 'd set) set"   and n1 :: "'c"   and n2 :: "'c" assumes "a \<in> allocationsUniverse"   and "n1 \<in> Domain a"   and "n2 \<in> Domain a"   and "n1 \<noteq> n2" shows "a ,,, n1 \<inter> a ,,, n2 = {}"
one: shows "\<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV} \<cdot> \<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV} = ((1::'b) + \<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV}) \<cdot> ((1::'b) + \<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV})"
commute: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<inter># b = b \<inter># a"
NOTRIGHT_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and B :: "ty"   and X :: "ntrm set" shows "pi \<bullet> NOTRIGHT (NOT B) X = NOTRIGHT (NOT B) (pi \<bullet> X)"
assms_4: shows "f (deg R f) \<otimes> g (deg R g) [^] deg R f \<noteq> \<zero>"
implicit_pres_empty: fixes effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list" shows "implicit_pres (map rem_effect_implicit_pres effs) = []"
hyps_1: fixes i :: "nat" shows "subst_db (Suc i) x s1 \<ge>\<^sub>t subst_db i x s1"
h2: shows "sortedTree h t2 \<longrightarrow> sortedTree h (remove h e t2)"
plossless_inline: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('d, 'e) \<I>"   and s :: "('a, 'b, 'd, 'e) converter" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>plossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> plossless_gpv \<I>' (run_converter s x)"   and "plossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "plossless_gpv \<I>' (inline run_converter gpv s)"
llist_all2_llengthD: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'f llist" assumes "llist_all2 P xs ys" shows "llength xs = llength ys"
Group_VectorSpaceHomSet: fixes smult' :: "'f \<Rightarrow> 'a \<Rightarrow> 'a"   and W :: "'a set" assumes "VectorSpace smult' W" shows "Group (VectorSpaceHomSet (\<cdot>) V smult' W)"
dist: shows "vars\<^sub>l\<^sub>s C \<inter> vars\<^sub>l\<^sub>s D = {}"
optimize_matches_option_normalized_nnf_match: fixes rs :: "'b rule list"   and f :: "'b match_expr \<Rightarrow> 'b match_expr option" assumes "\<And>r. r \<in> set rs \<Longrightarrow> normalized_nnf_match (get_match r)"   and "\<And>m m'. \<lbrakk>normalized_nnf_match m; f m = Some m'\<rbrakk> \<Longrightarrow> normalized_nnf_match m'" shows "\<forall>r\<in>set (optimize_matches_option f rs). normalized_nnf_match (get_match r)"
Mp_ident_product: fixes n :: "int"   and f :: "int poly" assumes "0 < n"   and "Mp f = f" shows "poly_mod.Mp (m * n) f = f"
descartes_sign_zero: fixes p :: "real poly"   and i :: "nat" assumes "\<And>x. poly (map_poly complex_of_real p) x = 0 \<Longrightarrow> Re x \<le> 0"   and "monic p" shows "0 \<le> coeff p i"
dist_triangle2: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist x z + dist y z"
gR_rel_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq>\<^sub>R y"   and "y \<preceq>\<^sub>R z" shows "x \<preceq>\<^sub>R z"
P: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and i :: "nat" assumes "Ionescu_Tulcea P M" shows "P i \<in> Pi\<^sub>M {0..<i} M \<rightarrow>\<^sub>M subprob_algebra (M i)"
Limit_Vfrom_eq: fixes i :: "V"   and a :: "V" assumes "Limit i" shows "Vfrom a i = \<Union>\<^sub>\<circ> (VLambda i (Vfrom a) `\<^sub>\<circ> i)"
assms: shows "map (\<lambda>(a, y). (mset a, y)) p = map (\<lambda>(a, y). (mset a, y)) s"
bangPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto>[Rel] Q"   and "eqvt Rel" shows "!P \<leadsto>[bangRel Rel] !Q"
collapse: fixes insert_message :: "('a, 'b) insert_message" shows "InsertMessage (P insert_message) (I insert_message) (S insert_message) (\<Sigma> insert_message) = insert_message"
fgs_single_span: fixes h :: "'a"   and x :: "'a" assumes "h \<in> carrier M"   and "x \<in> fgs R M {h}" shows "\<exists>a\<in>carrier R. x = a \<cdot>\<^sub>s h"
vertical_composite_axioms: shows "vertical_composite (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom t) (Cod t) (Cod t') (Map t) (Map t')"
line_integral_exists: shows "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"
wf_proc_def: fixes x :: "char list \<times> char list list \<times> char list list \<times> cmd" shows "wf_proc x \<equiv> let (p, ins, outs, c) = x in p \<noteq> Main \<and> distinct ins \<and> distinct outs"
incomplete_design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" shows "incomplete_design point_set block_collection u_block_size \<equiv> block_design point_set block_collection u_block_size \<and> incomplete_design_axioms point_set u_block_size"
matrix_vector_mul_lid: fixes x :: "('a, 'b) vec" shows "mat (1::'a) *v x = x"
tr_wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s: fixes A' :: "('fun, 'var) strand_step list"   and A :: "('fun, 'var) stateful_strand_step list" assumes "A' \<in> set (tr A [])"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t A)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t A')"
init_Val_PD: fixes P :: "(char list list \<times> char list exp) prog"   and C' :: "char list"   and Cs :: "char list list"   and b :: "bool"   and s :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>INIT C' (Cs,b) \<leftarrow> unit,s\<rangle> \<Rightarrow> \<langle>Val v,s'\<rangle>"   and "iconf (shp s) (INIT C' (Cs,b) \<leftarrow> unit)" shows "\<exists>sfs i. shp s' C' = \<lfloor>(sfs, i)\<rfloor> \<and> (i = Done \<or> i = Processing)"
set_prems_of: fixes \<iota> :: "'b inference" shows "set (prems_of \<iota>) = (if prems_of \<iota> = [] then {} else {main_prem_of \<iota>} \<union> set (side_prems_of \<iota>))"
LIMSEQ_root_const: fixes c :: "real" assumes "0 < c" shows "(\<lambda>n. root n c) \<longlonglongrightarrow> 1"
Units_2: shows "invertible u2 \<and> u2 \<in> M"
Im_Ln_le_pi: fixes z :: "complex" assumes "z \<noteq> 0" shows "Im (ln z) \<le> pi"
ent_star_mono: fixes P :: "assn"   and P' :: "assn"   and Q :: "assn"   and Q' :: "assn" assumes "P \<Longrightarrow>\<^sub>A P'"   and "Q \<Longrightarrow>\<^sub>A Q'" shows "P * Q \<Longrightarrow>\<^sub>A P' * Q'"
simps_2: fixes at :: "('c, 'd) atm" shows "varsL (Neg at) = varsA at"
senior_le_token: fixes x :: "nat"   and n :: "nat" shows "senior x n \<le> x"
tc_monomial: fixes t :: "'a"   and c :: "'c" shows "punit.tc (monomial c t) = c"
simps: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "sfirst P \<omega> = (if P \<omega> then 0 else eSuc (sfirst P (stl \<omega>)))"
Ord_in_in_VsetI: fixes \<alpha> :: "V"   and a :: "V" assumes "Ord \<alpha>"   and "a \<in>\<^sub>\<circ> \<alpha>" shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
C: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "cmod (f' (\<gamma> x)) \<le> C"
is_idirprod_generate: fixes A :: "'a set"   and gs :: "'a set" assumes "A = generate G gs"   and "gs \<subseteq> carrier G"   and "compl_fam (\<lambda>g. generate G {g}) gs" shows "is_idirprod A (\<lambda>g. generate G {g}) gs"
simple_linux_router_nol12_def: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme" shows "simple_linux_router_nol12 rt fw p \<equiv> let rd = routing_table_semantics rt (p_dst p); p = p\<lparr>p_oiface := output_iface rd\<rparr>; fd = simple_fw fw p in (case fd of Decision FinalAllow \<Rightarrow> Some () | Decision FinalDeny \<Rightarrow> None) \<bind> (\<lambda>_. Some p)"
ind_in_set_minus_subset_bound: fixes B :: "nat set"   and A :: "nat set"   and M :: "nat" assumes "B \<subseteq> A" shows "ind_in_set A ` ({0..<M} \<inter> A) - ind_in_set A ` B = ind_in_set A ` ({0..<M} \<inter> A) \<inter> ind_in_set A ` (A - B)"
assms_1: shows "\<C> \<turnstile> es : t1s _> t2s"
has_codomain_iff_arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C" shows "(partial_magma.codomains C f \<noteq> {}) = partial_magma.arr C f"
P: fixes l :: "'a" assumes "is_lub L l (insert x A)" shows "P l"
set_mset_Abs_multiset: fixes f :: "'a \<Rightarrow> nat" assumes "f \<in> multiset" shows "set_mset (Abs_multiset f) = {x. 0 < f x}"
AG_I: fixes q :: "'b set"   and p :: "'b set" assumes "q \<subseteq> p \<inter> \<^bold>A\<^bold>X q" shows "q \<subseteq> \<^bold>A\<^bold>G p"
W2_def: shows "W2 = local.span (set (cols B))"
coeff_map_poly_hom: fixes p :: "'b poly"   and i :: "nat" shows "coeff (map_poly to_fract p) i = to_fract (coeff p i)"
eq_id_iff: fixes f :: "'k \<Rightarrow> 'k" shows "(\<forall>x. f x = x) = (f = id)"
LeftDerives1_append_leftmost_unique: fixes a :: "symbol list"   and b :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and c :: "symbol list"   and j :: "nat" assumes "LeftDerives1 (a @ b) i r c"   and "leftmost j a" shows "i = j"
total_order_nsets_2_eq: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "total_on A r"   and "irrefl r" shows "[A]\<^bsup>2\<^esup> = {{x, y} |x y. x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
fds_eq_fds_iff': fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "f 0 = g 0" shows "(fds f = fds g) = (f = g)"
reachable_V: fixes x :: "'v" assumes "reachable x" shows "vertex \<Gamma> x"
elims_1: fixes x :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and xa :: "('c, 'd) fmap list"   and y :: "bool" assumes "valid_path x xa = y" obtains Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" where "x = Pi__" and "xa = []" and "y"   | Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" and s :: "('c, 'd) fmap" where "x = Pia__" and "xa = [s]" and "y = (s \<in> valid_states Pia__)"   | Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" and s1 :: "('c, 'd) fmap" and s2 :: "('c, 'd) fmap" and rest :: "('c, 'd) fmap list" where "x = Pi" and "xa = s1 # s2 # rest" and "y = (s1 \<in> valid_states Pi \<and> (\<exists>a. a \<in> Pi \<and> exec_plan s1 [a] = s2) \<and> valid_path Pi (s2 # rest))"
thing3: shows "sint (SCAST(32 \<rightarrow> 64) w2) \<le> 9223372034707292160"
nested_prop_atoms\<^sub>\<mu>_card: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "card (nested_prop_atoms\<^sub>\<mu> \<phi> X) \<le> card (nested_prop_atoms \<phi>)"
homeomorphic_map_subtopologies: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "homeomorphic_map X Y f"   and "f ` (topspace X \<inter> S) = topspace Y \<inter> T" shows "homeomorphic_map (subtopology X S) (subtopology Y T) f"
fminus_finter: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" shows "A |-| (B |\<inter>| C) = A |-| B |\<union>| (A |-| C)"
map_ran_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "map_ran f = map (\<lambda>(k, v). (k, f k v))"
Guard: fixes P :: "'d set"   and g :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f" assumes "P \<subseteq> g \<inter> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g c Q,A"
measurable_equality_set: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{x \<in> space M. f x = g x} \<in> sets M"
bexE: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" obtains x :: "'d" where "x \<in> A" and "P x"
purgeIdle_purge_ex: fixes sl :: "('St, 'U, 'C) state list"   and ucl :: "('U \<times> 'C) list"   and ss' :: "('St, 'U, 'C) state" assumes "wffp sl"   and "list_all isState sl"   and "map getGMUserCom (tl sl) = ucl" shows "\<exists>sl'. hd sl' = ss' \<and> wffp sl' \<and> list_all2 f (tl sl) (tl sl') \<and> map getGMUserCom (purgeIdle (tl sl')) = purge GH ucl"
initial_abs: shows "\<alpha> ` wa_initial WAC \<subseteq> wa_initial WAA"
simps: fixes xs :: "('b btree \<times> 'b) list" shows "split_half xs = (take (length xs div 2) xs, drop (length xs div 2) xs)"
mpj_mem: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and m :: "'e" assumes "R module M"   and "submodule R M H"   and "m \<in> carrier M" shows "mpj M H m \<in> carrier (M /\<^sub>m H)"
crel_vs1_executeD: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f Heap"   and heap :: "heap" assumes "crel_vs1 R a b"   and "P heap"   and "state_dp_consistency.cmem heap" obtains x :: "'f" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "state_dp_consistency.cmem heap'" and "R a x"
induct: fixes P2 :: "('c, 'd) bintree list \<Rightarrow> bool"   and P1 :: "('c, 'd) bintree \<Rightarrow> bool"   and bintree :: "('c, 'd) bintree" assumes "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (Node x1 x2 x3)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 bintree"
Compose: fixes T :: "('fun, 'var) Term.term list"   and f :: "'fun"   and M :: "('fun, 'var) Term.term set" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> M \<turnstile> t" shows "M \<turnstile> Fun f T"
g1: fixes x :: "'a"   and y :: "'a"   and S :: "'a set" assumes "\<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S" shows "geodesic_segment_between (g x S y) x y \<and> g x S y \<subseteq> S"
square_free_factorization_def: fixes p :: "'a poly"   and cbs :: "'a \<times> ('a poly \<times> nat) list" shows "square_free_factorization p cbs \<equiv> case cbs of (c, bs) \<Rightarrow> p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i) \<and> (p = 0 \<longrightarrow> c = (0::'a) \<and> bs = []) \<and> (\<forall>a i. (a, i) \<in> set bs \<longrightarrow> square_free a \<and> 0 < degree a) \<and> (\<forall>a i b j. (a, i) \<in> set bs \<longrightarrow> (b, j) \<in> set bs \<longrightarrow> (a, i) \<noteq> (b, j) \<longrightarrow> coprime a b) \<and> distinct bs"
hd_le_set: fixes xs :: "'a list"   and x :: "'a" assumes "sorted xs"   and "xs \<noteq> []"   and "x \<in> set xs" shows "hd xs \<le> x"
trace_decomp_tail: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv c (t @ [ev]) c'" shows "\<exists>c''. distributed_system.trace channel trans send recv c t c'' \<and> distributed_system.next channel trans send recv c'' ev c'"
one_Nat_def: shows "1 \<equiv> box\<cdot>(1::'a)"
monom_mult_monomial: fixes c :: "'b"   and s :: "'x \<Rightarrow>\<^sub>0 nat"   and v :: "'x \<Rightarrow>\<^sub>0 nat"   and d :: "'b" shows "local.punit.monom_mult c s (monomial d v) = monomial (c * d) (local.punit.splus s v)"
meta_allE: fixes P :: "'d \<Rightarrow> prop"   and x :: "'d" assumes "\<And>x. PROP P x" obtains "PROP P x"
Await: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and c' :: "('d, 'e, 'f) com"   and s' :: "'d" assumes "s \<in> b"   and "\<Gamma> \<turnstile> (c, Normal s) \<rightarrow>\<^sup>* (c', Normal s')"   and "atom_com c"   and "c' = Skip \<or> c' = Throw" shows "\<Gamma>\<turnstile> (Await b c, Normal s) \<rightarrow> (c', Normal s')"
ogets_def: fixes f :: "'c \<Rightarrow> 'd" shows "ogets f \<equiv> \<lambda>s. Some (f s)"
cnf2bhv_lnth_lappend: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and t' :: "nat \<Rightarrow> 'cmp" assumes "\<exists>i. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<not> (\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)" shows "lnth (\<pi>\<^bsub>c\<^esub>inf_llist t @\<^sub>l inf_llist t') (\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n) = lnth (inf_llist t') (n - \<langle>c \<and> t\<rangle> - 1)"
balanced_bdt: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and var :: "ref \<Rightarrow> nat"   and bdt1 :: "bdt"   and no :: "ref" assumes "Dag p low high t"   and "bdt t var = Some bdt1"   and "no \<in> set_of t" shows "(low no = Null) = (high no = Null)"
insert_ops_rem_cons: fixes x :: "'a \<times> 'a option"   and xs :: "('a \<times> 'a option) list" assumes "insert_ops (x # xs)" shows "insert_ops xs"
proots_count_monom: fixes A :: "'b set"   and d :: "nat" assumes "(0::'b) \<notin> A" shows "proots_count (monom (1::'b) d) A = 0"
all_final_exceptI: fixes s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('x2 \<times> 'b \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'd wait_set_status option) \<times> 'a set"   and Ts :: "'a set" assumes "\<And>t. r2.not_final_thread s t \<Longrightarrow> t \<in> Ts" shows "r2.all_final_except s Ts"
simps_7: fixes x31 :: "comb"   and x32 :: "comb" shows "x31 \<bullet> x32 \<noteq> S"
RDirProd_iso5: fixes f :: "'a \<Rightarrow> 'b"   and S :: "('a, 'c) ring_scheme"   and T :: "('b, 'd) ring_scheme"   and R :: "('e, 'f) ring_scheme" assumes "f \<in> ring_iso S T" shows "(\<lambda>(r, s). (r, f s)) \<in> ring_iso (RDirProd R S) (RDirProd R T)"
SubstTermP_unique: fixes v :: "tm"   and tm :: "tm"   and t :: "tm"   and u :: "tm"   and u' :: "tm" shows "{SubstTermP v tm t u, SubstTermP v tm t u'} \<turnstile> u' EQ u"
that: fixes c :: "nat"   and q :: "int poly" assumes "c \<noteq> 0"   and "p = Polynomial.smult (inverse (of_nat c)) (of_int_poly q)" shows "thesis"
row_swaprows_1: fixes l :: "nat"   and n :: "nat" assumes "l < n" shows "row (swaprows_mat n l l) l = unit_vec n l"
circ_circ_sup: fixes x :: "'a" shows "((1::'a) \<squnion> x)\<^sup>\<star> = x\<^sup>\<star>\<^sup>\<star>"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Compound F Fs, m) \<in> derivable R* \<longrightarrow> (\<forall>r'\<in>R. rightPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable R*)"
gt_zero: fixes x :: "'a" assumes "x \<in> E"   and "x \<noteq> (0::'a)" shows "0 < \<parallel>x\<parallel>"
subseqs_refl: fixes xs :: "'c list" shows "xs \<in> set (subseqs xs)"
simps_4: fixes bs :: "real list"   and a :: "Ferrack.num" shows "Inum bs (Neg a) = - Inum bs a"
Un_set_offending_flows_bound_minus_subseteq': fixes m :: "'b SecurityInvariant_configured"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and X :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "\<Union> (c_offending_flows m \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> X" shows "\<Union> (c_offending_flows m \<lparr>nodes = V, edges = E - E'\<rparr>) \<subseteq> X - E'"
diffs_minus: fixes c :: "nat \<Rightarrow> 'a" shows "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
Iod_less: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T" shows "a \<prec>\<^bsub>Iod D T\<^esub> b = a \<prec>\<^bsub>D\<^esub> b"
assms: shows "GEN_ALGO_tag (is_map_to_list Rk unit_rel R it)"
scaleR_add_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "real"   and x :: "'c"   and y :: "'c" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (plus x y) = plus (scaleR a x) (scaleR a y)"
retractionE: fixes g :: "'a" assumes "retraction g" obtains f :: "'a" where "ide (g \<cdot> f)"
parent_bounds: fixes l :: "int"   and x :: "int"   and r :: "int" assumes "l < x"   and "x < r" shows "bounded l r (parent l x)"
closure_closure: fixes S :: "'d set" shows "closure (closure S) = closure S"
inv_translationD_1: fixes r :: "nat set" assumes "inv_translation r" shows "2 \<le> card r"