hcju/isabelleps huggingface dataset:
Rule_UndecidedE: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and a :: "action" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided" obtains "\<not> matches \<gamma> m p"   | "a = Log \<or> a = Empty"   | c :: "char list" where "a = Call c" and "matches \<gamma> m p"
R': fixes e :: "real" assumes "0 < e" shows "\<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e"
HFail_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HFail s s' p" shows "HInv5_inner s' p"
tsnd: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and P :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and \<sigma> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> P : TPair \<tau> \<sigma>" shows "\<Gamma> \<turnstile> Snd P : \<sigma>"
toVariable_variable_inv': fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and v :: "varDecl"   and a :: "String.literal"   and b :: "variable"   and c :: "channel list" assumes "gState_inv prog g"   and "toVariable g p v = (a, b, c)" shows "variable_inv b"
line_integral_around_boundary: shows "one_chain_line_integral F {i} (boundary twoC) = line_integral F {i} (\<lambda>x. twoC (x, 0)) + line_integral F {i} (\<lambda>y. twoC (1, y)) - line_integral F {i} (\<lambda>x. twoC (x, 1)) - line_integral F {i} (\<lambda>y. twoC (0, y))"
empt: fixes x :: "'a" assumes "isNode x" shows "\<not> memb x empt"
self_le_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq one a"   and "0 < n" shows "less_eq a (power.power one times a n)"
distinct_295: fixes x151 :: "floatarith"   and x152 :: "nat"   and x16 :: "floatarith" shows "Power x151 x152 \<noteq> Floor x16"
ipurge_tr_all: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list" shows "(ipurge_tr I D u xs = xs) = (\<not> (\<exists>v\<in>D ` set xs. (u, v) \<in> I))"
S_def: shows "S \<equiv> \<Sum>i\<in>{a<..b}. f (real_of_int i)"
fds_subseries_def: fixes P :: "nat \<Rightarrow> bool"   and f :: "'b fds" shows "fds_subseries P f = fds (\<lambda>n. if P n then fds_nth f n else (0::'b))"
possafety_closed_iff: fixes P :: "'a llist set" assumes "P \<subseteq> A\<^sup>\<spadesuit>" shows "possafety A P = (P closed)"
ct_prefix_snoc: fixes xs :: "ct list"   and ys :: "ct list"   and y :: "ct" shows "ct_prefix xs (ys @ [y]) = (ct_list_eq xs (ys @ [y]) \<or> ct_prefix xs ys)"
erlang_distributed_gt: fixes X :: "'a \<Rightarrow> real"   and k :: "nat"   and l :: "real"   and a :: "real" assumes "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))"   and "0 < l"   and "0 \<le> a" shows "prob {x \<in> space M. a < X x} = 1 - erlang_CDF k l a"
hyps_2: shows "P \<turnstile> (C, Cs) has overrider M = mthd via Cs'"
assms_4: fixes r1 :: "'a"   and r2 :: "'a" assumes "r1 \<in> carrier R"   and "r2 \<in> carrier R" shows "P (r1 \<oplus> r2)"
diff_cancel: fixes a :: "'d" shows "a - a = (0::'d)"
run_snth: fixes r :: "'c stream"   and p :: "'a"   and k :: "nat" assumes "run r p" shows "r !! k \<in> en (target (stake k r) p)"
shift1_invI: fixes B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and d :: "nat" assumes "B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}"   and "shift2_inv d qs" shows "shift1_inv d (qs, B)"
filter_mono: fixes F1 :: "'a filter"   and F2 :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "F1 \<le> F2"   and "f \<in> O[F2](g)" shows "f \<in> O[F1](g)"
finite_set_mset: fixes M :: "'c multiset" shows "finite (set_mset M)"
tail_neq_head: shows "tail G a \<noteq> head G a"
card_Suc_k: shows "card {i. i < dim_row A \<and> i \<in> insert a I} = Suc k"
to': shows "h' \<turnstile> to_tree_order (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr) \<rightarrow>\<^sub>r to'"
col_transpose: fixes i :: "nat"   and A :: "'a mat" assumes "i < dim_row A" shows "col A\<^sup>t i = row A i"
to_int_mod_ring_of_int_M: fixes m :: "int"   and x :: "int" assumes "poly_mod_type TYPE('b) m" shows "to_int_mod_ring (of_int x) = poly_mod.M m x"
simps_4: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and v :: "'v node" shows "lift f (v, SOURCE) = 0"
the_inv_into_f_im_f_im: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b set" assumes "inj_on f A"   and "order.greater_eq A x" shows "the_inv_into A f ` f ` x = x"
assms_4: shows "\<FF> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> = dghm_id \<BB>"
Ex_3: fixes b :: "name list"   and ba :: "fm" shows "(trans_fm b fm = trans_fm b ba) = (fm = ba)"
nats_of_boolss_zeros: fixes n :: "nat"   and m :: "nat" shows "nats_of_boolss n (zeros m n) = replicate n 0"
insertl_len: fixes n :: "nat"   and x :: "'a"   and vs :: "'a list" shows "length (insertl n x vs) = Suc (length vs)"
norm_power_ineq: fixes x :: "'c"   and n :: "nat" shows "norm (x ^ n) \<le> norm x ^ n"
lfilter_eq_LNil: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "(lfilter P xs = []\<^sub>l) = (\<forall>x\<in>lset xs. \<not> P x)"
outer_measure_of_mono: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure" assumes "A \<subseteq> B" shows "outer_measure_of M A \<le> outer_measure_of M B"
hom_componentsI: fixes q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "q = hom_component p (deg_pm t)"   and "t \<in> keys p" shows "q \<in> hom_components p"
sp_upred_def: fixes p :: "'c upred"   and Q :: "('c, 'b) urel" shows "p sp Q = \<lfloor>\<lceil>p\<rceil>\<^sub>> ;; Q\<rfloor>\<^sub>>"
block_sizes_non_empty_set: assumes "\<B> \<noteq> {#}" shows "sys_block_sizes \<noteq> {}"
lg_exists: fixes A :: "'p"   and B :: "'p" shows "\<exists>l. QCong l \<and> l A B"
DEFL_eq_btree: shows "DEFL('a\<cdot>btree) = DEFL('a\<^sub>\<bottom> \<oplus> ('a\<cdot>btree)\<^sub>\<bottom> \<otimes> ('a\<cdot>btree)\<^sub>\<bottom>)"
map_snd_sorted_lesseq: fixes xs :: "('b \<times> 'a) list"   and i :: "nat"   and j :: "nat" assumes "sorted_less (separators xs)"   and "i \<le> j"   and "j < length xs" shows "snd (xs ! i) \<le> snd (xs ! j)"
assms_2: shows "cs \<noteq> [] \<or> es \<noteq> []"
DA_is_deny: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set" assumes "x \<in> dom (C (DenyAllFromTo a b \<oplus> DenyAllFromTo b a \<oplus> DenyAllFromTo a b))" shows "C (DenyAllFromTo a b \<oplus> DenyAllFromTo b a \<oplus> DenyAllFromTo a b) x = \<lfloor>deny ()\<rfloor>"
clop_idem: shows "cl_op \<circ> cl_op = cl_op"
nth_Cons': fixes x :: "'g"   and xs :: "'g list"   and n :: "nat" shows "(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))"
X_mean: fixes n :: "nat" shows "expectation (X n) = m"
balR_def: fixes A :: "('b \<times> nat) tree"   and a :: "'b"   and BC :: "('b \<times> nat) tree" shows "balR A a BC = (if ht BC = ht A + m + 1 then case BC of \<langle>B, (b, x), C\<rangle> \<Rightarrow> if ht B \<le> ht C then node (node A a B) b C else case B of \<langle>B\<^sub>1, (ab, x), B\<^sub>2\<rangle> \<Rightarrow> node (node A a B\<^sub>1) ab (node B\<^sub>2 b C) else node A a BC)"
monomial_is_Zp_continuous: fixes c :: "nat \<Rightarrow> int"   and n :: "nat" assumes "c \<in> carrier Zp" shows "is_Zp_continuous (monomial c n)"
Image_iff: fixes b :: "'h"   and r :: "('i \<times> 'h) set"   and A :: "'i set" shows "(b \<in> r `` A) = (\<exists>x\<in>A. (x, b) \<in> r)"
linorder_neqE_linordered_idom: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "x \<noteq> y" obtains "less x y"   | "less y x"
vimage_inter_cong: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and y :: "'b set" assumes "\<And>w. w \<in> S \<Longrightarrow> f w = g w" shows "f -` y \<inter> S = g -` y \<inter> S"
finite_ta_def: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)" shows "finite_ta A \<equiv> finite (clkp_set A) \<and> finite (collect_clkvt (trans_of A)) \<and> (\<forall>(uu_, m)\<in>clkp_set A. m \<in> \<nat>) \<and> clk_set A \<noteq> {} \<and> - clk_set A \<noteq> {}"
normal_range_eq: fixes F :: "ordinal \<Rightarrow> ordinal"   and G :: "ordinal \<Rightarrow> ordinal" assumes "normal F"   and "normal G"   and "range F = range G" shows "F = G"
swap_self: fixes a :: "'a"   and b :: "'a" shows "\<langle>a \<leftrightarrow> b\<rangle> * \<langle>a \<leftrightarrow> b\<rangle> = 1"
uniformly_convergent_eq_cauchy: fixes P :: "'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" shows "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e) = (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e)"
V\<^sub>2: shows "\<V> f\<^sub>2 = {u, v} \<union> set (between (vertices f) v u) \<union> set vs"
unique_connecting_path_set_1: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "v \<in> set (v \<leadsto> w)"
map_poly_monom: fixes f :: "'b \<Rightarrow> 'a"   and c :: "'b"   and n :: "nat" assumes "f (0::'b) = (0::'a)" shows "map_poly f (monom c n) = monom (f c) n"
dghm_id_ObjMap_vdomain: fixes \<CC> :: "V" shows "\<D>\<^sub>\<circ> (dghm_id \<CC>\<lparr>ObjMap\<rparr>) = \<CC>\<lparr>Obj\<rparr>"
hyps_for_simp: fixes h :: "'outPort"   and n :: "'node"   and p :: "'inPort" shows "(h |\<in>| hyps_for n p) = (hyps n h = Some p)"
HT_strengthen_modset: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT \<pi> P c Q" shows "HT \<pi> P c (\<lambda>s\<^sub>0 s. Q s\<^sub>0 s \<and> modifies (lhsv \<pi> c) s s\<^sub>0)"
inj_on_span_iff_independent_image: fixes B :: "'b set" assumes "vs2.independent (f ` B)" shows "inj_on f (vs1.span B) = inj_on f B"
num_eq_card_orbit: fixes P :: "'a set" assumes "P \<in> subgroups_of_size (p ^ a)" shows "subgroups_of_size (p ^ a) = conj.orbit P"
NullT_leaf_abs: fixes C :: "ATypeId" shows "NullT \<le> AClassT C"
less_eq_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "(x \<le> y) = (\<forall>i. x $ i \<le> y $ i)"
word_of_nat_le: fixes n :: "nat"   and x :: "'a word" assumes "n \<le> unat x" shows "word_of_nat n \<le> x"
conf_Null: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'b"   and T :: "ty" shows "heap_base.conf typeof_addr P h Null T = P \<turnstile> NT \<le> T"
f1: shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (s' \<otimes> r \<oplus> s \<otimes> r' |\<^bsub>rel\<^esub> s \<otimes> s')"
id: shows "factors_of_factor_tree (product_factor_tree p (Factor_Node i l r)) = factors_of_factor_tree (Factor_Node i l r)"
Heap_Merge_def: fixes merge :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> 'b tree" shows "Heap_Merge merge \<equiv> (\<forall>q1 q2. heap q1 \<longrightarrow> heap q2 \<longrightarrow> mset_tree (merge q1 q2) = mset_tree q1 + mset_tree q2) \<and> (\<forall>q1 q2. heap q1 \<longrightarrow> heap q2 \<longrightarrow> heap (merge q1 q2))"
matrix_to_iarray_interchange_rows: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c" shows "matrix_to_iarray (interchange_rows A i j) = interchange_rows_iarray (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j)"
ball_UNIV: fixes P :: "'a \<Rightarrow> bool" shows "Ball UNIV P = All P"
bdd_below_def: fixes A :: "'c multiset set" shows "subset_mset.bdd_below A = (\<exists>m. \<forall>x\<in>A. m \<subseteq># x)"
abs_uexpr_def: fixes u :: "('c, 'd) uexpr" shows "\<bar>u\<bar> = uop abs u"
exec_seq: shows "\<Gamma>\<turnstile> \<langle>sequence Seq (flatten c),s\<rangle> \<Rightarrow> t"
cases: fixes x :: "'b tree" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>l, xa, r\<rangle>"
intro: fixes sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" assumes "\<And>D f \<theta>. \<lbrakk>sop = (D, f); D \<subseteq> dom \<theta>\<rbrakk> \<Longrightarrow> f \<theta> = f (\<theta> |` D)" shows "valid_sop sop"
f_range_onD': fixes c :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and A :: "'a set" assumes "c \<in> f B"   and "f_range_on A f"   and "B \<subseteq> A" shows "c \<in> B"
inj_on_inv_into: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "B \<subseteq> f ` A" shows "inj_on (inv_into A f) B"
dist_nz: fixes x :: "'b"   and y :: "'b" shows "(x \<noteq> y) = (0 < dist x y)"
not_essential_inverse: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "not_essential f z"   and "isolated_singularity_at f z" shows "not_essential (\<lambda>w. inverse (f w)) z"
simps_1: fixes uu :: "'b \<times> 'b"   and uv :: "'b" shows "sd_path uu uv [] = []"
finprod_infinite: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "infinite A" shows "finprod G f A = \<one>"
openin_open: fixes U :: "'c set"   and S :: "'c set" shows "openin (top_of_set U) S = (\<exists>T. open T \<and> S = U \<inter> T)"
prop_atoms_nested_prop_atoms: fixes \<phi> :: "'a ltln" shows "prop_atoms \<phi> \<subseteq> nested_prop_atoms \<phi>"
iii_def: shows "iii = hcomplex_of_complex \<i>"
wt_App_ge_fun: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s \<le> kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym (App s t)"
assms_5: fixes e' :: "pexpr" assumes "e = PExpr1 (PNeg e')" shows "P"
is_constant_seq_imp_closed: fixes R :: "('a, 'b) ring_scheme"   and s :: "nat \<Rightarrow> 'a" assumes "is_constant_seq R s" shows "s \<in> carrier R\<^bsup>\<omega>\<^esup>"
in_carrier_def: fixes xs :: "'c list" shows "cring_class.in_carrier xs = (\<forall>x\<in>set xs. x \<in> carrier cring_class_ops)"
abs_eq: fixes y :: "'a \<Rightarrow> 'b option" shows "Mapping.lookup (Mapping y) = y"
lim_proc_is_lub3b: fixes X :: "'a process set"   and u :: "'a process"   and xa :: "'a process"   and x :: "'a event list" assumes "\<forall>x. x \<in> X \<longrightarrow> (\<forall>xa. xa \<in> D x \<and> (\<forall>t. t \<in> D x \<longrightarrow> \<not> t < xa) \<longrightarrow> xa \<in> T u)"   and "xa \<in> X"   and "\<forall>xa. xa \<in> X \<longrightarrow> x \<in> D xa"   and "\<forall>t. (\<forall>x. x \<in> X \<longrightarrow> t \<in> D x) \<longrightarrow> \<not> t < x" shows "x \<in> T u"
trms_proj\<^sub>l\<^sub>s\<^sub>t_append: fixes l :: "'f"   and A :: "('f strand_label \<times> ('d, 'e) strand_step) list"   and B :: "('f strand_label \<times> ('d, 'e) strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>t (proj l (A @ B)) = trms\<^sub>l\<^sub>s\<^sub>t (proj l A) \<union> trms\<^sub>l\<^sub>s\<^sub>t (proj l B)"
vrange_vfsequence_of_vlist: fixes xs :: "V list" shows "\<R>\<^sub>\<circ> (vfsequence_of_vlist xs) = set (list.set xs)"
inj: shows "inj_on (\<lambda>x. x + a) S"
simps_1: fixes cmp :: "'b comparator"   and ys :: "'b list" shows "Sorting_Algorithms.merge cmp [] ys = ys"
seqE: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>\<^sub>SC.seq g f" obtains "\<F>\<^sub>SC.arr f" and "\<F>\<^sub>SC.arr g" and "\<F>\<^sub>SC.dom g = \<F>\<^sub>SC.cod f"
emeasure_eq_sum_singleton: fixes S :: "'b set"   and M :: "'b measure" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
distinct_30: fixes ccs' :: "ccs"   and name :: "name"   and ccs :: "ccs" shows "!ccs' \<noteq> \<lparr>\<nu>name\<rparr>ccs"
that: fixes c :: "'a" assumes "poly p (x - y) = poly p x - c * y" shows "thesis"
image_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and r :: "'e set" shows "\<P> f (\<P> g r) = \<P> (f \<circ> g) r"
eq_inf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "traces.eq_inf ind w\<^sub>1 w\<^sub>2"   and "traces.eq_inf ind w\<^sub>2 w\<^sub>3" shows "traces.eq_inf ind w\<^sub>1 w\<^sub>3"
conv_abscissa_diff_le: fixes f :: "'a fds"   and g :: "'a fds" shows "conv_abscissa (f - g) \<le> max (conv_abscissa f) (conv_abscissa g)"
degree_mult_eq: fixes p :: "'a poly"   and q :: "'a poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "degree (p * q) = degree p + degree q"
nn_integral_add: fixes f :: "'k \<Rightarrow> ennreal"   and M :: "'k measure"   and g :: "'k \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x + g x \<partial>M = integral\<^sup>N M f + integral\<^sup>N M g"
distributedI_borel_atMost: fixes X :: "'b \<Rightarrow> real"   and M :: "'b measure"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "X \<in> borel_measurable M"   and "f \<in> borel_measurable borel"   and "AE x in lborel. 0 \<le> f x"   and "\<And>a. \<integral>\<^sup>+ x. ennreal (f x * indicat_real {..a} x) \<partial>lborel = ennreal (g a)"   and "\<And>a. emeasure M {x \<in> space M. X x \<le> a} = ennreal (g a)" shows "distributed M lborel X (\<lambda>x. ennreal (f x))"
induced_map_stabilizes_A: fixes x :: "'a"   and A :: "'a set" shows "(x \<in> A) = (induced_map A x \<in> A)"
nth_rewrite_neq: fixes i :: "nat"   and j :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i \<noteq> j" shows "rewrite xs i x ! j = xs ! j"
sfis_def: shows "sfis \<equiv> \<lambda>f M. {x. sfisp f M x}"
simp_eq_emptyset: fixes X :: "'a set" shows "(X = {}) = (\<forall>x. x \<notin> X)"
monad_fail_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and fail :: "'c" assumes "monad_fail return bind fail" shows "monad_fail (return_state return) (bind_state bind) (fail_state fail)"
def_lfp_unfold: fixes h :: "'b::complete_lattice"   and f :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice" assumes "h::'b::complete_lattice \<equiv> lfp (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)"   and "mono (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)" shows "(h::'b::complete_lattice) = (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) h"
simps_2: fixes t' :: "'b"   and n :: "nat"   and t :: "'b" shows "release_all \<lfloor>(t', n)\<rfloor> t = (if t = t' then None else \<lfloor>(t', n)\<rfloor>)"
sqrt_Sup_power2_eq_Sup_abs: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "finite A"   and "A \<noteq> {}" shows "sqrt (Sup {(f i)\<^sup>2 |i. i \<in> A}) = Sup {\<bar>f i\<bar> |i. i \<in> A}"
get_shadow_root_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "ShadowRootClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_shadow_root element_ptr"
fold_coeffs_pCons_not_0_0_eq: fixes p :: "'b poly"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b" assumes "p \<noteq> 0" shows "fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
abs_ereal_zero: shows "\<bar>0\<bar> = 0"
closed_segment_neq_empty: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b \<noteq> {}"
not_less: fixes x :: "'a"   and y :: "'a" shows "(\<not> lt x y) = le y x"
comp_mult: fixes i :: "'e"   and I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and x :: "'e \<Rightarrow> 'f"   and y :: "'e \<Rightarrow> 'f" assumes "i \<in> I" shows "(x \<otimes>\<^bsub>DirProds G I\<^esub> y) i = x i \<otimes>\<^bsub>G i\<^esub> y i"
rawpsubstT_same_Var: fixes r :: "'trm"   and xs :: "'var list" assumes "r \<in> atrm"   and "set xs \<subseteq> var" shows "rawpsubstT r (map (\<lambda>x. (Var x, x)) xs) = r"
intros: fixes l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "distinct l"   and "rec_join l = \<gamma>"   and "valid_chain_list l" shows "chain_subdiv_path \<gamma> (set l)"
that: fixes x :: "'a" assumes "x \<in> A" shows "thesis"
uniformly_Cauchy_eval_fds_aux: fixes f :: "'a fds"   and s0 :: "'a"   and B :: "'a set" assumes "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)"   and "compact B"   and "\<And>z. z \<in> B \<Longrightarrow> s0 \<bullet> (1::'a) < z \<bullet> (1::'a)" shows "uniformly_Cauchy_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
opF: fixes T :: "'a set" assumes "T \<in> \<C>" shows "open (F T)"
atUV_union: fixes U :: "'b set"   and V :: "'b set"   and c1 :: "'b list multiset"   and c2 :: "'b list multiset" shows "atUV U V (c1 + c2) = (atUV U V c1 \<or> atUV U V c2 \<or> atU U c1 \<and> atU V c2 \<or> atU V c1 \<and> atU U c2)"
C_def: fixes n :: "nat" shows "C n = avg (A n) (B n)"
simps_9: fixes x4 :: "32 word" shows "Intg x4 \<noteq> Unit"