hcju/isabelleps huggingface dataset:
add_inv_exists_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "\<exists>w\<in>carrier_vec n. w + v = 0\<^sub>v n \<and> v + w = 0\<^sub>v n"
has_fields_declaring_classes: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs" shows "\<exists>pre FDTs'. FDTs = pre @ FDTs' \<and> (C \<noteq> Object \<longrightarrow> (\<exists>D fs ms. class P C = \<lfloor>(D, fs, ms)\<rfloor> \<and> P \<turnstile> D has_fields FDTs')) \<and> set (map (\<lambda>t. snd (fst t)) pre) \<subseteq> {C} \<and> set (map (\<lambda>t. snd (fst t)) FDTs') \<subseteq> {C'. C' \<noteq> C \<and> P \<turnstile> C \<preceq>\<^sup>* C'}"
select_convs_4: fixes tdir :: "'b"   and xdir :: "'b"   and ydir :: "'b"   and zdir :: "'b"   and more :: "'c" shows "zdir \<lparr>tdir = tdir, xdir = xdir, ydir = ydir, zdir = zdir, \<dots> = more\<rparr> = zdir"
reduced_GB_D1: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_reduced_GB G" shows "punit.is_Groebner_basis G"
map_of_remap_SomeD: fixes t :: "(('d \<times> 'b) \<times> 'c) list"   and k :: "'d"   and k' :: "'b"   and x :: "'c" assumes "map_of (map (\<lambda>((k, k'), x). (k, k', x)) t) k = \<lfloor>(k', x)\<rfloor>" shows "map_of t (k, k') = \<lfloor>x\<rfloor>"
simps_24: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (Times e1 e2) (Plus v va) = False"
funpower_upperbound: fixes G :: "'b set"   and f :: "'b set \<Rightarrow> 'b set"   and I :: "'b set"   and n :: "nat" assumes "\<And>I. I \<subseteq> G \<Longrightarrow> f I \<subseteq> G"   and "I \<subseteq> G" shows "funpower f n I \<subseteq> G"
case_2: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and x21 :: "'c"   and x22 :: "('c, 'b) tllist" shows "(case TCons x21 x22 of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
induct: fixes P :: "('b \<rightarrow> ['b] \<rightarrow> tr) \<Rightarrow> bool" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> xa v. Fixrec.run\<cdot>(match_Nil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>FF) +++ match_Cons\<cdot>v\<cdot>(\<Lambda> y ys. Fixrec.succeed\<cdot>(eq\<cdot>y\<cdot>xa orelse x\<cdot>xa\<cdot>ys))))" shows "P elem"
assms_3: fixes x :: "'a" assumes "x \<in> S" shows "f x = g x"
strict_implies_order: fixes b :: "'e list"   and a :: "'e list" assumes "strict_prefix b a" shows "prefix b a"
qmpt_axioms_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "qmpt_axioms M T \<equiv> T \<in> quasi_measure_preserving M M"
final_finished: fixes F_empty :: "'a"   and F_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) instr fundef option"   and F_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) instr fundef \<Rightarrow> 'a"   and F_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f) instr fundef) list"   and heap_empty :: "'g"   and heap_get :: "'g \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'g \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'g"   and heap_to_list :: "'g \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'f \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'f option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'e"   and s :: "('a, 'g, ('b, 'c) frame) state" assumes "inca F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"   and "inca.final F_get s" shows "finished (inca.step F_get F_add heap_get heap_add is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>) s"
remove_subset: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "S - {x} \<subset> S"
Int_mono2: fixes A :: "'e set"   and C :: "'e set"   and B :: "'e set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<inter> B \<subseteq> C"
mod_exp_aux_def: fixes m :: "'a" shows "mod_exp_aux m = efficient_funpow (\<lambda>x y. x * y mod m)"
map_eq_consE: fixes f :: "'b \<Rightarrow> 'a"   and ls :: "'b list"   and fa :: "'a"   and fl :: "'a list" assumes "map f ls = fa # fl" obtains a :: "'b" and l :: "'b list" where "ls = a # l" and "f a = fa" and "map f l = fl"
iprev_neq_Max: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "\<exists>x\<in>I. x < n" shows "iprev n I \<noteq> Max I"
unfold: shows "llast_body = (\<Lambda> f v. Fixrec.run\<cdot>(match_lcons\<cdot>v\<cdot>(\<Lambda> x yys. Fixrec.succeed\<cdot>(case yys of lnil \<Rightarrow> x | y :@ ys \<Rightarrow> f\<cdot>yys))))"
set_ConsD: fixes y :: "'e"   and x :: "'e"   and xs :: "'e list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
representation_basis: fixes basis :: "'b set"   and b :: "'b" assumes "independent basis"   and "b \<in> basis" shows "representation basis b = (\<lambda>v. if v = b then 1::'a else (0::'a))"
word_sle_msb_le: fixes x :: "'a word"   and y :: "'a word" shows "(x \<le>s y) = ((msb y \<longrightarrow> msb x) \<and> (msb x \<and> \<not> msb y \<or> x \<le> y))"
comp: fixes x :: "'a" assumes "x \<in> M" shows "x \<in> iterates_above (lub M') \<or> lub M' \<in> iterates_above x"
next_inv: fixes s :: "('c, 'd) configuration stream" assumes "next s" obtains "next_performop s"   | "next_sendupd s"   | "next_recvupd s"   | "shd (stl s) = shd s"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "S.map f = S.null"
sum_f': shows "(\<Sum>\<^sup>+ x. ennreal (f' x)) = integral\<^sup>N (count_space A) f"
matches_MatchNotAnd_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m' :: "'c match_expr"   and a :: "action"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule (MatchAnd (MatchNot m) m') a], Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[], Undecided\<rangle> \<Rightarrow> t"
4: shows "v\<^sub>1 @ l @ w\<^sub>2 =\<^sub>F w"
simps: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and V'' :: "('d \<times> 'e) list set"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list" shows "Rep_Cov M2 M1 V'' vs xs = (\<exists>xs' vs'. xs' \<noteq> [] \<and> prefix xs' xs \<and> vs' \<in> V'' \<and> (\<exists>s2. io_targets M2 (initial M2) (vs @ xs') = {s2} \<and> io_targets M2 (initial M2) vs' = {s2}) \<and> (\<exists>s1. io_targets M1 (initial M1) (vs @ xs') = {s1} \<and> io_targets M1 (initial M1) vs' = {s1}))"
set: fixes x :: "'a list" shows "set_iarray (IArray x) = set x"
leq: shows "P \<turnstile> C \<preceq>\<^sup>* D"
translation_unit_eq: shows "identity G = (\<one>)\<^sub>R"
val: fixes vs :: "exp list"   and C :: "nat" assumes "vals(vs)" shows "val(New C vs)"
ennreal_zero_less_mult_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(0 < a * b) = (0 < a \<and> 0 < b)"
set_cases: fixes e :: "'b"   and a :: "'b list" assumes "e \<in> set a" obtains z2 :: "'b list" where "a = e # z2__"   | z1 :: "'b" and z2 :: "'b list" where "a = z1 # z2" and "e \<in> set z2"
inj_on_mapI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a list set" assumes "inj_on f (\<Union> (set ` A))" shows "inj_on (map f) A"
decseq_SucI: fixes X :: "nat \<Rightarrow> 'a" assumes "\<And>n. X (Suc n) \<le> X n" shows "decseq X"
cons: fixes a :: "'action"   and p :: "'state"   and r :: "'action list" assumes "ren a p"   and "R.path r (ex a p)" shows "R.path (a # r) p"
prems_1: shows "\<nexists>eva. eva \<in> set (t @ [ev]) \<and> regular_event eva \<and> occurs_on eva = p"
bex_UNIV: fixes P :: "'c \<Rightarrow> bool" shows "Bex UNIV P = Ex P"
inputSwapFrameSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and y :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "x \<sharp> P"   and "y \<sharp> P" shows "[(x, y)] \<bullet> \<Psi> \<rhd> P \<longmapsto> ([(x, y)] \<bullet> M)\<lparr>N\<rparr> \<prec> P'"
final: shows "nfa.final M \<subseteq> nfa.states M"
convergence_norm_U: fixes x :: "'s"   and y :: "'s"   and z :: "'a" assumes "convergence_G x y z" shows "summable (\<lambda>n. u x y n * norm z ^ Suc n)"
set_dlist_induct: fixes P :: "'a set_dlist \<Rightarrow> bool"   and dxs :: "'a set_dlist" assumes "P DList_Set.empty"   and "\<And>a dxs. \<lbrakk>\<not> member dxs a; P dxs\<rbrakk> \<Longrightarrow> P (DList_Set.insert a dxs)" shows "P dxs"
simps_1: fixes a :: "'e" shows "replicate_raw a 0 = Done"
next_non_stuttering: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n\<^sub>p :: "nat"   and n :: "nat"   and sn\<^sub>0 :: "nat" assumes "dilating f sub r"   and "f n\<^sub>p < n \<and> (\<forall>k. f n\<^sub>p < k \<and> k \<le> n \<longrightarrow> (\<nexists>k\<^sub>0. f k\<^sub>0 = k))"   and "f sn\<^sub>0 = Suc n" shows "sn\<^sub>0 = Suc n\<^sub>p"
effect_apply_eqvt_aux: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and p :: "perm" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "p \<bullet> effect_apply = effect_apply"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (G f) = A.arr f"
bdd_below_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a set"   and A :: "'a set" assumes "class.preorder less_eq less"   and "preorder.bdd_below less_eq B"   and "A \<subseteq> B" shows "preorder.bdd_below less_eq A"
left_commute: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry"   and c :: "'c DBMEntry" shows "dbm_add b (dbm_add a c) = dbm_add a (dbm_add b c)"
UN_finite2_eq: fixes A :: "nat \<Rightarrow> 'b set"   and B :: "nat \<Rightarrow> 'b set"   and k :: "nat" assumes "\<And>n. \<Union> (A ` {0..<n}) = \<Union> (B ` {0..<n + k})" shows "\<Union> (range A) = \<Union> (range B)"
consistent_length_tcsigs_imp_exe_consistent_length_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "exe_ars_conds arss"   and "consistent_length_tcsigs (translate_ars arss)" shows "exe_consistent_length_tcsigs arss"
subst_lsst_append: fixes A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and B :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "A @ B \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) @ (B \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
poly_mod_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c poly"   and z :: "'c poly" assumes "finite A" shows "sum f A mod z = (\<Sum>i\<in>A. f i mod z)"
c_of0: fixes A :: "'a" shows "(production N (CA_of A) = {A}) = (A \<in> set As)"
assms: shows "A \<noteq> []\<^sub>\<circ>"
p: shows "root_unity n = monom (1::'a) n + - 1"
r_sub_smult: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K " shows "K \<subseteq> H \<diamondop> K"
matrix_to_iarray_ncols: fixes A :: "(('a, 'b) vec, 'c) vec" shows "ncols A = ncols_iarray (matrix_to_iarray A)"
path_root_iff: fixes x :: "'a" shows "(\<exists>r. path_root r x) = path x"
nlistListLCons: fixes xs :: "('a, 'b) Node list"   and t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.toList (FingerTreeStruc.lconsNlist xs t) = FingerTreeStruc.nlistToList xs @ FingerTreeStruc.toList t"
drinks2_end: fixes r :: "nat \<Rightarrow>f value option"   and a :: "String.literal"   and b :: "value list" shows "possible_steps drinks2 3 r a b = {||}"
bisimilar_def: fixes P :: "'state"   and Q :: "'state" shows "P \<sim>\<cdot> Q \<equiv> \<exists>R. is_bisimulation R \<and> R P Q"
subst_atm_list_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and As :: "'b list"   and \<sigma> :: "'c" shows "substitution_ops.subst_atm_list subst_atm As \<sigma> = map (\<lambda>A. subst_atm A \<sigma>) As"
minus_eq_not_minus_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "uminus a = not (minus a one)"
assms_2: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "(P, Q) \<in> Rel" shows "(<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"
tvs_mk_all_list: fixes vs :: "(variable \<times> typ) list"   and B :: "Core.term" shows "tvs (mk_all_list vs B) = tvs B \<union> tvsT_Set (snd ` set vs)"
comp_in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VoV.in_hom f a b"   and "VoV.in_hom g b c" shows "VoV.in_hom (VoV.comp g f) a c"
l4_16R2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" assumes "A B C D FSC A' B' C' D'"   and "Bet B C A" shows "Cong C D C' D'"
ah_interleavable2: fixes \<alpha> :: "'d \<Rightarrow> 'c set \<times> 'c set"   and w1 :: "'d list"   and w2 :: "'d list" assumes "\<alpha>ah (map \<alpha> w1) [*] \<alpha>ah (map \<alpha> w2)" shows "w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2 \<noteq> {}"
Let_2: fixes \<rho> :: "Vars.var \<Rightarrow> Value" shows "eq (fst (a, Aheap \<Gamma> e\<cdot>a)) (\<lbrakk> \<T>\<^bsub>fst (a, Aheap \<Gamma> e\<cdot>a)\<^esub> e \<rbrakk>\<^bsub>\<rho>\<^esub>) (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)"
ParCptnComp: fixes P :: "'b com option list"   and s :: "'b"   and Q :: "'b com option list"   and t :: "'b"   and xs :: "('b com option list \<times> 'b) list" assumes "(P, s) -pc\<rightarrow> (Q, t)"   and "(Q, t) # xs \<in> par_cptn" shows "(P, s) # (Q, t) # xs \<in> par_cptn"
reduce_basis_cost_N: fixes Lg :: "nat" assumes "nat \<lceil>log (real_of_rat (4 * \<alpha> / (4 + \<alpha>))) (real N)\<rceil> \<le> Lg"   and "0 < Lg" shows "cost (reduce_basis_cost fs_init) \<le> 49 * m ^ 3 * n * Lg"
eqpoll_trans: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set" assumes "A \<approx> B"   and "B \<approx> C" shows "A \<approx> C"
assms_2: assumes "\<pi>l \<noteq> []" shows "stateOf \<pi> = stateOf (last \<pi>l)"
assms_2: shows "fresh zs z (X #[y1 // y]_ys)"
1: shows "\<not> hamlet (Rep_run sub 0 c\<^sub>2) \<and> (\<forall>n. tick_count sub c\<^sub>2 (Suc n) \<le> tick_count sub c\<^sub>1 n)"
simps_2: fixes f :: "'a \<Rightarrow> 'a" shows "redexes (Basic f) = {Basic f}"
eval_monom_neg: fixes basis :: "(real \<Rightarrow> real) list"   and monom :: "real \<times> real list" assumes "basis_wf basis"   and "fst monom < 0" shows "\<forall>\<^sub>F x in at_top. eval_monom monom basis x < 0"
class_add_app\<^sub>i: fixes i :: "instr"   and P :: "jvm_prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "app\<^sub>i (i, P, pc, mxs, T\<^sub>r, ST, LT)"   and "\<not> is_class P C" shows "app\<^sub>i (i, class_add P (C, cdec), pc, mxs, T\<^sub>r, ST, LT)"
deletion: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "\<not> reduced_word (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) ss" shows "\<exists>a b as bs cs. ss = as @ [a] @ bs @ [b] @ cs \<and> sum_list ss = sum_list (as @ bs @ cs)"
mset_factorsD_2: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and F :: "'a multiset"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F p" shows "F \<noteq> {#}"
intro: fixes final :: "'g \<Rightarrow> bool"   and r :: "('j, 'h, 'g, 'i, 'k, 'l) semantics"   and P :: "'h \<Rightarrow> 'g \<Rightarrow> 'i \<Rightarrow> bool" assumes "multithreaded final r"   and "lifting_wf_axioms r P" shows "lifting_wf final r P"
env_restr_join2: fixes S :: "'a set"   and m :: "'a \<Rightarrow> 'b"   and S' :: "'a set" shows "m f|` S \<squnion> m f|` S' = m f|` (S \<union> S')"
is_empty: fixes q :: "'b tree" assumes "braun q \<and> heap q" shows "(q = \<langle>\<rangle>) = (mset_tree q = {#})"
distinct_164: fixes x13 :: "Syntax.term"   and x6 :: "Syntax.term" shows "Roll x13 \<noteq> Inj2 x6"
Par_memb_rep: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "arr f"   and "t \<in> f" shows "Par t (rep f)"
pause_env_def: fixes pause :: "'f \<Rightarrow> ('g \<Rightarrow> 'e) \<Rightarrow> 'e"   and out :: "'f"   and c :: "'g \<Rightarrow> ('d, 'e) envT" shows "pause_env pause out c = EnvT (\<lambda>r. pause out (\<lambda>i. run_env (c i) r))"
exec_all_def: fixes P :: "jvm_prog"   and \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>' = ((\<sigma>, \<sigma>') \<in> {(\<sigma>, \<sigma>'). exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>}\<^sup>*)"
pochhammer_Suc0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc 0) = a"
ginit_approx: fixes prog :: "program" shows "init_gc prog \<in> gc_approx prog"
inject_5: fixes x5 :: "'a \<times> nat"   and y5 :: "'a \<times> nat" shows "(Actual_out x5 = Actual_out y5) = (x5 = y5)"
conf_fs_shupd: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and C\<^sub>0 :: "char list"   and M :: "char list"   and n :: "nat"   and T :: "ty"   and frs :: "frame list"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state" assumes "conf_fs P h sh \<Phi> C\<^sub>0 M n T frs"   and "distinct (C # clinit_classes frs)" shows "conf_fs P h (sh(C \<mapsto> (sfs, i))) \<Phi> C\<^sub>0 M n T frs"
Incremental_NS_Constraint_Ops_axioms: shows "Incremental_NS_Constraint_Ops init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc"
Bseq_mono_convergent: fixes X :: "nat \<Rightarrow> real" assumes "Bseq X"   and "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n" shows "convergent X"
Rad_of_def: fixes w :: "floatarith" shows "Rad_of w = w * (Pi / Num 180)"
confT_OK: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and x :: "val"   and T :: "ty" shows "(P,h \<turnstile> x :\<le>\<^sub>\<top> OK T) = (P,h \<turnstile> x :\<le> T)"
real_scalar_mult2: fixes x :: "('a, 'n) vec"   and A :: "'a" shows "real_of (inner x x) *\<^sub>R A = inner x x * A"
cases: fixes K :: "'a \<Rightarrow> 'a pmf set"   and a1 :: "'a"   and a2 :: "'a cfg" assumes "Markov_Decision_Process K"   and "Markov_Decision_Process.cfg_onp K a1 a2" obtains cfg :: "'a cfg" and s :: "'a" where "a1 = s" and "a2 = cfg" and "state cfg = s" and "action cfg \<in> K s" and "\<And>t. t \<in> set_pmf (action cfg) \<Longrightarrow> Markov_Decision_Process.cfg_onp K t (cont cfg t)"
Derives1_leftmost: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "\<exists>j. leftmost j a \<and> j \<le> i"
closed_map_compose: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "closed_map X X' f"   and "closed_map X' X'' g" shows "closed_map X X'' (g \<circ> f)"
sr_2: shows "(s \<circ> r) n \<notin> range (flow0 y)"
fmrel_on_fset_fmmerge2: fixes xm :: "'f \<rightharpoonup>\<^sub>f 'd"   and f :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'g"   and ym :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "\<And>x y. x \<in> fmran' xm \<Longrightarrow> f x (g x y)" shows "fmrel_on_fset (fmdom ym) f xm (fmmerge g xm ym)"
sum_right: fixes x :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and g :: "'d \<Rightarrow> 'c"   and S :: "'d set" shows "x *\<^sub>V sum g S = (\<Sum>i\<in>S. x *\<^sub>V g i)"
rename_formula_def: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a set"   and S :: "'a Literal set set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.rename_formula A S = {C. \<exists>CC. CC \<in> S \<and> C = propositional_atoms.rename_clause A CC}"
R36_d: shows "pmf (sds R36) d = 1 / 2 - pmf (sds R36) c"
re_leq_two: shows "\<forall>c. |re c| \<le> 2"
ub_add_nat: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a"   and p :: "nat" assumes "m \<le> n + 1" shows "prod g {m..n + p} = prod g {m..n} * prod g {n + 1..n + p}"
PiE_singleton: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "f \<in> extensional A" shows "(\<Pi>\<^sub>E x\<in>A. {f x}) = {f}"
distinct_9: fixes x61 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and x62 :: "bexp"   and x63 :: "acom" shows "SKIP \<noteq> {x61} WHILE x62 DO x63"
homom4: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<triangleright> N"   and "G \<guillemotright> H " shows "gbij\<^bsub>(\<natural>\<^bsub>G\<^esub>H / (H \<inter> N)),(\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N / N)\<^esub> (Pj (\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N) N \<circ>\<^bsub>(\<natural>\<^bsub>G\<^esub>H)\<^esub> \<iota>\<^bsub>(\<natural>\<^bsub>G\<^esub>H)\<^esub>\<dieresis>\<^bsub>(\<natural>\<^bsub>G\<^esub>H),(\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N / N)\<^esub>)"
dg_def: shows "\<BB> = [\<BB>\<lparr>Obj\<rparr>, \<BB>\<lparr>Arr\<rparr>, \<BB>\<lparr>Dom\<rparr>, \<BB>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
e2ennreal_enn2ereal: fixes x :: "ennreal" shows "e2ennreal (enn2ereal x) = x"
either_empty1: fixes t :: "'b ttree" shows "TTree.empty \<oplus>\<oplus> t = t"
hyps: shows "set (vwalk_arcs (rev xs)) = {a. case a of (v, u) \<Rightarrow> (u, v) \<in> set (vwalk_arcs xs)}"
in_set_impl_in_set_zip1: fixes xs :: "'d list"   and ys :: "'e list"   and x :: "'d" assumes "length xs = length ys"   and "x \<in> set xs" obtains y :: "'e" where "(x, y) \<in> set (zip xs ys)"
assms_1: shows "P \<Longrightarrow>\<^sub>A Q"