hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'c \<Rightarrow> 'd"   and a2 :: "'c option"   and a3 :: "'c option"   and a4 :: "'d DBMEntry" assumes "dbm_entry_val a1 a2 a3 a4" obtains u :: "'c \<Rightarrow> 'd" and r :: "'c" and d :: "'d" where "a1 = u__" and "a2 = Some r__" and "a3 = None" and "a4 = Le da__" and "u__ r__ \<le> da__"   | u :: "'c \<Rightarrow> 'd" and c :: "'c" and d :: "'d" where "a1 = ua__" and "a2 = None" and "a3 = Some ca__" and "a4 = Le db__" and "- ua__ ca__ \<le> db__"   | u :: "'c \<Rightarrow> 'd" and r :: "'c" and d :: "'d" where "a1 = ub__" and "a2 = Some ra__" and "a3 = None" and "a4 = Lt dc__" and "ub__ ra__ < dc__"   | u :: "'c \<Rightarrow> 'd" and c :: "'c" and d :: "'d" where "a1 = uc__" and "a2 = None" and "a3 = Some cb__" and "a4 = Lt dd__" and "- uc__ cb__ < dd__"   | u :: "'c \<Rightarrow> 'd" and r :: "'c" and c :: "'c" and d :: "'d" where "a1 = ud__" and "a2 = Some rb__" and "a3 = Some cc__" and "a4 = Le de__" and "ud__ rb__ - ud__ cc__ \<le> de__"   | u :: "'c \<Rightarrow> 'd" and r :: "'c" and c :: "'c" and d :: "'d" where "a1 = u" and "a2 = Some r" and "a3 = Some c" and "a4 = Lt d" and "u r - u c < d"   | uu_ :: "'c \<Rightarrow> 'd" and uv_ :: "'c option" and uw_ :: "'c option" where "a1 = uu_" and "a2 = uv_" and "a3 = uw_" and "a4 = \<infinity>"
generated_topology_subset: fixes S :: "'a set"   and B :: "'a set set"   and U :: "'a set" assumes "generated_topology S B U" shows "U \<subseteq> S"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog" assumes "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs"   and "l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs"   and "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"   and "l_set_disconnected_nodes_get_host set_disconnected_nodes_locs get_host_locs" shows "l_set_disconnected_nodes_get_ancestors_si\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_parent get_parent_locs type_wf known_ptr known_ptrs get_child_nodes get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
closure_of_subtopology_subset: fixes X :: "'b topology"   and U :: "'b set"   and S :: "'b set" shows "subtopology X U closure_of S \<subseteq> X closure_of S"
degree_pCons_0: fixes a :: "'b" shows "degree [:a:] = 0"
es_def_4: shows "\<S>\<bullet>\<C>' \<turnstile> es : [] _> tls"
neg_pos_kauff_mat: shows "kauff_mat (basic [brick.under] \<circ> basic [over]) = kauff_mat (basic [vert, vert] \<circ> basic [vert, vert])"
preallocated_dom: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and C :: "char list" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "addr_of_sys_xcpt C \<in> dom h"
map_sel_3: fixes a :: "'b tree"   and f :: "'b \<Rightarrow> 'c" assumes "a \<noteq> Leaf" shows "val (map_tree f a) = map_option f (val a)"
t_loop_1: shows "t1s _> t2s = tfn _> tfm"
uhnf_def: fixes P :: "pi" shows "uhnf P \<equiv> hnf P \<and> (\<forall>R\<in>summands P. \<forall>R'\<in>summands P. R \<noteq> R' \<longrightarrow> \<not> R \<equiv>\<^sub>e R')"
emeasure_buffon_set_short: shows "emeasure lborel (buffon_set l d) = 4 * ennreal l"
IH: fixes A :: "'a set" assumes "finite A"   and "mset_set A = mset xs" shows "distinct xs"
axioms_2: fixes V\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>C :: "'b \<Rightarrow> 'b"   and src\<^sub>C :: "'b \<Rightarrow> 'b"   and trg\<^sub>C :: "'b \<Rightarrow> 'b"   and V\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>D :: "'c \<Rightarrow> 'c"   and src\<^sub>D :: "'c \<Rightarrow> 'c"   and trg\<^sub>D :: "'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and \<Phi> :: "'b \<times> 'b \<Rightarrow> 'c" assumes "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
assms_3: shows "\<forall>\<^sub>F n in F. u n \<in> S"
r_step_def: shows "r_step \<equiv> Cn 1 r_ifz [r_e2stack, Cn 1 r_prod_encode [Z, r_e2rv], Cn 1 r_ifz [r_e2i, Cn 1 r_prod_encode [r_e2tail, r_const 1], Cn 1 r_ifeq [r_e2i, r_const 1, Cn 1 r_prod_encode [r_e2tail, Cn 1 S [Cn 1 S [Cn 1 r_hd [r_e2xs]]]], Cn 1 r_ifeq [Cn 1 r_kind [r_e2i], r_const 2, Cn 1 r_prod_encode [r_e2tail, Cn 1 S [Cn 1 r_nth [r_e2xs, Cn 1 r_pdec22 [r_e2i]]]], Cn 1 r_ifeq [Cn 1 r_kind [r_e2i], r_const 3, r_step_Cn, Cn 1 r_ifeq [Cn 1 r_kind [r_e2i], r_const 4, r_step_Pr, Cn 1 r_ifeq [Cn 1 r_kind [r_e2i], r_const 5, r_step_Mn, Z]]]]]]]"
preserves_reflects_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "arr (\<alpha>.map f) = VVV.arr f"
fmdom_of_list: fixes m :: "('a \<times> 'b) list" shows "fmdom (fmap_of_list m) = fst |`| fset_of_list m"
r: shows "rec' \<in> KBPAlg.reachable (set (k_frontier a))"
nqfree_nnf: fixes \<phi> :: "atom fm" assumes "qfree \<phi>" shows "nqfree (R.nnf \<phi>)"
Liminf_min_eq_min_Liminf: fixes F :: "nat filter"   and u :: "nat \<Rightarrow> 'a"   and v :: "nat \<Rightarrow> 'a" shows "Liminf F (\<lambda>n. min (u n) (v n)) = min (Liminf F u) (Liminf F v)"
absorb_iff2: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a + b = b)"
c5: assumes "ip \<in> kD rt"   and "\<pi>\<^sub>3 r = unk" shows "P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r, \<pi>\<^sub>7 (addpre r (\<pi>\<^sub>7 (the (rt ip)))))))"
elims: fixes x :: "('b, 'c) result"   and xa :: "('b, 'c) result"   and y :: "('b, 'c) result" assumes "x <|> xa = y" obtains x :: "'c" and uu_ :: "('b, 'c) result" where "xa__ = Ok xb__" and "xa = uu_" and "y = Ok xb__"   | v :: "'b" and x :: "'c" where "xa__ = Error v__" and "xa = Ok x" and "y = Ok x"   | e :: "'b" and v :: "'b" where "xa__ = Error e" and "xa = Error v" and "y = Error e"
ordinal_times_monoR: fixes y :: "ordinal"   and y' :: "ordinal"   and x :: "ordinal" assumes "y \<le> y'" shows "x * y \<le> x * y'"
add_mset_eq_single: fixes b :: "'d"   and M :: "'d multiset"   and a :: "'d" shows "(add_mset b M = {#a#}) = (b = a \<and> M = {#})"
fmrestrict_set_null: fixes m :: "('a, 'b) fmap" shows "fmrestrict_set {} m = fmempty"
reachable_blackD: fixes s :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state"   and m :: "'a"   and r :: "'d" assumes "no_grey_refs s"   and "mut_m.reachable_snapshot_inv m s"   and "mut_m.reachable m r s" shows "black r s"
can_lock_devreserp: fixes t :: "'thread_id"   and e :: "'addr expr"   and l :: "String.literal \<Rightarrow> 'addr val option"   and h' :: "'heap"   and L :: "('addr + 'thread_id + 'thread_id) set"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "red_mthr.can_sync P t (e, l) h' L"   and "P,E,h \<turnstile> e : T"   and "P,h \<turnstile> t \<surd>t"   and "hconf h"   and "h \<unlhd> h'" shows "red_mthr.can_sync P t (e, l) h L"
Sup_3': fixes \<xi> :: "'a" assumes "\<parallel>\<xi>\<parallel> < r" shows "\<parallel>f *\<^sub>v (x + \<xi>)\<parallel> \<in> (\<lambda>\<xi>. \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>) ` ball (0::'a) r"
le0: fixes n :: "nat" shows "zero_class.zero \<le> n"
as_needed_asses_submap_exec_viii: fixes f :: "('a, 'b) fmap"   and g :: "('a, 'b) fmap" assumes "f \<subseteq>\<^sub>f g" shows "\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v"
valid_7: shows "valid_sops ts\<^sub>s\<^sub>b'"
assms_4: shows "\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y"
mat_of_row_carrier_1: fixes y :: "'a vec"   and n :: "nat" assumes "y \<in> carrier_vec n" shows "mat_of_row y \<in> carrier_mat 1 n"
insert_insert_body_commute: fixes i :: "'c"   and e1 :: "'c \<times> 'd \<times> bool"   and e2 :: "'c \<times> 'd \<times> bool"   and xs :: "('c \<times> 'd \<times> bool) list" assumes "i \<noteq> fst e1"   and "fst e1 \<noteq> fst e2" shows "insert (insert_body xs e1) e2 (Some i) = insert xs e2 (Some i) \<bind> (\<lambda>y. Some (insert_body y e1))"
res: shows "basis_reduction_swap i fs = (upw', i', fs')"
inducts: fixes x :: "'c llist"   and P :: "'c llist \<Rightarrow> bool" assumes "lfinite x"   and "P LNil"   and "\<And>xs x. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (LCons x xs)" shows "P x"
w\<^sub>f_is_map: shows "is_left_adjoint w\<^sub>f"
integral_pmf: fixes X :: "'d set"   and M :: "'d pmf" shows "integral\<^sup>L (count_space X) (pmf M) = measure_pmf.prob M X"
enat_ord_code_3: fixes q :: "enat" shows "(q \<le> \<infinity>) = True"
sp_append_one: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and e :: "('c \<times> nat) edge"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.subpath g v1 (es @ [e]) v2 subs = (SubRel.subpath g v1 es (src e) subs \<and> e \<in> edges g \<and> (tgt e = v2 \<or> (tgt e, v2) \<in> subs\<^sup>+))"
simps_13: fixes V :: "'a"   and T :: "ty"   and e :: "'a exp" shows "max_vars {V:T; e} = max_vars e + 1"
hyps_2: fixes b :: "trm" assumes "t \<longmapsto> b" shows "\<Gamma> \<turnstile> b : S"
IH: assumes "sem\<^sub>e\<^sub>s\<^sub>t_d {} \<I> (to_est \<A>1)" shows "\<exists>\<A>\<^sub>d'. \<A>1 = to_st (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d') \<and> (\<S>, \<A>\<^sub>d) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>1, \<A>\<^sub>d') \<and> sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> \<A>\<^sub>d'"
assms_1: shows "a * 2 powr real_of_int p = real_of_int ai"
a: shows "\<forall>A\<in>carrier_mat 1 2. admits_diagonal_reduction A"
matrix_mul_lid: fixes A :: "(('b, 'c) vec, 'd) vec" shows "mat (1::'b) ** A = A"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "rotate1 (x # xs) = xs ## x"
assms_4: shows "v * - v\<^sup>T \<sqinter> g = bot"
2: fixes d' :: "('test, 'atom) com"   and t' :: "'state" assumes "(d, t) \<rightarrow>*c (d', t')"   and "s' \<approx> t'"   and "discr d'" shows "thesis"
nodes_trans: fixes p :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and q :: "'d" assumes "p \<in> nodes A"   and "q \<in> reachable A p" shows "q \<in> nodes A"
elims_2: fixes P :: "ast_problem"   and x :: "ground_action" assumes "ast_problem.wf_ground_action P x" obtains pre :: "object atom formula" and eff :: "object ast_effect" where "x = Ground_Action pre eff" and "ast_domain.wf_fmla (domain P) (ast_problem.objT P) pre \<and> ast_domain.wf_effect (domain P) (ast_problem.objT P) eff"
aux_lemma: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "[a = b] (mod c)"   and "b < c" shows "\<exists>k. a = c * k + b"
a: shows "cod\<^bsub>CatCod F\<^esub> (F ## f) = F @@ cod\<^bsub>CatDom F\<^esub> f"
COUNTER_def: fixes M :: "nat" shows "COUNTER M = SymSystem {u. u 0 = 0} (prec_count M) rel_count"
convex_on_segmentI: fixes C :: "'c set"   and a :: "'c"   and j :: "real"   and b :: "'c"   and i :: "real" assumes "convex C"   and "a \<in> C"   and "a + j *\<^sub>R b \<in> C"   and "0 \<le> i"   and "i \<le> j" shows "a + i *\<^sub>R b \<in> C"
B_imp_A: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> B m n" shows "x \<in> A m n"
region_set'_id: fixes R :: "('a \<Rightarrow> real) set"   and X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and v :: "'a \<Rightarrow> real"   and c :: "nat"   and r :: "'a list" assumes "R \<in> {Regions.region X I r |I r. Regions.valid_region X k I r}"   and "v \<in> R"   and "finite X"   and "0 \<le> c"   and "\<forall>x\<in>set r. c \<le> k x"   and "set r \<subseteq> X" shows "[[r\<rightarrow>real c]v]\<^sub>{Regions.region X I r |I r. Regions.valid_region X k I r} = region_set' R r (real c) \<and> [[r\<rightarrow>real c]v]\<^sub>{Regions.region X I r |I r. Regions.valid_region X k I r} \<in> {Regions.region X I r |I r. Regions.valid_region X k I r} \<and> [r\<rightarrow>real c]v \<in> [[r\<rightarrow>real c]v]\<^sub>{Regions.region X I r |I r. Regions.valid_region X k I r}"
P_def: shows "P \<equiv> \<lambda>t s. \<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> Unifier \<delta> t s"
inf1: fixes A :: "'a"   and k :: "'a"   and a :: "'a"   and b :: "'a" assumes "A \<equiv> k \<sqinter> a" shows "A \<sqinter> b \<equiv> k \<sqinter> (a \<sqinter> b)"
cospan_of_maps_in_bicategory_of_spans_axioms: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg s\<^sub>1 r\<^sub>0"
rv_X: fixes i :: "'i" assumes "i \<in> I" shows "random_variable borel (X i)"
listE_length: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set" assumes "xs \<in> list n A" shows "length xs = n"
join_connection2: fixes x :: "nat"   and y :: "nat" shows "(x \<le> y) = (nat.join x y = y)"
gb_schema_aux_term1_def: shows "gb_schema_aux_term1 = {(a, b). fst ` set a \<sqsupset>p fst ` set b} <*lex*> measure (card \<circ> set)"
sum_pos: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> {#} \<subset># f i" shows "{#} \<subset># comm_monoid_add.sum (+) {#} f I"
complete_3: fixes \<sigma> :: "(char list \<times> 'a list) trace" assumes "prefix_of \<pi> \<sigma>" shows "MFOTL.sat \<sigma> (map the v) i \<phi>"
fvc: fixes v :: "variable \<Rightarrow> real"   and w :: "variable \<Rightarrow> real" assumes "Vagree v w (- {z})"   and "v \<in> fml_sem I e" shows "w \<in> fml_sem I e"
decode_Var_inject: fixes v :: "hf"   and v' :: "hf" assumes "is_Var v"   and "is_Var v'" shows "(decode_Var v = decode_Var v') = (v = v')"
assms_1: shows "\<forall>x\<in>L. \<forall>y\<in>L. x \<le> y \<longrightarrow> f x \<le> f y"
wf_tuple_restrict: fixes n :: "nat"   and B :: "nat set"   and v :: "'a option list"   and A :: "nat set"   and C :: "nat set" assumes "wf_tuple n B v"   and "A \<inter> B = C" shows "wf_tuple n C (restrict A v)"
cobounded1: fixes a :: "'i"   and b :: "'i" shows "a \<le> max a b"
translation_length_le: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "translation_length f \<le> dist x (f x)"
infsetsum_of_real: fixes f :: "'b \<Rightarrow> real"   and A :: "'b set" shows "(\<Sum>\<^sub>ax\<in>A. of_real (f x)) = of_real (infsetsum f A)"
dec_word_enc: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list" assumes "formula \<Sigma>" shows "dec_word (formula.enc (w, I)) = w"
directly_attracted_empty: fixes p :: "Player" shows "directly_attracted p {} = {}"
cInf_le_cSup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq A"   and "preorder.bdd_below less_eq A" shows "less_eq (Inf A) (Sup A)"
absorb2: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" shows "a \<sqinter> b = b"
a_rcos_zero: fixes I :: "'c set"   and i :: "'c" assumes "ideal I S"   and "i \<in> I" shows "I +>\<^bsub>S\<^esub> i = I"
hom_add_eq_zero: fixes x :: "int"   and y :: "int" assumes "x + y = 0" shows "of_int x + of_int y = (0::'a)"
subst_renameLCs_cong: fixes s :: "'subst"   and f1 :: "'var \<Rightarrow> 'var"   and f2 :: "'var \<Rightarrow> 'var" assumes "\<And>x. x \<in> subst_lconsts s \<Longrightarrow> f1 x = f2 x" shows "subst_renameLCs f1 s = subst_renameLCs f2 s"
lsided_le4: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.ab_quantale times Inf Sup inf less_eq less sup bot top"   and "quantale.lsd times less_eq top x" shows "less_eq (times (inf x y) z) (inf x (times y z))"
zero_in_bigtheta_iff: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "((\<lambda>_. 0::'c) \<in> \<Theta>[F](f)) = (\<forall>\<^sub>F x in F. f x = (0::'c))"
fin_dg_tiny_digraph: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "finite_digraph \<alpha> \<CC>" shows "tiny_digraph \<alpha> \<CC>"
arg_cis: fixes \<phi> :: "real" shows "arg (cis \<phi>) = \<downharpoonright>\<phi>\<downharpoonleft>"
reflexive: fixes a :: "'a" assumes "a \<in> S" shows "(a, a) \<in> Orbit_Relation"
cbelow_on_transitive: fixes A :: "'c set"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "cbelow_on A ord z rel a b"   and "cbelow_on A ord z rel b c" shows "cbelow_on A ord z rel a c"
sets': shows "sets (embed_measure (distr M N f') f) = sets (embed_measure N f)"
s_in_S: shows "s \<in> S \<or> D = (\<lambda>_. \<infinity>)(s := 0) \<and> S = {}"
r_neg: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<oplus> \<ominus> x = \<zero>"
Poly_append: fixes a :: "'b list"   and b :: "'b list" shows "Poly (a @ b) = Poly a + monom (1::'b) (length a) * Poly b"
PrefixListTransitive: fixes l1 :: "'a list"   and l2 :: "'a list"   and l3 :: "'a list" assumes "prefixList l1 l2"   and "prefixList l2 l3" shows "prefixList l1 l3"
viable_iff: shows "viable_market Mkt = (d < 1 + r \<and> 1 + r < u)"
acyclic_5g_def: fixes x :: "'a" shows "acyclic_5g x \<equiv> \<forall>y z. y \<squnion> z = x \<and> y \<sqinter> z = bot \<longrightarrow> y\<^sup>\<star> \<sqinter> z\<^sup>\<star> = (1::'a)"
listrelp_app: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and xs' :: "'a list"   and ys' :: "'b list" assumes "listrelp R xs ys"   and "listrelp R xs' ys'" shows "listrelp R (xs @ xs') (ys @ ys')"
LtK_Aenc: fixes X :: "ltkey"   and X' :: "msg"   and Y' :: "msg" shows "LtK X \<noteq> Aenc X' Y'"
t_wise_balance_axioms: shows "t_wise_balance V B t \<Lambda>\<^sub>t"
Nml_HcompNml_5: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Trg (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Trg t"
assms_2: shows "stream_all (\<lambda>u. u \<in> V) xs"
weakBisimE_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q" shows "\<Psi> \<rhd> P \<lessapprox><weakBisim> Q"
Gg: fixes X :: "real \<Rightarrow> 'a" assumes " D X = (\<lambda>\<tau>. f \<tau> (X \<tau>)) on U (X t\<^sub>0)" shows "\<forall>\<tau>\<in>U (X t\<^sub>0). t\<^sub>0 < \<tau> \<longrightarrow> G (X \<tau>) \<longrightarrow> \<nu>' (X \<tau>) \<le> \<mu>' (X \<tau>)"
measurable_bounded_by_integrable_imp_absolutely_integrable: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "f absolutely_integrable_on S"
Poly_append_replicate_0: fixes xs :: "'b list"   and n :: "nat" shows "Poly (xs @ replicate n (0::'b)) = Poly xs"
prems_2: shows "image_mset disj_of_clause (mset (g # G)) + \<Gamma> \<Rightarrow>\<^sub>n"
simps_7: fixes v :: "'b poly" shows "degreen (Neg v) = (\<lambda>m. 0)"
unl_lift2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'e \<Rightarrow> 'c"   and y :: "'e \<Rightarrow> 'd"   and w :: "'e" shows "(f<x, y>) w \<equiv> f (x w) (y w)"
s_cond_def_1: fixes t :: "real"   and a :: "real"   and v :: "real"   and s0 :: "real" assumes "t \<le> 0" shows "movement.s a v s0 t = s0"
eq2: shows "(\<exists>(c, t)\<in>set (uset p). \<exists>(d, s)\<in>set (uset p). Ifm vs (x # bs) (msubst2 p ((- 2)\<^sub>p *\<^sub>p c *\<^sub>p d) (tm.Add (tm.Mul d t) (tm.Mul c s)))) = (\<exists>(n, t)\<in>set (uset p). \<exists>(m, s)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'a) \<and> \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'a) \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'a) # bs) p)"
simps_14: fixes v :: "formula"   and va :: "\<I>"   and vb :: "formula" shows "get_and_list (Since v va vb) = [Since v va vb]"
leg1_in_hom_2: shows "\<guillemotleft>r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>"
SLPATH_1: shows "(([entry fg p], {#}), wsl, [v], ch) \<in> trcl (trss fg)"
select_convs_2: fixes name :: "mname"   and parTs :: "ty list"   and more :: "'b" shows "parTs \<lparr>name = name, parTs = parTs, \<dots> = more\<rparr> = parTs"
simps_1: fixes s1 :: "'c stream"   and s2 :: "'d stream" shows "shd (s1 ||| s2) = (shd s1, shd s2)"
alphaAbs_qAbs_iff_alphaAbs_ex_equal_or_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_equal_or_qAFresh xs x X xs' x' X'"
suffix_Cons: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "suffix xs (y # ys) = (xs = y # ys \<or> suffix xs ys)"
And_constraint: fixes \<phi> :: "formula"   and \<psi> :: "formula" assumes "safe_formula \<phi>"   and "\<not> safe_assignment (fv \<phi>) \<psi>"   and "\<not> safe_formula \<psi>"   and "fv \<psi> \<subseteq> fv \<phi>"   and "is_constraint \<psi>"   and "P \<phi>" shows "P (And \<phi> \<psi>)"