hcju/isabelleps huggingface dataset:
rt_fresherI1: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "nsqn rt1 dip < nsqn rt2 dip" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
non_empty_cycle_root_loop: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "r \<le> x\<^sup>T\<^sup>+ ; r"
simps_4: fixes x :: "'a"   and xs :: "[:'a:]"   and y :: "'a"   and ys :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>"   and "y \<noteq> \<bottom>"   and "ys \<noteq> \<bottom>" shows "eq\<cdot>(x :# xs)\<cdot>(y :# ys) = (eq\<cdot>x\<cdot>y andalso eq\<cdot>xs\<cdot>ys)"
ex1E: fixes P :: "'c \<Rightarrow> bool" assumes "\<exists>!x. P x" obtains x :: "'c" where "P x" and "\<forall>y. P y \<longrightarrow> y = x"
subst_cls_list_def: fixes Cs :: "('a, 'b) Term.term literal multiset list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_cls_list Cs \<sigma> = map (\<lambda>A. subst_cls A \<sigma>) Cs"
inorder_del: fixes t :: "'b tree234"   and x :: "'b" assumes "bal t"   and "Sorted_Less.sorted (inorder t)" shows "inorder (tree\<^sub>d (del x t)) = del_list x (inorder t)"
\<mu>LTL_implies_valid_prefix: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<phi> \<in> \<mu>LTL"   and "w \<Turnstile>\<^sub>n \<phi>" shows "\<exists>i. af \<phi> (prefix i w) \<sim>\<^sub>C true\<^sub>n"
simps: fixes a1 :: "binding list"   and a2 :: "(char list \<times> trm) list"   and a3 :: "(char list \<times> POPLmarkRecord.type) list" shows "(a1 \<turnstile> a2 [:] a3) = ((\<exists>\<Gamma>. a1 = \<Gamma> \<and> a2 = [] \<and> a3 = [] \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f) \<or> (\<exists>\<Gamma> t T fs fTs l. a1 = \<Gamma> \<and> a2 = (l, t) \<Colon> fs \<and> a3 = (l, T) \<Colon> fTs \<and> \<Gamma> \<turnstile> t : T \<and> \<Gamma> \<turnstile> fs [:] fTs \<and> fs\<langle>l\<rangle>\<^sub>? = \<bottom>))"
one_go_def: shows "one_go = (\<lambda>B is. mat (dim_row B) (dim_col B) (\<lambda>(i, j). if i \<noteq> I \<and> i \<in> set is then B $$ (i, j) - A $$ (i, J) * B $$ (I, j) else B $$ (i, j)))"
simps_9: fixes v :: "Formula.formula list" shows "remove_neg (formula.Ands v) = formula.Ands v"
isotoneI: fixes L1 :: "('c, 'd) gorder_scheme"   and L2 :: "('e, 'f) gorder_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "weak_partial_order L1"   and "weak_partial_order L2"   and "\<And>x y. \<lbrakk>x \<in> carrier L1; y \<in> carrier L1; x \<sqsubseteq>\<^bsub>L1\<^esub> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq>\<^bsub>L2\<^esub> f y" shows "isotone L1 L2 f"
test_eq1: fixes y :: "'b"   and x :: "'b"   and p :: "'a" shows "(y \<le> x) = (\<iota> p \<cdot> y \<le> x \<and> ! (\<iota> p) \<cdot> y \<le> x)"
DomainE: fixes a :: "'d"   and r :: "('d \<times> 'e) set" assumes "a \<in> Domain r" obtains b :: "'e" where "(a, b) \<in> r"
brw_step_invar: fixes \<delta> :: "('d, 'e) ta_rule set"   and \<Sigma> :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('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)" assumes "finite \<delta>"   and "\<Sigma> \<in> brw_invar_add \<delta>"   and "brw_\<alpha> \<Sigma> \<in> br'_invar \<delta>"   and "(\<Sigma>, \<Sigma>') \<in> brw_step \<delta>" shows "\<Sigma>' \<in> brw_invar_add \<delta>"
image_image_fixed_card_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and k :: "nat" assumes "inj_on f A" shows "(`) f ` {X. X \<subseteq> A \<and> card X = k} = {X. X \<subseteq> f ` A \<and> card X = k}"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_reader return bind ask" shows "monad_reader_axioms bind ask"
node_lift_anycast: fixes T :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "nat"   and I :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and R\<^sub>i :: "nat set" assumes "T \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg I), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). anycast (Q \<sigma> \<sigma>') a)"   and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'" shows "\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (otherwith S {i} (oarrivemsg I), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (Q \<sigma> \<sigma>') a)"
squarefree_imp_rsquarefree: fixes p :: "'a poly" assumes "squarefree p" shows "rsquarefree p"
ipresIGSwapAll_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'd, 'e, 'j) model_scheme" shows "ipresIGSwapAll h hA termMOD MOD = ipresSwapAll h hA MOD"
qSwap_qSwapInp: fixes xs :: "'f"   and x :: "'g"   and y :: "'g"   and delta :: "'h"   and qinp :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and qbinp :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "qOp delta qinp qbinp #[[x \<and> y]]_xs = qOp delta (qSwapInp xs x y qinp) (qSwapBinp xs x y qbinp)"
finfun_apply_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> finfun" obtains x :: "'a \<Rightarrow>f 'b" where "y = ($) x"
length_tm_even: fixes ap :: "abc_inst list" shows "is_even (length (tm_of ap))"
fmlookup_of_list: fixes m :: "('c \<times> 'd) list" shows "fmlookup (fmap_of_list m) = map_of m"
poly_prod_mset_zero_iff: fixes F :: "'a poly multiset"   and x :: "'a" shows "(poly (\<Prod>\<^sub># F) x = (0::'a)) = (\<exists>f\<in>#F. poly f x = (0::'a))"
DomainpE: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a" assumes "Domainp r a" obtains b :: "'b" where "r a b"
adjugate_def: fixes A :: "('b, 'c) sq_matrix" shows "adjugate A = Square_Matrix.transpose (cofactor A)"
Inf_trans_def: fixes S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set" shows "Inf_trans S = (\<lambda>P. Inf_exp {t P |t. t \<in> S})"
arc_to_block_sym: fixes e1 :: "'b"   and u :: "'a"   and v :: "'a"   and e2 :: "'b" assumes "arc_to_ends G e1 = (u, v)"   and "arc_to_ends G e2 = (v, u)" shows "arc_to_block e1 = arc_to_block e2"
sin_zero_pi_iff: fixes x :: "real" assumes "\<bar>x\<bar> < pi" shows "(sin x = 0) = (x = 0)"
error_free_supd: fixes s :: "state"   and f :: "st \<Rightarrow> st" assumes "error_free s" shows "error_free (supd f s)"
wf_dickson_less_v: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "wfP (dickson_less_v d m)"
f: shows "ide f \<and> \<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
normal_restrict_supergroup: fixes S :: "'a set"   and N :: "'a set" assumes "subgroup S (G\<lparr>carrier := S\<rparr>)"   and "N \<lhd> G\<lparr>carrier := S\<rparr>"   and "N \<subseteq> S" shows "N \<lhd> G\<lparr>carrier := S, carrier := S\<rparr>"
update_convs_4: fixes obj' :: "(obj_id_t \<Rightarrow> obj_t) \<Rightarrow> obj_id_t \<Rightarrow> obj_t"   and sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "obj_update obj' \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj' obj, thread = thread, \<dots> = more\<rparr>"
set_insort_key: fixes less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and f :: "'m \<Rightarrow> 'l"   and x :: "'m"   and xs :: "'m list" assumes "class.linorder less_eq less" shows "set (linorder.insort_key less_eq f x xs) = insert x (set xs)"
minus_of_nat: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g1 :: "'b \<Rightarrow> 'b"   and gn1 :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "periodic_fun f g gm g1 gn1" shows "f (gm x (of_nat n)) = f x"
v_space: shows "apply_bcontfun v \<in> space\<^sub>N bcontfun\<^sub>N"
assms: shows "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
is_bisimulation_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "nominal_ts satisfies transition" shows "nominal_ts.is_bisimulation satisfies transition R \<equiv> symp R \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<phi>. satisfies P \<phi> \<longrightarrow> satisfies Q \<phi>)) \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> transition P \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. transition Q \<langle>\<alpha>,Q'\<rangle> \<and> R P' Q')))"
SUP_img_rule_complete: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" shows "(\<forall>x. x \<in> S \<longrightarrow> f x \<le> SPEC \<Phi>) = (Sup (f ` S) \<le> SPEC \<Phi>)"
assms_2: shows "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt p"
wordinterval_intersection_set_eq: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_intersection r1 r2) = wordinterval_to_set r1 \<inter> wordinterval_to_set r2"
trans_strong: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ext_irrefl_trans_strong ext"   and "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "ext gt zs ys"   and "ext gt ys xs" shows "ext gt zs xs"
gfp_pulldown: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "\<And>u P. unitary P \<Longrightarrow> T u P = t P (u P)"   and "\<And>t P. \<lbrakk>\<And>Q. unitary Q \<Longrightarrow> unitary (t Q); unitary P\<rbrakk> \<Longrightarrow> unitary (T t P)"   and "\<And>P Q. \<lbrakk>unitary P; unitary Q\<rbrakk> \<Longrightarrow> unitary (t P Q)"   and "\<And>P Q R. \<lbrakk>unitary P; unitary Q; unitary R; Q \<tturnstile> R\<rbrakk> \<Longrightarrow> t P Q \<tturnstile> t P R"   and "unitary P"   and "\<And>t u. \<lbrakk>le_utrans t u; \<And>P. unitary P \<Longrightarrow> unitary (t P); \<And>P. unitary P \<Longrightarrow> unitary (u P)\<rbrakk> \<Longrightarrow> le_utrans (T t) (T u)" shows "gfp_trans T P = gfp_exp (t P)"
insert_spec_nonex: fixes ref :: "'a"   and xs :: "'a list"   and oid :: "'a" assumes "ref \<notin> set xs" shows "insert_spec xs (oid, Some ref) = xs"
182: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> - - (y \<sqinter> z) \<sqinter> y) \<sqinter> z = (x \<squnion> y) \<sqinter> z"
not_is_TNil_conv: fixes xs :: "('d, 'e) tllist" shows "(\<not> is_TNil xs) = (\<exists>x xs'. xs = TCons x xs')"
r_prefixes_recfn: shows "recfn 2 r_prefixes"
P'_le_post_R: shows "P' \<le>\<^sub>L tensor_P post (proj_k R)"
convex_polychain_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "1 < length (half_segments_of_aform X)" shows "convex_polychain (segments_of_aform X)"
image_comp: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and r :: "'f set" shows "f ` g ` r = (f \<circ> g) ` r"
deg_le_sect_deg_sect_disjoint: fixes d1 :: "nat"   and d2 :: "nat"   and Y :: "'b set"   and X :: "'b set" assumes "d1 < d2" shows "deg_le_sect Y d1 \<inter> deg_sect X d2 = {}"
sum_list_distinct_selsort: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'a list"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.comm_monoid_add plus zero"   and "distinct xs" shows "monoid_add.sum_list plus zero (linorder_list0.selsort le xs) = monoid_add.sum_list plus zero xs"
sum_monoms_P: fixes n :: "nat"   and b :: "nat"   and g :: "nat \<Rightarrow> 'a mod_ring" assumes "monom 1 1 \<in> carrier R"   and "monom 1 1 [^]\<^bsub>R\<^esub> CARD('a) ^ n = monom 1 1"   and "b < degree f" shows "(\<Sum>i\<le>b. monom (g i) i) [^]\<^bsub>R\<^esub> CARD('a) ^ n = (\<Sum>i\<le>b. monom (g i) i)"
A1: shows "n_leaves (bt_map f t1) = n_leaves t1"
t_reflD_2: fixes x :: "'a"   and y :: "'a" assumes "t x y" shows "y \<in> A"
SubstXinE: fixes e :: "('a, 'b) Expression" shows "sub Vx in e = e"
par_subst_def: fixes \<sigma>\<^sub>1 :: "'e \<Rightarrow> 'e"   and A :: "'f \<Longrightarrow> 'e"   and B :: "'g \<Longrightarrow> 'e"   and \<sigma>\<^sub>2 :: "'e \<Rightarrow> 'e" shows "\<sigma>\<^sub>1 [A|B]\<^sub>s \<sigma>\<^sub>2 = (\<lambda>s. s \<triangleleft>\<^bsub>A\<^esub> \<sigma>\<^sub>1 s \<triangleleft>\<^bsub>B\<^esub> \<sigma>\<^sub>2 s)"
Func_is_emp: fixes A :: "'c set"   and B :: "'d set" shows "(Func A B = {}) = (A \<noteq> {} \<and> B = {})"
aux0_6: fixes xs :: "('b, 'c) Combinators list"   and x :: "('b, 'c) Combinators" assumes "singleCombinators xs"   and "\<nexists>a b. x = a \<oplus> b" shows "singleCombinators (x # xs)"
conth: shows "continuous_map (prod_topology (top_of_set {0..1}) X) Y h"
extf_map2: fixes ys :: "('s, 'v) tm list"   and xs :: "('s, 'v) tm list"   and h :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm"   and f :: "'s" assumes "\<forall>y\<in>set ys \<union> set xs. \<forall>x\<in>set ys \<union> set xs. y >\<^sub>t x \<longrightarrow> h y >\<^sub>t h x"   and "extf f (>\<^sub>t) ys xs" shows "extf f (>\<^sub>t) (map h ys) (map h xs)"
bin_nth_Bit0: fixes w :: "num"   and n :: "nat" shows "bit (numeral (num.Bit0 w)) n = (\<exists>m. n = Suc m \<and> bit (numeral w) m)"
corr: fixes L :: "(nat \<times> int \<times> nat) list" assumes "uGraph_from_list_invar L" shows "<emp> kruskal_algo L <\<lambda>r. \<up> (uGraph_from_list_invar r \<and> set r \<subseteq> set L \<and> uGraph.minSpanningForest (uGraph_from_list_\<alpha>_edges L) (uGraph_from_list_\<alpha>_weight L) (uGraph_from_list_\<alpha>_edges r))>\<^sub>t"
eq_mrexp2_imp_eq: fixes r :: "'c mrexp2"   and s :: "'c mrexp2" assumes "eq_mrexp2 r s"   and "ok2 r"   and "ok2 s" shows "r = s"
simps_3: fixes uv :: "real list" shows "eval FalseF uv = False"
comm_monoid_add_ow: shows "class.comm_monoid_add = comm_monoid_add_ow UNIV"
fn_norm_def: fixes f :: "'a \<Rightarrow> real"   and V :: "'a set" shows "\<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>B V f"
wfs_s\<^sub>H\<^sub>0: shows "wfs \<Gamma>\<^sub>0 s\<^sub>H\<^sub>0"
lnorm_nonneg: fixes M :: "'a measure"   and p :: "real"   and f :: "'a \<Rightarrow> real" shows "0 \<le> lnorm M p f"
distinct_1: shows "TernaryTrue \<noteq> TernaryFalse"
x0_hb_bound7: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and x :: "real"   and i :: "nat" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "x\<^sub>1 \<le> x"   and "i < k" shows "x\<^sub>0 < bs ! i * x + (hs ! i) x"
mset_2dist2_cases: fixes a :: "'f"   and b :: "'f"   and A :: "'f multiset"   and B :: "'f multiset" assumes "{#a, b#} \<subseteq># A + B" obtains "{#a, b#} \<subseteq># A"   | "{#a, b#} \<subseteq># B"   | "a \<in># A" and "b \<in># B"   | "a \<in># B" and "b \<in># A"
simps_1: shows "wbalanced \<langle>\<rangle> = True"
assms_5: shows "r \<cdot> l \<le> l \<cdot> r"
fmadd_drop_left_dom: fixes n :: "('c, 'd) fmap"   and m :: "('c, 'd) fmap" shows "fmdrop_fset (fmdom n) m ++\<^sub>f n = m ++\<^sub>f n"
measurable_translation: fixes S :: "'a set"   and a :: "'a" assumes "S \<in> lmeasurable" shows "(+) a ` S \<in> lmeasurable"
simps_2: fixes s :: "'a"   and r :: "'b \<times> 'c"   and rs :: "('b \<times> 'c) list" shows "s \<star> Cons r rs = (s \<star> rs) \<bullet> r"
antisym_conv: fixes x :: "('d, 'e) Term.term"   and y :: "('d, 'e) Term.term" assumes "x \<unrhd> y" shows "y \<unrhd> x = (x = y)"
size_arg: fixes t :: "'a"   and s :: "'a" assumes "t \<in> trees"   and "s \<in> set (args t)" shows "size s < size t"
remove_def: fixes x :: "'l"   and A :: "'l set" shows "Set.remove x A = A - {x}"
assms_2: shows "\<forall>c. c el F' \<longrightarrow> clauseTautology c"
lems_3: fixes t :: "'b tree"   and a :: "'b"   and b :: "'b" assumes "consistent t"   and "a \<in> alphabet t"   and "b \<in> alphabet t"   and "freq t a \<le> freq t b"   and "depth t a \<le> depth t b" shows "cost (swapSyms t a b) \<le> cost t"
wo_rel_def: fixes r :: "('a \<times> 'a) set" shows "wo_rel r \<equiv> Well_order r"
REFL: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "Refl r"
mapset_none: fixes t :: "(int \<times> 'b) Tree"   and i :: "int" assumes "valid_tmap t" shows "(mapOf t i = None) = (\<forall>a. (i, a) \<notin> setOf t)"
assms_3: shows "subtopology (Euclidean_space n) S homeomorphic_space subtopology (Euclidean_space n) T"
S_def: shows "S = {n. [\<one>, \<ominus> x] [^]\<^bsub>poly_ring R\<^esub> n pdivides p}"
sumSym: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<sim>\<^sup>s\<^sub>l Q \<oplus> P"
II: fixes xa :: "'a" shows "- xa = - (- (- xa \<squnion> - (- (- xa))))"
seqE: fixes g :: "'a"   and f :: "'a" assumes "Left.seq g f" obtains "Left.arr f" and "Left.arr g" and "Left.dom g = Left.cod f"
1: assumes "local.iso \<mu>" shows "arr \<mu> \<and> B.iso (Map \<mu>)"
no_VVp_deadends: fixes v :: "'a" assumes "v \<in> winning_region p"   and "v \<in> VV p" shows "\<not> deadend v"
fminus_finsert_absorb: fixes x :: "'c"   and A :: "'c fset" assumes "x |\<notin>| A" shows "finsert x A |-| {|x|} = A"
map_def: fixes f :: "'b" shows "\<mu>\<sigma>.map f = (if \<tau>.B.arr f then D (\<mu> (\<tau>.B.cod f)) (\<sigma> f) else \<sigma>.B.null)"
finite_measure_subalgebra_axioms_def: fixes M :: "'b measure"   and F :: "'b measure" shows "finite_measure_subalgebra_axioms M F \<equiv> subalgebra M F"
compat_f: shows "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt (f y) (f x)"
set_eqI2: fixes A :: "'a set"   and B :: "'a set"   and R :: "('a, 'b) eq_object_scheme" assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a .=\<^bsub>R\<^esub> b"   and "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b .=\<^bsub>R\<^esub> a" shows "A {.=}\<^bsub>R\<^esub> B"
csl1: shows "cs\<^bsup>\<pi>\<^sub>1\<^esup> l = [\<pi>\<^sub>1 l]"
sat_\<epsilon>: fixes \<phi> :: "WS1S_Nameful.ws1s"   and bs1 :: "WS1S_Nameful.fo list"   and bs2 :: "WS1S_Nameful.so list"   and \<AA> :: "interp" assumes "set (fos \<phi>) \<subseteq> set bs1"   and "set (sos \<phi>) \<subseteq> set bs2"   and "ok_I \<AA> bs1" shows "WS1S.sat \<AA> (\<epsilon> bs1 bs2 \<phi>) = satisfies (dec_I1 \<AA> bs1) (dec_I2 \<AA> bs2) \<phi>"
solution_usolves_ode: shows "(solution usolves_ode f from t) (cball t et) (cball (flow t0 x0 t) ex)"
Rep_mini_alg_unique_induct: fixes y :: "mini_alg"   and P :: "mini_alg \<Rightarrow> bool" assumes "y \<in> {r. ma_coeff r = 0 \<and> ma_base r = 0 \<or> ma_coeff r \<noteq> 0 \<and> prime_product (ma_base r)}"   and "\<And>x. P (Rep_mini_alg_unique x)" shows "P y"
assms_2: assumes "(0::'a) < \<bar>a\<bar>" shows "thesis"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c option"   and b :: "'d option" assumes "rel_option R a b" obtains "a = None" and "b = None"   | x :: "'c" and y :: "'d" where "a = Some x" and "b = Some y" and "R x y"
that_6: fixes x :: "'a" assumes "x \<in> W" shows "f x = 0"
fresher: fixes ip :: "nat"   and ip' :: "nat" assumes "(ip, ip') \<in> (rt_graph \<sigma> dip)\<^sup>+"   and "dip \<in> vD (rt (\<sigma> ip'))"   and "ip' \<noteq> dip" shows "rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> ip')"
indep_in_supI: fixes X :: "'a set"   and \<E>' :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>'"   and "\<E>' \<subseteq> \<E>"   and "indep_in \<E> X" shows "indep_in \<E>' X"
not_in_set_insert: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "List.insert x xs = x # xs"
R26_a: shows "pmf (sds R26) a = 1 - pmf (sds R8) a"
tr'_par_sem_equiv: fixes D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and M :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>(l, t, s)\<in>set D. (fv t \<union> fv s) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "ground M"   and "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> i = j"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<lbrakk>M; set (unlabel D) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>; unlabel A\<rbrakk>\<^sub>s \<I> = (\<exists>B\<in>set (tr'\<^sub>p\<^sub>c A D). \<lbrakk>M; unlabel B\<rbrakk>\<^sub>d \<I>)"
known_ptrs_known_ptr: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_known_ptrs known_ptr known_ptrs"   and "known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "known_ptr ptr"
rel_funD2: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'c" assumes "(A ===> B) f g"   and "A x x" shows "B (f x) (g x)"
f1: shows "Deliver (mo, Append mo e) \<in> set pre \<or> (\<exists>mo2. Deliver (mo, Store e mo2 mo) \<in> set pre)"
less_add_same_cancel2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subset># b + a) = ({#} \<subset># b)"
n_value_x_1_nat: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> vp K v \<^bsup>\<diamondsuit>Vr K v n\<^esup>" shows "an n \<le> n_val K v x"
sorted_upd_list: fixes ps :: "('c \<times> 'd) list"   and x :: "'c"   and y :: "'d" assumes "sorted1 ps" shows "sorted1 (upd_list x y ps)"
iso_map\<^sub>1_ide: fixes f :: "'d" assumes "D.ide f" shows "D.iso (\<eta>\<^sub>1 f)"
inject_2: fixes x2 :: "('a, 'b, 'c, 'd) Tree"   and y2 :: "('a, 'b, 'c, 'd) Tree" shows "(tNot x2 = tNot y2) = (x2 = y2)"
renaming_distr_subst: fixes \<alpha> :: "'r \<Rightarrow> 'r"   and \<beta> :: "'l \<Rightarrow> 'l"   and e :: "('r, 'l, 'v) expr"   and x :: "'v"   and e' :: "('r, 'l, 'v) expr" shows "\<R>\<^sub>E \<alpha> \<beta> (subst e x e') = subst (\<R>\<^sub>E \<alpha> \<beta> e) x (\<R>\<^sub>E \<alpha> \<beta> e')"
rel_sel: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "('d, 'f, 'h) writerT"   and b :: "('e, 'g, 'i) writerT" shows "rel_writerT R1 R2 R3 a b = R3 (run_writer a) (run_writer b)"
\<ll>'_ide_simp: fixes a :: "'d" assumes "ide a" shows "\<ll>' a = \<l>\<^sup>-\<^sup>1[a]"