hcju/isabelleps huggingface dataset:
balance_def: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "balance l k v r = (if cl l = R then let ll = lsub l; rl = rsub l in if cl ll = R then Node (Node (lsub ll) B (key ll) (val ll) (rsub ll)) R (key l) (val l) (Node (rsub l) B k v r) else if cl rl = R then Node (Node (lsub l) B (key l) (val l) (lsub rl)) R (key rl) (val rl) (Node (rsub rl) B k v r) else balanceR l k v r else balanceR l k v r)"
A: shows "p\<^sub>0 \<noteq> p\<^sub>1"
assms: shows "A \<subseteq>\<^sub>\<circ> B -\<^sub>\<circ> C"
diff_Integer_eq_bottom_iff: fixes x :: "Integer"   and y :: "Integer" shows "(x - y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
cCons_def: fixes x :: "'c"   and xs :: "'c list" shows "cCons x xs = (if xs = [] \<and> x = (0::'c) then [] else x # xs)"
induct: fixes P :: "'a gexp \<Rightarrow> 'a gexp \<Rightarrow> bool"   and a0 :: "'a gexp"   and a1 :: "'a gexp" assumes "\<And>v l v' l'. P (Eq (V v) (L l)) (Eq (V v') (L l'))"   and "\<And>v l v' l'. P (In v l) (Eq (V v') (L l'))"   and "\<And>v' l' v l. P (Eq (V v') (L l')) (In v l)"   and "\<And>v l v' l'. P (In v l) (In v' l')"   and "\<And>v uv_. P (Bc v) uv_"   and "\<And>vb va uv_. P (Eq (L vb) va) uv_"   and "\<And>vb vc va uv_. P (Eq (Plus vb vc) va) uv_"   and "\<And>vb vc va uv_. P (Eq (Minus vb vc) va) uv_"   and "\<And>vb vc va uv_. P (Eq (Times vb vc) va) uv_"   and "\<And>v vb uv_. P (Eq v (V vb)) uv_"   and "\<And>v vb vc uv_. P (Eq v (Plus vb vc)) uv_"   and "\<And>v vb vc uv_. P (Eq v (Minus vb vc)) uv_"   and "\<And>v vb vc uv_. P (Eq v (Times vb vc)) uv_"   and "\<And>v va uv_. P (Gt v va) uv_"   and "\<And>v va vb. P (In v va) (Bc vb)"   and "\<And>v va vd vc. P (In v va) (Eq (L vd) vc)"   and "\<And>v va vd ve vc. P (In v va) (Eq (Plus vd ve) vc)"   and "\<And>v va vd ve vc. P (In v va) (Eq (Minus vd ve) vc)"   and "\<And>v va vd ve vc. P (In v va) (Eq (Times vd ve) vc)"   and "\<And>v va vb vd. P (In v va) (Eq vb (V vd))"   and "\<And>v va vb vd ve. P (In v va) (Eq vb (Plus vd ve))"   and "\<And>v va vb vd ve. P (In v va) (Eq vb (Minus vd ve))"   and "\<And>v va vb vd ve. P (In v va) (Eq vb (Times vd ve))"   and "\<And>v va vb vc. P (In v va) (Gt vb vc)"   and "\<And>v va vb vc. P (In v va) (Nor vb vc)"   and "\<And>v va uv_. P (Nor v va) uv_"   and "\<And>uu_ v. P uu_ (Bc v)"   and "\<And>uu_ vb va. P uu_ (Eq (L vb) va)"   and "\<And>uu_ vb vc va. P uu_ (Eq (Plus vb vc) va)"   and "\<And>uu_ vb vc va. P uu_ (Eq (Minus vb vc) va)"   and "\<And>uu_ vb vc va. P uu_ (Eq (Times vb vc) va)"   and "\<And>uu_ v vb. P uu_ (Eq v (V vb))"   and "\<And>uu_ v vb vc. P uu_ (Eq v (Plus vb vc))"   and "\<And>uu_ v vb vc. P uu_ (Eq v (Minus vb vc))"   and "\<And>uu_ v vb vc. P uu_ (Eq v (Times vb vc))"   and "\<And>uu_ v va. P uu_ (Gt v va)"   and "\<And>vb v va. P (Bc vb) (In v va)"   and "\<And>vd vc v va. P (Eq (L vd) vc) (In v va)"   and "\<And>vd ve vc v va. P (Eq (Plus vd ve) vc) (In v va)"   and "\<And>vd ve vc v va. P (Eq (Minus vd ve) vc) (In v va)"   and "\<And>vd ve vc v va. P (Eq (Times vd ve) vc) (In v va)"   and "\<And>vb vd v va. P (Eq vb (V vd)) (In v va)"   and "\<And>vb vd ve v va. P (Eq vb (Plus vd ve)) (In v va)"   and "\<And>vb vd ve v va. P (Eq vb (Minus vd ve)) (In v va)"   and "\<And>vb vd ve v va. P (Eq vb (Times vd ve)) (In v va)"   and "\<And>vb vc v va. P (Gt vb vc) (In v va)"   and "\<And>vb vc v va. P (Nor vb vc) (In v va)"   and "\<And>uu_ v va. P uu_ (Nor v va)" shows "P a0 a1"
red1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and x :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and h' :: "'heap"   and x' :: "'addr val list"   and exs :: "('addr expr1 \<times> 'addr val list) list" assumes "uf,P,t \<turnstile>1 \<langle>e,(h, x)\<rangle> -ta\<rightarrow> \<langle>e',(h', x')\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>(e, x)/exs,h\<rangle> -extTA2J1 P ta\<rightarrow> \<langle>(e', x')/exs,h'\<rangle>"
monad_left_unit: fixes x :: "'c"   and f :: "'c \<rightarrow> 'a\<cdot>'b" shows "return\<cdot>x \<bind> f = f\<cdot>x"
Pi_empty: fixes B :: "'c \<Rightarrow> 'd set" shows "Pi {} B = UNIV"
comm_monoid_add_hom_axioms: shows "comm_monoid_add_hom (pCons (0::'c))"
fv_trms\<^sub>s\<^sub>s\<^sub>t_subset_2: fixes S :: "('d, 'c) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t S \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S)"
image_mset_cong2: fixes M :: "'c multiset"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and N :: "'c multiset" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x"   and "M = N" shows "f `# M = g `# N"
simps_2: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b"   and r :: "'a"   and rs :: "'a list" shows "Partial_Cost_Model.config' A s (r # rs) = Partial_Cost_Model.config' A (Partial_Cost_Model.Step A s r) rs"
induceES_yields_ES: fixes SES :: "('a, 'b) SES_rec" assumes "SES_valid SES" shows "ES_valid (induceES SES)"
simps: fixes rects :: "'a rectangle list"   and i :: "nat" shows "del_op rects i = DEL (high (yint (rects ! i))) i (xint (rects ! i))"
fimage_flrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<bullet> A) `\<^sub>\<bullet> B = r `\<^sub>\<bullet> (A \<inter>\<^sub>\<circ> B)"
memberdecl_is_static_fdecl_simp: fixes f :: "fdecl" shows "is_static (fdecl f) = is_static f"
Card_order_ordIso: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Card_order r"   and "r' =o r" shows "Card_order r'"
CK: shows "finite K \<and> (\<forall>i\<in>K. \<forall>j\<in>K. i \<noteq> j \<longrightarrow> C i \<inter> C j = {}) \<and> (\<forall>i\<in>K. C i \<in> measurable_sets M) \<and> \<Union> (C ` K) = UNIV"
in_qtableI: fixes n :: "nat"   and A :: "nat set"   and P :: "'b option list \<Rightarrow> bool"   and Q :: "'b option list \<Rightarrow> bool"   and X :: "'b option list set"   and x :: "'b option list" assumes "qtable n A P Q X"   and "wf_tuple n A x"   and "P x"   and "Q x" shows "x \<in> X"
path_connectedin_path_components_of: fixes C :: "'b set"   and X :: "'b topology" assumes "C \<in> path_components_of X" shows "path_connectedin X C"
lm006: fixes G :: "'a set" assumes "G \<noteq> {}" shows "{G} \<in> all_partitions G"
assms_2: shows "S \<subseteq> \<Union> \<T>"
hyps: shows "mat n n (mat_powI ze on pl ti n m i)"
exe_instr_all: fixes s :: "'b sparc_state" assumes "good_context s" shows "snd (execute_instruction () s) = False"
bound: fixes x :: "'a"   and y :: "'b" shows "h x y \<noteq> \<top>"
span_unit_vecs_is_carrier: fixes n :: "nat" shows "LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (set (unit_vecs n)) = carrier_vec n"
ceq_prod_def: shows "CEQ('c \<times> 'd) \<equiv> case ID CEQ('c) of None \<Rightarrow> None | Some eq_'a \<Rightarrow> case ID CEQ('d) of None \<Rightarrow> None | Some eq_'b \<Rightarrow> Some (equality_prod eq_'a eq_'b)"
zero_compare_simps_7: fixes a :: "'a"   and b :: "'a" shows "((0::'a) < a / b) = ((0::'a) < a \<and> (0::'a) < b \<or> a < (0::'a) \<and> b < (0::'a))"
D_1: shows "\<forall>x\<in>s. (f has_derivative (\<lambda>z. z *\<^sub>R D x)) (at x)"
simps_4: fixes x21 :: "'a"   and x22 :: "('a, 'b) tllist"   and x1 :: "'b" shows "TCons x21 x22 \<noteq> TNil x1"
F_T: fixes s :: "'a event list"   and X :: "'a event set"   and P :: "'a process" assumes "(s, X) \<in> F P" shows "s \<in> T P"
prj_chine_assoc_1: shows "gfg.Prj\<^sub>1 \<cdot> gfg.chine_assoc = gfg.Prj\<^sub>1\<^sub>1"
degree_leI: fixes y :: "'b pdevs"   and x :: "'c pdevs" assumes "\<And>i. pdevs_apply y i = (0::'b) \<Longrightarrow> pdevs_apply x i = (0::'c)" shows "degree x \<le> degree y"
antisymmetric_conv_closed: fixes x :: "'a" assumes "antisymmetric x" shows "antisymmetric (x\<^sup>T)"
SubstForm_subst_dbfm: fixes u :: "dbtm"   and i :: "name"   and e :: "(name, hf) finfun"   and A :: "dbfm" assumes "wf_dbtm u" shows "SubstForm (q_Var i) \<lbrakk>quot_dbtm u\<rbrakk>e \<lbrakk>quot_dbfm A\<rbrakk>e \<lbrakk>quot_dbfm (subst_dbfm u i A)\<rbrakk>e"
bigtheta_trans2': fixes g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "g \<in> \<Theta>[F](f)"   and "g \<in> \<omega>[F](h)" shows "f \<in> \<omega>[F](h)"
less_multiset\<^sub>H\<^sub>O_def: fixes M :: "'b multiset"   and N :: "'b multiset" shows "less_multiset\<^sub>H\<^sub>O M N = (M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x)))"
phi_in_O: fixes V :: "'c set"   and q :: "'d" assumes "is_open\<^sub>Y V"   and "q \<in> \<O>\<^sub>Y V" shows "\<phi>\<^sub>f V q \<in> \<O>\<^sub>X (f \<^sup>\<inverse> X V)"
internal_case_prod_conv: fixes c :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and a :: "'f"   and b :: "'g" shows "??.Product_Type.internal_case_prod c (a, b) = c a b"
heap_upds_okE: fixes \<Gamma> :: "(Vars.var \<times> 'b) list"   and S :: "stack_elem list"   and x :: "Vars.var" assumes "heap_upds_ok (\<Gamma>, S)"   and "x \<in> domA \<Gamma>" shows "x \<notin> upds S"
norm_insert_def: fixes x :: "'a \<times> 'b \<times> 'a"   and xs :: "('a \<times> 'b \<times> 'a) list" shows "norm_insert x xs \<equiv> let (i, x') = norm_fold x xs in if i then (i, xs) else (i, x' # xs)"
skip_r_def: shows "II = \<langle>id\<rangle>\<^sub>a"
almost_full_on_hom: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q (h x) (h y)"   and "almost_full_on P A" shows "almost_full_on Q (h ` A)"
member_set_of_pred: fixes x :: "'a"   and P :: "'a pred" shows "(x \<in> set_of_pred P) = eval P x"
simps_3: fixes x :: "real"   and y :: "real" shows "R x + R y = R (max x y)"
card_univ_subtract: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "mono \<tau>"   and "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subset> (\<tau> ^^ i) UNIV" shows "\<forall>i. card ((\<tau> ^^ i) UNIV) \<le> card UNIV - i"
assms_5: fixes w :: "complex" assumes "w \<in> s"   and "w \<noteq> z" shows "f w = g w * (w - z) powr of_int n"
unreachable_Null: fixes s :: "Store"   and l :: "Location"   and x :: "Value" assumes "s\<turnstile> l reachable_from x" shows "x \<noteq> nullV"
match_ex_table_SomeD2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and D :: "char list"   and pc :: "nat"   and C :: "char list"   and M :: "char list"   and pc' :: "nat"   and d' :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and I :: "nat set"   and d :: "nat" assumes "match_ex_table P D pc (ex_table_of P C M) = \<lfloor>(pc', d')\<rfloor>"   and "P,C,M \<rhd> xt / I,d"   and "\<forall>x\<in>set xt. \<not> matches_ex_entry P D pc x"   and "pc \<in> I" shows "d' \<le> d"
set_empty2: fixes xs :: "'f list" shows "({} = set xs) = (xs = [])"
terminates_append_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and h :: "'g \<Rightarrow> ('f, 'g) step"   and sh_start :: "'g" assumes "terminates g"   and "terminates h" shows "terminates (append_raw g h sh_start)"
simps_1: fixes f :: "ref \<Rightarrow> ref" shows "dag_map f Tip = Tip"
finsert_def: shows "finsert \<equiv> map_fun id (map_fun fset Abs_fset) insert"
semi_complete_onE: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "semi_complete_on r A" obtains "WN_on r A" and "CR_on r A"
negligible_setdiff: fixes T :: "'c set"   and S :: "'c set" assumes "T \<subseteq> S" shows "negligible (T - S)"
respection_of_binary_predicates_and_closures_1: fixes Rel :: "('a \<times> 'a) set"   and Pred :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_respects_binary_pred Rel Pred" shows "rel_respects_binary_pred (Rel\<^sup>=) Pred"
minus_fset_def: shows "(|-|) \<equiv> map_fun fset (map_fun fset Abs_fset) (-)"
mantissa_not_dvd: fixes f :: "float" assumes "f \<noteq> 0" shows "odd (mantissa f)"
Unifier_dom_restrict: fixes t :: "('a, 'b) Term.term"   and s :: "'b \<Rightarrow> ('a, 'b) Term.term"   and t' :: "('a, 'b) Term.term"   and S :: "'b set" assumes "Unifier s t t'"   and "fv t \<union> fv t' \<subseteq> S" shows "Unifier (rm_vars (UNIV - S) s) t t'"
linear': fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "linorder_list le S"   and "a \<in> S"   and "b \<in> S"   and "a \<noteq> b" shows "le a b \<or> le b a"
dickson_leD2: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "dickson_le d m s t" shows "d t \<le> m"
tm_append_first_steps_eq: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and A :: "(action \<times> nat) list"   and off :: "nat"   and stp :: "nat"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and B :: "(action \<times> nat) list" assumes "steps (s, l, r) (A, off) stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps (s, l, r) (A @ B, off) stp = (s', l', r')"
mult_vec_minus_mult_right: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_vec pair_of_term term_of_pair p (- q) = - term_powerprod.mult_vec pair_of_term term_of_pair p q"
sum_wt_Par_sub: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set"   and s :: "'state" assumes "properL cl"   and "n < length cl"   and "I \<subseteq> {..<brn (cl ! n)}" shows "sum (wt (Par cl) s) ((+) (brnL cl n) ` I) = 1 / real (length cl) * sum (wt (cl ! n) s) I"
zero_i2: shows "is_zero_row_upt_k i (mod_type_class.to_nat (mod_type_class.from_nat k)) A"
proj_preserves: fixes x :: "'d"   and A :: "'d set"   and r :: "('d \<times> 'd) set" assumes "x \<in> A" shows "Equiv_Relations.proj r x \<in> A // r"
non_empty_cycle_root_msc_plus: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "x\<^sup>+ ; r = x\<^sup>T\<^sup>+ ; r"
ofilter_linord: fixes r :: "('c \<times> 'c) set"   and A :: "'c set"   and B :: "'c set" assumes "wo_rel r"   and "ofilter r A"   and "ofilter r B" shows "A \<subseteq> B \<or> B \<subseteq> A"
expand: fixes literal :: "'b literal"   and literal' :: "'b literal" assumes "is_pos literal = is_pos literal'"   and "\<lbrakk>is_pos literal; is_pos literal'\<rbrakk> \<Longrightarrow> atm_of literal = atm_of literal'"   and "\<lbrakk>is_neg literal; is_neg literal'\<rbrakk> \<Longrightarrow> atm_of literal = atm_of literal'" shows "literal = literal'"
apex_def: shows "SPN_fgh.\<mu>.apex \<equiv> Maps.dom SPN_fgh.\<mu>.leg0"
rel_compile_fundef: fixes fd :: "('c, 'd, 'e, 'op) Std.instr fundef" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd (compile_fundef fd)"
order_ow_axioms: shows "order_ow U (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w)"
derangements_formula: fixes n :: "nat"   and S :: "nat set" assumes "n \<noteq> 0"   and "finite S"   and "card S = n" shows "int (card (derangements S)) = round (fact n / exp 1)"
primitive_part_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "primitive_part (p * q) = primitive_part p * primitive_part q"
lex: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "f \<in> SEQ A \<or> g \<in> SEQ A"   and "LEX P f g"   and "\<forall>i<k. f i = g i"   and "f k \<noteq> g k" shows "P (f k) (g k)"
construct_rhs_vector_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and Is :: "(nat list \<times> nat list) list" shows "v_vec_R p qs Is = vec_of_list (map (\<lambda>(I1, I2). construct_NofI_R p (retrieve_polys qs I1) (retrieve_polys qs I2)) Is)"
upd_inv_Some_eq: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ta :: "('f, 'i, 'j) new_thread_action"   and t :: "'f"   and x :: "'g \<times> 'h \<Rightarrow>f nat"   and I :: "'f \<Rightarrow> 'k option"   and Q :: "'k \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool" assumes "thread_ok ts ta"   and "ts t = \<lfloor>x\<rfloor>" shows "upd_inv I Q ta t = I t"
under_underS_trans: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "trans r"   and "antisym r"   and "a \<in> under r b"   and "b \<in> underS r c" shows "a \<in> underS r c"
nset_Suc: fixes n :: "nat" shows "nset (Suc 0) (Suc (Suc n)) = nset (Suc 0) (Suc n) \<union> {Suc (Suc n)}"
C_restr_C_cong: fixes r :: "C"   and f :: "C \<rightarrow> 'a"   and g :: "C \<rightarrow> 'a" assumes "\<And>r'. r' \<sqsubseteq> r \<Longrightarrow> f\<cdot>(C\<cdot>r') = g\<cdot>(C\<cdot>r')"   and "f\<cdot>\<bottom> = g\<cdot>\<bottom>" shows "f|\<^bsub>C\<cdot>r\<^esub> = g|\<^bsub>C\<cdot>r\<^esub>"
\<alpha>_def: shows "\<alpha> = (\<lambda>y'. if Var y' \<in> \<delta> ` (subst_domain \<delta> \<inter> subst_domain \<theta>) then Var (inv_into (subst_domain \<delta> \<inter> subst_domain \<theta>) \<delta> (Var y')) else Var y')"
assms_2: fixes v :: "'t" shows "fst (rep_nat_term v) = rep_nat_pp (fst (pair_of_term v))"
isUIP_def: fixes l :: "Literal"   and c :: "Literal list"   and M :: "(Literal \<times> bool) list" shows "isUIP l c M \<equiv> isLastAssertedLiteral (opposite l) (oppositeLiteralList c) (elements M) \<and> (\<forall>l'. l' el c \<and> l' \<noteq> l \<longrightarrow> elementLevel (opposite l') M < elementLevel (opposite l) M)"
induct: fixes P :: "'a poly \<Rightarrow> bool"   and a0 :: "'a poly" assumes "\<And>c. P (c)\<^sub>p"   and "\<And>c n p. \<lbrakk>P c; P p\<rbrakk> \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (poly.Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
subprob_measure_le_1: fixes p :: "'d spmf"   and X :: "'d set" shows "Sigma_Algebra.measure (measure_spmf p) X \<le> 1"
simps_2: fixes p :: "('Var, 'AExp, 'BExp) Stmt"   and q :: "('Var, 'AExp, 'BExp) Stmt" shows "has_annotated_stop (p ;; q) = (has_annotated_stop p \<or> has_annotated_stop q)"
disj: fixes a :: "'c"   and a' :: "'c" assumes "a \<in> A"   and "a' \<in> A"   and "a \<noteq> a'" shows "B a \<inter> B a' = {}"
put_env_def: fixes put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'c"   and m :: "('a, 'b) envT" shows "put_env put s m = EnvT (\<lambda>r. put s (run_env m r))"
permutes_inv_inv: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "inv (inv p) = p"
mat_multI_def: fixes ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and nr :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list" shows "mat_multI ze pl ti nr m1 m2 \<equiv> map (matT_vec_multI ze pl ti (Matrix_Legacy.transpose nr m1)) m2"
preservation: shows "rel_preserves_binary_pred (indRelRTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
scale_right_distrib_on: fixes x :: "'b"   and y :: "'b"   and a :: "'a" assumes "x \<in> U\<^sub>M"   and "y \<in> U\<^sub>M" shows "a *s (x + y) = a *s x + a *s y"
distinct_17: fixes x3 :: "int"   and x5 :: "loc" shows "Intg x3 \<noteq> Addr x5"
span_mono: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "Modules.module scale"   and "A \<subseteq> B" shows "module.span scale A \<subseteq> module.span scale B"
intros_5: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F \<^bold>\<and> G, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
delete_Some_set_of: fixes x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "delete x t = Some t'" shows "set_of t' \<subseteq> set_of t"
Mf_def: fixes cfs :: "int \<times> int poly multiset" shows "q.Mf cfs \<equiv> case cfs of (c, fs) \<Rightarrow> (q.M c, image_mset q.Mp fs)"
indecomposable_ordertype_ge: fixes \<alpha> :: "V"   and A :: "V set"   and B :: "V set" assumes "indecomposable \<alpha>"   and "\<alpha> \<le> ordertype A VWF"   and "small A"   and "small B" shows "\<alpha> \<le> ordertype B VWF \<or> \<alpha> \<le> ordertype (A - B) VWF"
error_free_abrupt_if_Xcpt1: fixes a :: "abopt"   and s :: "st"   and p :: "bool"   and x :: "xcpt" assumes "error_free (a, s)" shows "error_free (abrupt_if p (Some (Xcpt x)) a, s)"
Rep_upset: fixes x :: "'a upset" shows "Rep_upset x \<in> range \<up>"
Shift_def: fixes Kl :: "'a list set"   and k :: "'a" shows "BNF_Greatest_Fixpoint.Shift Kl k = {kl. k # kl \<in> Kl}"
finite_multisets_of_size: fixes A :: "'a set"   and n :: "nat" assumes "finite A" shows "finite {X. set_mset X \<subseteq> A \<and> size X = n}"
holomorphic_on_imp_differentiable_at: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and x :: "complex" assumes "f holomorphic_on s"   and "open s"   and "x \<in> s" shows "f field_differentiable at x"
less_TC_imp_not_le: fixes x :: "V"   and a :: "V" assumes "x \<sqsubset> a" shows "\<not> a \<le> x"
simps_4: fixes \<phi> :: "'c fm" shows "qfree (Neg \<phi>) = qfree \<phi>"
ahm_delete_correct_2: fixes hm :: "('a, 'b) ArrayHashMap_Impl.hashmap"   and k :: "'a" assumes "ArrayHashMap_Impl.ahm_invar hm" shows "ArrayHashMap_Impl.ahm_\<alpha> (ArrayHashMap_Impl.ahm_delete k hm) = ArrayHashMap_Impl.ahm_\<alpha> hm |` (- {k})"
mult_sign_intros_1: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(0::'a) \<le> a \<cdot> b"
plus_hf_def: fixes A :: "hf"   and B :: "hf" shows "A + B = \<lbrace>0\<rbrace> * A \<squnion> \<lbrace>1\<rbrace> * B"
wt_jvm_prog_impl_wt_start: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C" shows "0 < length ins \<and> wt_start P C b Ts mxl\<^sub>0 (\<Phi> C M)"
imp_igSwapBinpIPresIGWlsBinpSTR: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" assumes "igSwapAbsIPresIGWlsAbsSTR MOD" shows "igSwapBinpIPresIGWlsBinpSTR MOD"
i_last_message_hold_i_take: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and k :: "nat" shows "f \<Down> n \<longmapsto> k = f \<longmapsto> k \<Down> n"
mult_locally_cancelL: fixes s :: "('b \<times> 'b) set"   and X :: "'b multiset"   and Z :: "'b multiset"   and Y :: "'b multiset" assumes "trans s"   and "locally_irrefl s (X + Z)"   and "locally_irrefl s (Y + Z)" shows "((Z + X, Z + Y) \<in> mult s) = ((X, Y) \<in> mult s)"
simps_1: fixes n :: "int" shows "int_option_enumeration (Some n) = (if 0 \<le> n then nat (2 * (n + 1)) else nat (2 * - n + 1))"
existence_ivl_subsetI: fixes Y :: "'a set"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "Y = Y"   and "S = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> g t x = g t x"   and "t \<in> Y.existence_ivl t0 x0" shows "t \<in> ll_on_open.existence_ivl S g Y t0 x0"
D_props: shows "\<forall>x\<in>{0..1} - s \<inter> {0..1}. (\<phi> has_vector_derivative D x) (at x)"
cm_enorm: shows "continuous_map (Euclidean_space n) euclideanreal enorm"