hcju/isabelleps huggingface dataset:
CondReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>if (e') e\<^sub>1 else e\<^sub>2,s',b'\<rangle>"
cases: fixes x :: "'b acom" obtains a :: "'b" where "x = SKIP {a__}"   | xa :: "char list" and e :: "aexp" and a :: "'b" where "x = xa ::= e {aa__}"   | C1 :: "'b acom" and C2 :: "'b acom" where "x = C1__;; C2__"   | b :: "bexp" and C1 :: "'b acom" and C2 :: "'b acom" and a :: "'b" where "x = IF b__ THEN C1 ELSE C2 {ab__}"   | i :: "'b" and b :: "bexp" and C :: "'b acom" and a :: "'b" where "x = {i} WHILE b DO C {a}"
eoutM_def: fixes sP :: "specID"   and M :: "chanID set"   and E :: "Expression" shows "eoutM sP M E \<equiv> \<exists>ch. ch \<in> out sP \<and> ch \<in> M \<and> exprChannel ch E"
sup_inf_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y \<cdot> z = (x + y) \<cdot> (x + z)"
sublist_antisym: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "subseq l1 l2"   and "subseq l2 l1" shows "l1 = l2"
powr_zero_eq_one: fixes x :: "'a" shows "x powr (0::'a) = (if x = (0::'a) then 0::'a else (1::'a))"
conv_callee_parallel: fixes callee1 :: "'i \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'i, 'e, 'g) gpv"   and callee2 :: "'j \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'j, 'f, 'h) gpv"   and s :: "'i"   and s' :: "'j" shows "CNV (callee1 \<ddagger>\<^sub>I callee2) (s, s') = CNV callee1 s |\<^sub>= CNV callee2 s'"
rel_OO: shows "rel_T L1 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y"
Ord_induct3: fixes \<alpha> :: "V"   and P :: "V \<Rightarrow> bool" assumes "Ord \<alpha>"   and "P 0"   and "\<And>\<alpha>. \<lbrakk>Ord \<alpha>; P \<alpha>\<rbrakk> \<Longrightarrow> P (ZFC_in_HOL.succ \<alpha>)"   and "\<And>\<alpha>. \<lbrakk>Limit \<alpha>; \<And>\<xi>. \<xi> \<in> elts \<alpha> \<Longrightarrow> P \<xi>\<rbrakk> \<Longrightarrow> P (SUP \<xi>\<in>elts \<alpha>. \<xi>)" shows "P \<alpha>"
l_assigned_nodes_def: fixes assigned_nodes :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_assigned_nodes assigned_nodes \<equiv> ((\<forall>slot h. pure (assigned_nodes slot) h) \<and> (\<forall>h slot. h \<turnstile> ok assigned_nodes slot \<longrightarrow> slot |\<in>| element_ptr_kinds h)) \<and> (\<forall>h slot. h \<turnstile> ok assigned_nodes slot \<longrightarrow> h \<turnstile> get_tag_name slot \<rightarrow>\<^sub>r ''slot'') \<and> (\<forall>h slot nodes slot' nodes'. h \<turnstile> assigned_nodes slot \<rightarrow>\<^sub>r nodes \<longrightarrow> h \<turnstile> assigned_nodes slot' \<rightarrow>\<^sub>r nodes' \<longrightarrow> slot \<noteq> slot' \<longrightarrow> set nodes \<inter> set nodes' = {})"
circ_simulate_left_plus: fixes x :: "'a"   and z :: "'a"   and y :: "'a"   and w :: "'a" shows "x * z \<le> z * y\<^sup>\<circ> \<squnion> w \<longrightarrow> x\<^sup>\<circ> * z \<le> (z \<squnion> x\<^sup>\<circ> * w) * y\<^sup>\<circ>"
of_fun_def: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "IArray.of_fun f n = IArray (map f [0..<n])"
1: shows "1 + x + y \<le> (x + 1) * (y + 1)"
str_del_block_def: fixes b :: "'a set" shows "str_del_block b \<equiv> {#bl - b. bl \<in># {#bl \<in># \<B>. bl \<noteq> b#}#}"
degreeGood: shows "\<forall>p\<in>set A. MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "lam"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> a \<tau> y. \<lbrakk>valid \<Gamma>; (a, \<tau>) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var a) \<tau>"   and "\<And>\<Gamma> t1 \<tau> \<sigma> t2 y. \<lbrakk>\<Gamma> \<turnstile> t1 : \<tau> \<rightarrow> \<sigma>; \<And>z. P z \<Gamma> t1 (\<tau> \<rightarrow> \<sigma>); \<Gamma> \<turnstile> t2 : \<tau>; \<And>z. P z \<Gamma> t2 \<tau>\<rbrakk> \<Longrightarrow> P y \<Gamma> (App t1 t2) \<sigma>"   and "\<And>a \<Gamma> \<tau> t \<sigma> y. \<lbrakk>a \<sharp> y; a \<sharp> \<Gamma>; (a, \<tau>) # \<Gamma> \<turnstile> t : \<sigma>; \<And>z. P z ((a, \<tau>) # \<Gamma>) t \<sigma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Lam [a].t) (\<tau> \<rightarrow> \<sigma>)" shows "P z x1 x2 x3"
assms_1: shows "(succl, succ) \<in> \<langle>Id\<rangle>list_set_rel"
Quotient3_complex_homo: shows "Quotient3 (\<approx>) abs_complex_homo rep_complex_homo"
rsemimodule1: fixes next_behaviour :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and next_stimulus :: "'c \<times> 'd \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c"   and a :: "'d" assumes "c2ka next_behaviour next_stimulus" shows "next_stimulus (s \<oplus> t, a) = next_stimulus (s, a) \<oplus> next_stimulus (t, a)"
abs_sgn_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (sgn a) = (if a = zero then zero else one)"
inverse_map_identity: fixes S :: "'c set" shows "inverse_map (identity S) S S = identity S"
set_choose_def: fixes s :: "'a set" shows "set_choose s \<equiv> SOME x. x \<in> s"
ev\<^sub>A_eq': fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and P :: "'BExp set"   and e :: "'AExp"   and t :: "'BExp set" assumes "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "pred P mem\<^sub>1"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t"   and "P \<turnstile> t" shows "ev\<^sub>A mem\<^sub>1 e = ev\<^sub>A mem\<^sub>2 e"
neq_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "(x \<noteq> y) = (less x y \<or> less y x)"
find_first_le: fixes a :: "'a"   and xs :: "'a list" assumes "a \<in> set xs" shows "find_first a xs < length xs"
rep_eq: fixes x :: "interface" shows "asn x = conc_asn (rep_interface x)"
invariance_of_domain_ball: fixes a :: "'a"   and r :: "real"   and f :: "'a \<Rightarrow> 'a" assumes "continuous_on (cball a r) f"   and "0 < r"   and "inj_on f (cball a r)" shows "open (f ` ball a r)"
max2: fixes b :: "nat"   and y :: "nat"   and x :: "nat" assumes "b \<le> y" shows "b \<le> max x y"
step_has_target: fixes x :: "'a"   and r :: "'a" assumes "x ; r \<noteq> 0" shows "x\<^sup>T ; 1 \<noteq> 0"
exhaust: fixes y :: "'b memref" obtains x11 :: "bool" and x12 :: "nat" and x13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" and x14 :: "nat" and x15 :: "nat set" and x16 :: "nat set" and x17 :: "nat set" and x18 :: "nat set" where "y = Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18"   | x21 :: "bool" and x22 :: "nat" and x23 :: "nat" and x24 :: "nat" where "y = Read\<^sub>s\<^sub>b x21 x22 x23 x24"   | x31 :: "'b" and x32 :: "'b" and x33 :: "instr list" where "y = Prog\<^sub>s\<^sub>b x31 x32 x33"   | x41 :: "nat set" and x42 :: "nat set" and x43 :: "nat set" and x44 :: "nat set" where "y = Ghost\<^sub>s\<^sub>b x41 x42 x43 x44"
ring_hom: shows "ring_hom_ring (R\<lparr>carrier := K\<rparr>) (S\<lparr>carrier := h ` K\<rparr>) h"
cases: fixes a1 :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and a2 :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and r :: "('d, 'e) ta_rule" assumes "(a1, a2) \<in> brw_inner_step r" obtains c :: "nat" and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" and \<Sigma> :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" and Q :: "'d \<Rightarrow> 'e tree option" and W :: "'d set" and \<Sigma>' :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" and Q' :: "'d \<Rightarrow> 'e tree option" and W' :: "'d set" and rcm' :: "('d, 'e) ta_rule \<Rightarrow> nat option" where "a1 = \<Sigma>" and "a2 = \<Sigma>'" and "c = the (rcm r)" and "\<Sigma> = (Q, W, rcm)" and "\<Sigma>' = (Q', W', rcm')" and "if c \<le> 1 \<and> lhs r \<notin> dom Q then Q' = Q(lhs r \<mapsto> construct_witness Q r) else Q' = Q" and "if c \<le> 1 \<and> lhs r \<notin> dom Q then W' = insert (lhs r) W else W' = W" and "rcm' = rcm(r \<mapsto> c - 1)"
matricize_0: fixes I :: "nat set"   and ds :: "nat list" shows "matricize I (tensor0 ds) = 0\<^sub>m (dim_row (matricize I (tensor0 ds))) (dim_col (matricize I (tensor0 ds)))"
bin_rest_AND: fixes x :: "int"   and y :: "int" shows "(x AND y) div 2 = x div 2 AND y div 2"
prems_2: fixes x :: "'a node \<times> 'a node set" assumes "True" shows "f x \<le> SPEC (expand_rslt_exist_eq x)"
RRR_closure: shows "closure RRR = {0..2} \<union> {3} \<union> {5..7}"
prems: shows "\<alpha> \<in># r|\<beta> # as|"
K: shows "(\<lambda>n. norm (X n)) \<longlonglongrightarrow> K"
axioms_1: fixes M :: "'c measure" assumes "finite_measure M" shows "sigma_finite_measure M"
assms_2: shows "has_greatest_fixpoint (f \<circ> g)"
remove_child_children_subset: fixes h :: "(_) heap"   and parent :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and children' :: "(_) node_ptr list" assumes "h \<turnstile> remove_child parent child \<rightarrow>\<^sub>h h'"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'"   and "known_ptrs h"   and "type_wf h" shows "set children' \<subseteq> set children"
infinite_computation_extract_head_Catch: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "nat \<Rightarrow> ('b, 'a, 'c) com \<times> ('b, 'c) xstate"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and k :: "nat" assumes "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (i + 1)"   and "f 0 = (Catch c\<^sub>1 c\<^sub>2, s)"   and "\<forall>i<k. \<not> final (head (f i))" shows "\<forall>i<k. (\<exists>c' s'. f (i + 1) = (Catch c' c\<^sub>2, s')) \<and> \<Gamma>\<turnstile> head (f i) \<rightarrow> head (f (i + 1))"
cutY: shows "S \<inter> (X \<union> insert e Y) = {}"
shrK_neq_pubK: fixes b :: "keymode"   and C :: "agent"   and A :: "agent" shows "publicKey b C \<noteq> shrK A"
not_free_impl_id: fixes x :: "nat"   and S :: "nat \<Rightarrow> typ" assumes "x \<notin> free_tv S" shows "S x = TVar x"
uivar_convr: fixes x :: "'b \<Longrightarrow> 'd" shows "$x\<^sup>- = $x\<acute>"
approx_trans: fixes x :: "'b star"   and y :: "'b star"   and z :: "'b star" assumes "x \<approx> y"   and "y \<approx> z" shows "x \<approx> z"
array_length_array_map: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'b array" shows "array_length (array_map f a) = array_length a"
getFrN_distinct: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and n :: "nat" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla" shows "distinct (getFrN xs ts \<phi>s n)"
abs_perm: fixes pi :: "(name \<times> name) list"   and a :: "name"   and x :: "'a" shows "pi \<bullet> [a].x = [(pi \<bullet> a)].(pi \<bullet> x)"
fun_upd_idem: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and y :: "'d" assumes "f x = y" shows "f(x := y) = f"
cfidi: shows "f\<inverse> O d\<inverse> \<subseteq> d\<inverse>"
a: shows "find (\<lambda>n. vec_to_iarray A !! n \<noteq> (0::'b)) (rev [0..<IArray.length (vec_to_iarray A)]) = Some a"
object_more_extend_id: fixes x :: "(_) RObject_ext"   and y :: "'m" shows "RObject.more (RObject.extend x y) = y"
finite_subset_image: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "finite B"   and "B \<subseteq> f ` A" shows "\<exists>C\<subseteq>A. finite C \<and> B = f ` C"
P_1_def: fixes res :: "int \<times> int"   and x :: "int" shows "P_1 res x = (x mod int p = fst res \<and> x mod int q = snd res \<and> x \<in> {0..int p * int q - 1})"
case_5: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "('c, 'd) fm \<Rightarrow> 'b"   and f4 :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and f5 :: "('c, 'd) fm \<Rightarrow> 'b"   and f6 :: "'d \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and x5 :: "('c, 'd) fm" shows "(case \<^bold>\<diamond> x5 of Pro x \<Rightarrow> f1 x | Nom x \<Rightarrow> f2 x | \<^bold>\<not> x \<Rightarrow> f3 x | x \<^bold>\<or> xa \<Rightarrow> f4 x xa | \<^bold>\<diamond> x \<Rightarrow> f5 x | \<^bold>@ x xa \<Rightarrow> f6 x xa) = f5 x5"
l_set_val_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_shadow_root type_wf set_val set_val_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_val type_wf set_val set_val_locs \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_set_val_get_shadow_root_axioms set_val_locs get_shadow_root_locs"
base: fixes C :: "('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "([], C) \<in> R" shows "(C, 0) \<in> derivable R"
empty_iff: fixes c :: "'e" shows "(c \<in> {}) = False"
simps: fixes r :: "('g, 'e, 'f) gpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "the_gpv (r \<bind> f) = map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. r \<bind> f)))) (the_gpv r \<bind> case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x))) (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))"
ok_imp_VarB_closed_in: fixes \<Gamma> :: "binding list"   and x :: "vrs"   and T :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "VarB x T \<in> set \<Gamma>" shows "T closed_in \<Gamma>"
closure_equiv_eq: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(\<forall>A. (P \<longleftrightarrow> Q) A) = (P = Q)"
eq: shows "take (\<^bold>|w\<^bold>| - n) w = drop n w"
group_axioms: shows "Group.group \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
unch_eq1: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'd" shows "Unchanged (x, y) = (Unchanged x \<and> Unchanged y)"
preserves_id_def: fixes G :: "('a, 'b, 'a, 'b) functor" shows "preserves_id G \<equiv> \<forall>A\<in>Ob. G \<^bsub>\<a>\<^esub> (Id A) = Id\<^bsub>BB\<^esub> G \<^bsub>\<o>\<^esub> A"
admissibleI: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and lub :: "'a set \<Rightarrow> 'a" assumes "\<And>A. \<lbrakk>Complete_Partial_Order.chain ord A; A \<noteq> {}; \<forall>x\<in>A. P x\<rbrakk> \<Longrightarrow> P (lub A)" shows "ccpo.admissible lub ord P"
assms_3: shows "\<psi> \<in> old (\<sigma> 0)"
replay_code: fixes thy :: "exetheory"   and P :: "proofterm" assumes "exe_wf_theory' thy" shows "replay (translate_theory thy) P = exereplay thy P"
cod_simp: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.cod f = B.VxV.cod f"
fps_conv_radius_shift: fixes n :: "nat"   and f :: "'a fps" shows "fps_conv_radius (fps_shift n f) = fps_conv_radius f"
external_to_internal_invariant: fixes S :: "('f, 'e) spec"   and Q :: "'e set"   and P :: "'f set" assumes "oreach S \<subseteq> Q"   and "Q \<subseteq> obs S ` P"   and "observable (obs S) P" shows "reach S \<subseteq> P"
rep_eq: fixes x :: "perm"   and xa :: "'a fs_set" shows "Rep_fs_set (x \<bullet> xa) = x \<bullet> Rep_fs_set xa"
pivot_bound: fixes A :: "'c mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f n"   and "i + j < nr" shows "f (i + j) = n \<or> j + f i \<le> f (i + j)"
Derives1_terminals_stay: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list"   and t :: "symbol" assumes "Derives1 u i r v"   and "t \<in> set u"   and "is_terminal t" shows "t \<in> set v"
wf_mbuf2_add: fixes i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> 'a option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'a option list set \<Rightarrow> bool"   and buf :: "'a option list set list \<times> 'a option list set list"   and ja' :: "nat"   and xs :: "'a option list set list"   and jb' :: "nat"   and ys :: "'a option list set list" assumes "wf_mbuf2 i ja jb P Q buf"   and "list_all2 P [ja..<ja'] xs"   and "list_all2 Q [jb..<jb'] ys"   and "ja \<le> ja'"   and "jb \<le> jb'" shows "wf_mbuf2 i ja' jb' P Q (mbuf2_add xs ys buf)"
face_cycle_succ_closed: fixes G :: "('e, 'f) pre_digraph"   and M :: "'f pre_map"   and a :: "'f" assumes "digraph_map G M" shows "(pre_digraph_map.face_cycle_succ M a \<in> arcs G) = (a \<in> arcs G)"
sint_0: fixes x :: "'b word" shows "(sint x = 0) = (x = 0)"
is_singleton_the_elem: fixes A :: "'f set" shows "is_singleton A = (A = {the_elem A})"
cfun_cong: fixes f :: "'c \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "f = g"   and "x = y" shows "f\<cdot>x = g\<cdot>y"
simps_6: fixes s :: "'a rexp" shows "rexp_ewp (s\<^sup>\<star>\<^sub>r) = True"
ssubst_None: fixes z :: "char list"   and u :: "sterm" shows "ssubst_option z u None = None"
label_5: fixes i :: "'a"   and x :: "'a" assumes "i \<in> Basis"   and "x \<in> cbox (0::'a) One \<and> label x i = 1" shows "f x \<bullet> i \<le> x \<bullet> i"
rsp: shows "eq_onp (\<lambda>x. x \<in> {0..<int CARD('b)}) 0 0"
IH1: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "s \<in> reachable (pnet np p1) TT"   and "(s, i:deliver(d), s') \<in> automaton.trans (pnet np p1)" shows "i \<in> net_ips s"
part1: shows "preprocess_part_1 (map (Pair ()) nsl) = (t, ias, I)"
fterm_to_iterm_subst: fixes t1 :: "fterm"   and \<sigma> :: "char list \<Rightarrow> (char list, char list) Term.term" shows "fterm_to_iterm t1 \<cdot> \<sigma> = fterm_to_iterm (t1 \<cdot>\<^sub>t (\<lambda>x. iterm_to_fterm (\<sigma> x)))"
expectation_gpv_stop: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and fail :: "ennreal"   and c :: "ennreal" assumes "\<I> \<turnstile>g gpv \<surd>"   and "fail \<le> c" shows "expectation_gpv fail (stop_\<I> \<I>) (\<lambda>_. c) (gpv_stop gpv) = expectation_gpv fail \<I> (\<lambda>_. c) gpv"
le_2: shows "A *\<^sub>v y \<le> b"
assms_3: fixes x :: "nat" assumes "x \<in> A" shows "prime x"
WhileWpisPre: fixes c :: "com"   and b :: "bexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<And>Q. \<turnstile>\<^sub>3\<^sub>a {wp\<^sub>3\<^sub>' c Q} c { Q}" shows "\<turnstile>\<^sub>3\<^sub>a {wp\<^sub>3\<^sub>' (WHILE b DO c) Q} WHILE b DO c { Q}"
select_min_weight_clause_in: fixes P0 :: "'a literal list \<times> nat"   and P :: "('a literal list \<times> nat) list" shows "select_min_weight_clause P0 P \<in> set (P0 # P)"
f: shows "f = (\<lambda>stp. (steps0 (Suc 0, l, r) (mopup_a n @ shift mopup_b (2 * n)) stp, n))"
inj_to_bl: fixes cbl :: "(('b \<Rightarrow> bool list) \<times> (bool list \<Rightarrow> 'b)) option"   and to_bl :: "'b \<Rightarrow> bool list"   and from_bl :: "bool list \<Rightarrow> 'b" assumes "class.cbl cbl"   and "ID cbl = Some (to_bl, from_bl)" shows "inj to_bl"
T0: fixes x :: "'a" assumes "x \<in> U" shows "(setdist {x} T = 0) = (x \<in> T)"
pull_posneg_def: fixes x :: "wall"   and y :: "wall" shows "pull_posneg x y \<equiv> x = basic [over] \<circ> basic [brick.under] \<and> y = basic [vert, vert] \<circ> basic [vert, vert]"
conv_radius_altdef: fixes f :: "nat \<Rightarrow> 'a" shows "conv_radius f = liminf (\<lambda>n. inverse (ereal (root n (norm (f n)))))"
disc_13: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "isRecv (Recv x31 x32 x33 x34 x35 x36)"
is_Proj_algebraic: fixes P :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "is_Proj P = (P o\<^sub>C\<^sub>L P = P \<and> P = P*)"
a_heap_is_wellformed_def: fixes h :: "(_) heap" shows "Shadow_DOM.heap_is_wellformed h = (heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h \<and> acyclic (Shadow_DOM.parent_child_rel h \<union> host_shadow_root_rel h \<union> ptr_disconnected_node_rel h) \<and> Shadow_DOM.all_ptrs_in_heap h \<and> Shadow_DOM.distinct_lists h \<and> shadow_root_valid h)"
base: fixes v :: "'a" assumes "Q (LCons v LNil)" shows "v \<in> V"
S5RPC_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> ($S5 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> <RPCNext crCh rmCh rst p>_(r p) \<longrightarrow> S6 rmhist p$"
huntington1: fixes x :: "'a"   and y :: "'a" shows "!(!(t x \<cdot> !y) \<cdot> !(t x \<cdot> t y)) = t x"
float_zero2: fixes x :: "('e, 'f) IEEE.float" shows "is_zero (- x) = is_zero x"
BOOL_E: fixes v :: "val" assumes "val_type v = BOOL" obtains b :: "bool" where "v = BoolVal b"
seq_elim: fixes c\<^sub>1 :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>2 :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C"   and "c\<^sub>1 \<noteq> Stop" shows "\<exists>c\<^sub>1'. eval_abv\<^sub>C \<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>C \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>C \<and> c' = c\<^sub>1' ;; c\<^sub>2"
sets_point_measure_count_space: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal" shows "sets (point_measure A f) = sets (count_space A)"
CBasis_le_norm: fixes b :: "'a"   and x :: "'a" assumes "b \<in> CBasis" shows "cmod (x \<bullet>\<^sub>C b) \<le> norm x"
hyps: fixes \<sigma>' :: "'state"   and coll' :: "'coll"   and \<sigma> :: "'state"   and coll'' :: "'coll"   and n' :: "nat" assumes "(\<sigma>', coll') \<in> cbig P \<sigma>"   and "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> n1"   and "(\<sigma>', coll'') \<in> csmall_nstep P \<sigma> n'" shows "n1 = n'"
Smith_mxn_pinduct_carrier_ge_2: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "Smith_mxn_dom A"   and "A \<in> carrier_mat m n"   and "2 < m"   and "2 < n" shows "fst (Smith_mxn A) \<in> carrier_mat m m \<and> fst (snd (Smith_mxn A)) \<in> carrier_mat m n \<and> snd (snd (Smith_mxn A)) \<in> carrier_mat n n"
orthogonal_matrix_orthonormal_columns: fixes A :: "((real, 'b) vec, 'b) vec" shows "orthogonal_matrix A = ((\<forall>i. norm (column i A) = 1) \<and> (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (column i A) (column j A)))"
t: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> U" shows "t (f \<triangleright> n) \<down>= the (c (the (s (f \<triangleright> n))))"
idem_normalize: fixes a :: "'a" shows "a \<^bold>* a = normalize a"