hcju/isabelleps huggingface dataset:
linear: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "linorder_ow U le ls"   and "x \<in> U"   and "y \<in> U" shows "le x y \<or> le y x"
Gp_rcs: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and K :: "'c set"   and x :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> K "   and "H \<subseteq> K"   and "x \<in> K" shows "H \<bullet>\<^bsub>\<natural>\<^bsub>G\<^esub>K\<^esub> x = H \<bullet>\<^bsub>G\<^esub> x"
groupF_code: fixes f :: "'a \<Rightarrow> 'b"   and as :: "'a list" shows "groupF f as = ??.GroupF.groupF_code f as"
L: shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> wrap {} \<union> (P\<^sub>2 \<union> wrap B\<^sub>2) \<union> wrap {u}) = \<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> wrap {} \<union> (P\<^sub>2 \<union> wrap B\<^sub>2)) \<union> {u}"
p_def: fixes t :: "real" shows "ego_other.p t = s\<^sub>o - s\<^sub>e + (v\<^sub>o - v\<^sub>e) * t + 1 / 2 * (a\<^sub>o - a\<^sub>e) * t\<^sup>2"
ring_hom_add: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
power_up_nonpos_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(power_up prec b n \<le> 0) = (b < 0 \<and> odd n \<or> b = 0 \<and> n \<noteq> 0)"
set_iterator_image_def: fixes g :: "'e \<Rightarrow> 'f" shows "set_iterator_image g \<equiv> set_iterator_image_filter (\<lambda>x. Some (g x))"
A: shows "d' * e' = c * d' * (s' + a * b)"
source_UNIV: fixes x :: "'b \<Longrightarrow> 'c" assumes "wb_lens x" shows "\<S>\<^bsub>x\<^esub> = UNIV"
that: fixes d :: "'a" assumes "D = (\<lambda>x. cnj x *\<^sub>C d)" shows "thesis"
assms_3: shows "insertion (nth_default 0 (xs' @ x # xs)) c = C"
InternalSets_Compl: fixes X :: "'a star set" assumes "X \<in> InternalSets" shows "- X \<in> InternalSets"
drop_last_message: fixes evts :: "('a \<times> operation) event list"   and i :: "nat"   and msgs :: "('a \<times> operation) list"   and last_msg :: "'a \<times> operation" assumes "evts prefix of i"   and "node_deliver_messages evts = msgs @ [last_msg]" shows "\<exists>pre. pre prefix of i \<and> node_deliver_messages pre = msgs"
Rep_CoCall_cases: fixes y :: "(Vars.var \<times> Vars.var) set" assumes "y \<in> {G. sym G}" obtains x :: "CoCalls" where "y = Rep_CoCall x"
assoc_four_block_mat: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat" shows "four_block_mat A (0\<^sub>m (dim_row A) (dim_col (four_block_mat B (0\<^sub>m (dim_row B) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col B)) C))) (0\<^sub>m (dim_row (four_block_mat B (0\<^sub>m (dim_row B) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col B)) C)) (dim_col A)) (four_block_mat B (0\<^sub>m (dim_row B) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col B)) C) = four_block_mat (four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B) (0\<^sub>m (dim_row (four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B)) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col (four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B))) C"
hyps_2: shows "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
\<tau>inv: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and tl1 :: "'h"   and s1' :: "'g"   and tl2 :: "'j"   and s2' :: "'i" assumes "\<tau>inv trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "trsys1 s1 tl1 s1'"   and "trsys2 s2 tl2 s2'"   and "bisim s1' s2'"   and "tlsim tl1 tl2" shows "\<tau>move1 s1 tl1 s1' = \<tau>move2 s2 tl2 s2'"
lookup_lcs_fun: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (lcs s t) = lcs (lookup s) (lookup t)"
intros_3: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Drop], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
OrdI: fixes x :: "\<kappa>"   and v :: "i" assumes "\<exists>o\<^sub>1 y. Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<omega>\<nu> y" shows "[\<lparr>O!,x\<rparr> in v]"
iprod_a_def: fixes cs :: "real list"   and as :: "atom list" shows "cs \<odot>\<^sub>a as = sum_list (map2 (*\<^sub>a) cs as)"
inres_leof_mono: fixes m :: "'b nres"   and m' :: "'b nres"   and x :: "'b" assumes "m \<le>\<^sub>n m'"   and "nofail m"   and "inres m x" shows "inres m' x"
x_or_neighbors: fixes P :: "'a set set"   and S :: "'a set" assumes "\<forall>p\<in>P. \<exists>s\<in>V. p = {s} \<union> neighbors s"   and "independent_vertices E S" shows "\<forall>p\<in>P. card (S \<inter> p) \<le> \<Delta>"
igAbsRen_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igAbsRen MOD \<equiv> \<forall>xs y x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
cadd_0: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "0 \<oplus> \<kappa> = \<kappa>"
odd_of_intvl_intvl: fixes S :: "nat set"   and m :: "nat"   and n :: "nat" assumes "S = {m..n} \<inter> {i. odd i}" shows "\<exists>m' n'. S = Suc ` (\<lambda>i. i * 2) ` {m'..n'}"
termMOD_igAbsCongS: shows "igAbsCongS termMOD"
unfolding_sound_complete: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state" assumes "wf_chain \<Gamma> rs" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_call \<Gamma> rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
wf_data_prev': fixes as :: "'a option list" assumes "as \<in> \<Union> (snd ` set (linearize data_prev'))" shows "wf_tuple n R as"
ile_add2: fixes n :: "enat"   and m :: "enat" shows "n \<le> m + n"
OK_le_conv: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b err" shows "(OK x \<sqsubseteq>\<^bsub>Err.le r\<^esub> e) = (e = Err \<or> (\<exists>y. e = OK y \<and> x \<sqsubseteq>\<^bsub>r\<^esub> y))"
SUP_mult_right_ennreal: fixes f :: "'b \<Rightarrow> ennreal"   and I :: "'b set"   and c :: "ennreal" shows "\<Squnion> (f ` I) * c = (\<Squnion>i\<in>I. f i * c)"
group_inv_rel: fixes g :: "'a"   and m :: "'c"   and n :: "'c" assumes "g \<in> carrier G"   and "m \<in> M"   and "n \<in> M"   and "\<phi> g n = m" shows "\<phi> (inv g) m = n"
cc_n_minus_1: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F" shows "pm_powerprod.aa X f F (card X - 1) + pm_powerprod.bb ord X F (card X - 1) = 2 * poly_deg f"
multiplicity_same_power': fixes p :: "'a"   and n :: "nat" shows "multiplicity p (p ^ n) = (if p = (0::'a) \<or> is_unit p then 0 else n)"
subpath_def: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.subpath g v1 es v2 subs \<equiv> sub_rel_of g subs \<and> v1 \<in> Graph.vertices g \<and> SubRel.ces v1 es v2 subs \<and> set es \<subseteq> edges g"
open_path_component: fixes S :: "'a set"   and x :: "'a" assumes "open S" shows "open (path_component_set S x)"
k_2: fixes i :: "nat"   and G1 :: "'b poly"   and G2 :: "'b poly" assumes "i < k G1 G2" shows "F G1 G2 (Suc i) \<noteq> 0"
rpo_with_assms_axioms: shows "rpo_with_assms prc prl"
suff_nhd_base: fixes u :: "'a llist set"   and t :: "'a llist" assumes "u \<in> nhds t" obtains r :: "'a llist" where "r \<in> finpref A t" and "suff A r \<subseteq> u"
elims: fixes x :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and y :: "int vec" assumes "fi_state x = y" obtains f :: "int vec list \<times> int vec list" and mu :: "int iarray iarray" and d :: "int iarray" where "x = (f, mu, d)" and "y = get_nth_i f"
le_fininf_le_inf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
less_Measurement_System_ext_def: fixes x :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>"   and y :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
intros: fixes W :: "('a \<times> 'b) list"   and q1 :: "'a"   and q2 :: "'b"   and Wtl :: "('a \<times> 'b) list"   and Wn :: "('a \<times> 'b) list"   and \<delta>1 :: "('a, 'c) ta_rule set"   and \<delta>2 :: "('b, 'c) ta_rule set"   and Q :: "('a \<times> 'b) set"   and W' :: "('a \<times> 'b) list"   and Q' :: "('a \<times> 'b) set"   and \<delta>d' :: "('a \<times> 'b, 'c) ta_rule set"   and \<delta>d :: "('a \<times> 'b, 'c) ta_rule set" assumes "W = (q1, q2) # Wtl"   and "distinct Wn"   and "set Wn = f_succ (\<delta>_prod \<delta>1 \<delta>2) `` {(q1, q2)} - Q"   and "W' = Wn @ Wtl"   and "Q' = Q \<union> f_succ (\<delta>_prod \<delta>1 \<delta>2) `` {(q1, q2)}"   and "\<delta>d' = \<delta>d \<union> {r \<in> \<delta>_prod \<delta>1 \<delta>2. lhs r = (q1, q2)}" shows "((Q, W, \<delta>d), Q', W', \<delta>d') \<in> frp_step \<delta>1 \<delta>2"
characteristic_iso_3: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>(f \<star> \<iota>) \<cdot>\<^sub>S \<a>[f, a, a] : R (R f) \<Rightarrow>\<^sub>S R f\<guillemotright>"
expand: fixes stream :: "'b stream"   and stream' :: "'b stream" assumes "shd stream = shd stream' \<and> stl stream = stl stream'" shows "stream = stream'"
can_take_1: fixes a :: "nat"   and g :: "vname gexp list"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "can_take a g i r = (length i = a \<and> apply_guards g (join_ir i r))"
ik\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('a, 'b) stateful_strand_step list"   and B :: "('a, 'b) stateful_strand_step list" shows "ik\<^sub>s\<^sub>s\<^sub>t (A @ B) = ik\<^sub>s\<^sub>s\<^sub>t A \<union> ik\<^sub>s\<^sub>s\<^sub>t B"
adjacent_nodes_def: fixes u :: "nat" shows "adjacent_nodes u \<equiv> E `` {u} \<union> E\<inverse> `` {u}"
properSeparationOfViews_def: fixes ES1 :: "'a ES_rec"   and ES2 :: "'a ES_rec"   and \<V> :: "'a V_rec"   and \<V>1 :: "'a V_rec"   and \<V>2 :: "'a V_rec" shows "properSeparationOfViews ES1 ES2 \<V> \<V>1 \<V>2 \<equiv> V\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES1\<^esub> = V\<^bsub>\<V>1\<^esub> \<and> V\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES2\<^esub> = V\<^bsub>\<V>2\<^esub> \<and> C\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES1\<^esub> \<subseteq> C\<^bsub>\<V>1\<^esub> \<and> C\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES2\<^esub> \<subseteq> C\<^bsub>\<V>2\<^esub> \<and> N\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> = {}"
OtherMessagesOnlyGrowing: fixes cfg1 :: "('p, 'v, 's) configuration"   and m :: "('p, 'v) message"   and cfg2 :: "('p, 'v, 's) configuration"   and m' :: "('p, 'v) message" assumes "cfg1 \<turnstile> m \<mapsto> cfg2"   and "m \<noteq> m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
gv_1: shows "gv \<in> borel_measurable Invariants"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>c n p. (\<And>x. n = x \<Longrightarrow> P p) \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (C v)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
algebraic_int_altdef_ipoly: fixes x :: "'a" shows "algebraic_int x = (\<exists>p. ipoly p x = (0::'a) \<and> monic p)"
Lim_within_id: fixes a :: "'b"   and s :: "'b set" shows "(id \<longlongrightarrow> a) (at a within s)"
r_def: fixes p :: "PrIds" shows "r p = (rst!p, rtrner crCh!p, caller rmCh!p)"
in_hom_char: fixes f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr"   and b :: "('a, 'a set) Maps.arr" shows "Maps.in_hom f a b = (Maps.arr f \<and> Maps.ide a \<and> Maps.ide b \<and> Maps.Dom f = Maps.Dom a \<and> Maps.Cod f = Maps.Dom b)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x \<noteq> 0"
rel_mono: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr"   and "R \<le> S" shows "rel R \<le> rel S"
assms_3: assumes "x \<le> 0"   and "0 \<le> y" shows "P (lcm (- x) y)"
trancl_Image_unfold_left: fixes E :: "('a \<times> 'a) set"   and S :: "'a set" shows "E\<^sup>+ `` S = E\<^sup>* `` E `` S"
simps_2: shows "\<dd> One = (\<lambda>_. Zero)"
intro: fixes adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_ancestors_di :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "adopt_node = l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node get_owner_document get_parent remove_child get_disconnected_nodes set_disconnected_nodes get_ancestors_di"   and "adopt_node_locs = l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs remove_child_locs set_disconnected_nodes_locs" shows "l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_owner_document get_parent remove_child remove_child_locs get_disconnected_nodes set_disconnected_nodes set_disconnected_nodes_locs get_ancestors_di adopt_node adopt_node_locs"
reduction_6: fixes CTa :: "nat \<Rightarrow> classDef option"   and e :: "exp"   and m :: "nat"   and e' :: "exp list"   and e'' :: "exp list"   and eia :: "exp"   and ei' :: "exp"   and el :: "exp list"   and er :: "exp list" assumes "x = CTa"   and "xa = MethodInvk e m e'"   and "xb = MethodInvk e m e''"   and "CTa \<turnstile> eia \<rightarrow> ei'"   and "Execute.append el (eia # er) e'"   and "Execute.append el (ei' # er) e''" shows "thesis"
onode_disconnect1: fixes \<sigma>' :: "nat \<Rightarrow> 'c"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and R :: "nat set"   and i' :: "nat"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, NodeS i s R), disconnect(i, i'), \<sigma>', NodeS i s (R - {i'})) \<in> onode_sos S"
elementLevelLtLevelImpliesMemberPrefixToLevel: fixes e :: "'a"   and M :: "('a \<times> bool) list"   and level :: "nat" assumes "e \<in> set (elements M)"   and "elementLevel e M \<le> level" shows "e \<in> set (elements (prefixToLevel level M))"
lookup_append: fixes l :: "'a list"   and f :: "'a \<Rightarrow> bool"   and r :: "'a"   and l' :: "'a list" assumes "lookup l f = Some r" shows "lookup (l @ l') f = Some r"
analz_insert_freshK1: fixes evs :: "event list"   and K :: "nat"   and SesKey :: "nat" assumes "evs \<in> kerbIV"   and "K \<in> authKeys evs \<union> range shrK"   and "SesKey \<notin> range shrK" shows "(Key K \<in> analz (insert (Key SesKey) (knows Spy evs))) = (K = SesKey \<or> Key K \<in> analz (knows Spy evs))"
nextVertex_in_graph: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "f \<bullet> v \<in> \<V> g"
set_map0_3: fixes f1 :: "'l \<Rightarrow> 'p"   and f2 :: "'m \<Rightarrow> 'q"   and f3 :: "'n \<Rightarrow> 'o" shows "F'set2 \<circ> F'map f2 f3 f1 = (`) f3 \<circ> F'set2"
hyps: shows "A n = \<Union> (mkdisjoint A ` {..n})"
msb_nth: fixes w :: "'b word" shows "msb w = w !! (LENGTH('b) - 1)"
reducedly_statically_complete_reduced_calculus_axioms: shows "reducedly_statically_complete_reduced_calculus Bot Inf (\<Turnstile>) Red_I Red_F"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelR Enc"
CatchSkip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile> (Catch Skip c\<^sub>2, s) \<rightarrow> (Skip, s)"
wfp_on_imp_af: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P A" shows "af A (\<lambda>x y. \<not> P y x)"
stopType_nonempty: fixes a :: "nat \<Rightarrow> stopType list"   and t :: "nat" assumes "msg (Suc 0) a"   and "a t \<noteq> []" shows "a t = [stop_vc]"
"2_3": shows "decr_grading_p d n (p \<odot> q) = punit.decr_grading_p d n p \<odot> decr_grading_p d n q"
pseudo_divmod_def: fixes p :: "'c poly"   and q :: "'c poly" shows "pseudo_divmod p q \<equiv> if q = 0 then (0, p) else pseudo_divmod_main (lead_coeff q) 0 p q (degree p) (1 + length (coeffs p) - length (coeffs q))"
i: shows "\<And>s. PROP P \<Longrightarrow> PROP Q s"
s_cf_deg: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "deg_n R S X p = fst (s_cf R S X p)"
mojmir_accept_iff_rabin_accept: shows "accept\<^sub>M = accept\<^sub>R (\<delta>\<^sub>\<R>, q\<^sub>\<R>, {(fail\<^sub>R \<union> merge\<^sub>R j, succeed\<^sub>R j) |j. j < max_rank}) w"
complement_involution: fixes x :: "i" assumes "CEMC"   and "x \<noteq> u" shows "x = \<midarrow> (\<midarrow> x)"
fresh_symvar_def: fixes sv :: "'c \<times> nat"   and s :: "'c \<Rightarrow> nat" shows "fresh_symvar sv s = (sv \<notin> symvars s)"
applyEdge_def: fixes prog :: "program"   and e :: "edge"   and p :: "pState"   and g :: "gState\<^sub>I" shows "applyEdge prog e p g = (let (g', p') = evalEffect (effect e) prog g p in ASSERT ((g, g') \<in> gState_progress_rel prog) \<bind> (\<lambda>_. ASSERT (pState_inv prog p') \<bind> (\<lambda>_. ASSERT (cl_inv (g', p')) \<bind> (\<lambda>_. let p'' = case target e of Index t \<Rightarrow> if t < IArray.length (states prog !! pState.idx p') then p'\<lparr>pc := t\<rparr> else abort STR ''Edge target out of bounds'' (\<lambda>_. p') | LabelJump literal option \<Rightarrow> abort STR ''Edge target not Index'' (\<lambda>_. p') in ASSERT (pState_inv prog p'') \<bind> (\<lambda>_. let g'' = g'\<lparr>procs := (procs g')[pid p'' - 1 := p'']\<rparr> in ASSERT ((g', g'') \<in> gState_progress_rel prog) \<bind> (\<lambda>_. let g''' = if isAtomic e \<and> handshake g'' = 0 then g''\<lparr>exclusive := pid p''\<rparr> else g'' in ASSERT ((g', g''') \<in> gState_progress_rel prog) \<bind> (\<lambda>_. let g\<^sub>f = if pc p'' = 0 then checkDeadProcs g''' else g''' in ASSERT ((g''', g\<^sub>f) \<in> gState_progress_rel prog) \<bind> (\<lambda>_. RETURN g\<^sub>f))))))))"
priv_is_guar_priv_ignores_old_vars: shows "ignores_old_vars priv_is_guar_priv"
which_node_dyn_standard_control_dependence_source: fixes as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n :: "'node"   and as'' :: "'edge list"   and n' :: "'node"   and a' :: "'edge" assumes "(_Entry_) -as @ a # as'\<rightarrow>* n"   and "n -as''\<rightarrow>* (_Exit_)"   and "sourcenode a = n'"   and "sourcenode a' = n'"   and "n \<notin> set (sourcenodes (a # as'))"   and "valid_edge a'"   and "inner_node n"   and "\<not> n postdominates targetnode a'"   and "\<forall>ax ax'. ax \<in> set as' \<and> sourcenode ax = sourcenode ax' \<and> valid_edge ax' \<longrightarrow> n postdominates targetnode ax'" shows "n' controls\<^sub>s n via a # as'"
lim: shows "(f \<longlongrightarrow> flim) at_top"
heap_def: shows "class.heap TYPE('a) \<equiv> class.countable TYPE('a)"
next_subseqs1_set_def: fixes prevs :: "(nat \<times> 'b) list" shows "next_subseqs1_set prevs \<equiv> \<Union> {next_subseqs2_set v [0..<i] |v i. (i, v) \<in> set prevs}"
strongly_strategyproof_sds_axioms: shows "strongly_strategyproof_sds agents alts sds"
mult_monom: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and y :: "nat \<Rightarrow>\<^sub>0 nat"   and b :: "'b" shows "monom x a * monom y b = monom (x + y) (a * b)"
piecewise_differentiable_on_def: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'c set" shows "f piecewise_differentiable_on i \<equiv> continuous_on i f \<and> (\<exists>S. finite S \<and> (\<forall>x\<in>i - S. f differentiable at x within i))"
fixp_preserves_mono1: fixes F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. lfp.mono_body (\<lambda>f. F f x)"   and "f \<equiv> lfp.fixp_fun F"   and "\<And>f. monotone ord (\<le>) f \<Longrightarrow> monotone ord (\<le>) (F f)" shows "monotone ord (\<le>) f"
infdist_zero: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "infdist a A = 0"
in_spanI: fixes v :: "'b vec"   and n :: "nat"   and a :: "'b vec \<Rightarrow> 'b"   and A :: "'b vec set"   and W :: "'b vec set" assumes "v = module.lincomb (module_vec TYPE('b) n) a A"   and "finite A"   and "A \<subseteq> W" shows "v \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W"
intro: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring R"   and "ring S"   and "ring_hom_ring_axioms R S h" shows "ring_hom_ring R S h"
elims_3: fixes x :: "'b \<times> 'b" assumes "\<not> is_interval x" obtains l :: "'b" and u :: "'b" where "x = (l, u)" and "\<not> l < u"
elims_1: fixes x :: "'b trm"   and y :: "bool" assumes "is_compound x = y" obtains xa :: "'b" where "x = Var xa__" and "\<not> y"   | xa :: "'b" where "x = Const xaa__" and "\<not> y"   | xa :: "'b trm" and ya :: "'b trm" where "x = xa \<cdot> ya" and "y"
relcompE: fixes xz :: "'g \<times> 'h"   and r :: "('g \<times> 'i) set"   and s :: "('i \<times> 'h) set" assumes "xz \<in> r O s" obtains x :: "'g" and y :: "'i" and z :: "'h" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"