hcju/isabelleps huggingface dataset:
C_eq_RAD: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "p \<noteq> []" shows "C (list2FWpolicy p) = C (list2FWpolicy (removeAllDuplicates p))"
backward_terminating_path_root_aux: fixes r :: "'a"   and x :: "'a" assumes "backward_terminating_path_root r x" shows "x \<le> x\<^sup>T\<^sup>\<star> ; - (x\<^sup>T ; 1)"
preserves_arr: fixes f :: "'d \<times> 'c" assumes "DopxC.arr f" shows "S.arr (Hom_DopxG.map f)"
select_result_I2: fixes h :: "'aq"   and f :: "('aq, 'ar, 'as) prog"   and x :: "'as" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = x"
hyps_5: fixes b :: "(name \<times> trm) list" assumes "b closes \<Gamma>" shows "b
 \<in> RED (T \<sigma>)"
bisim1Sync7: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and a' :: "'addr"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (insync\<^bsub>V\<^esub> (a) Throw a', xs) \<leftrightarrow> ([Addr a'], xs, 6 + length (compE2 e1) + length (compE2 e2), None)"
insert_Diff_if: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" shows "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
f: fixes i :: "nat"   and c :: "bool"   and x :: "'a"   and y :: "nat" shows "fst (f i c x) = (c \<or> g y (snd (f i c x)) < g y x)"
mat_of_rows_carrier_1: fixes n :: "nat"   and vs :: "'a vec list" shows "mat_of_rows n vs \<in> carrier_mat (length vs) n"
FVar: fixes G :: "prog"   and s0 :: "st"   and statDeclC :: "qtname"   and s1 :: "state"   and e :: "expr"   and a :: "val"   and s2 :: "state"   and v :: "vvar"   and s2' :: "state"   and stat :: "bool"   and fn :: "vname"   and s3 :: "state"   and accC :: "qtname" assumes "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2"   and "(v, s2') = fvar statDeclC stat fn a s2"   and "s3 = check_field_access G accC statDeclC fn stat a s2'" shows "G\<turnstile>Norm s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>v\<rightarrow> s3"
binomial_ineq_1: fixes n :: "nat"   and i :: "nat"   and k :: "nat" shows "2 * (n + i choose k) \<le> n choose k + (n + 2 * i choose k)"
26: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<squnion> y) = x \<squnion> y"
snocs_def: fixes xs :: "'b list"   and ys :: "'b list" shows "snocs xs ys = xs @ ys"
intro: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'd set"   and \<phi>\<^sub>f :: "'d set set \<Rightarrow> ('d set \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> 'f" assumes "comm_ring R addition multiplication zero unit"   and "locally_ringed_space X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str"   and "iso_locally_ringed_spaces X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str (comm_ring.spectrum R addition multiplication zero unit) (comm_ring.is_zariski_open R addition multiplication zero unit) (comm_ring.sheaf_spec R addition multiplication zero unit) (comm_ring.sheaf_spec_morphisms R addition multiplication zero unit) comm_ring.\<O>b (comm_ring.add_sheaf_spec R addition multiplication zero) (comm_ring.mult_sheaf_spec R addition multiplication zero) (comm_ring.zero_sheaf_spec R addition multiplication zero unit) (comm_ring.one_sheaf_spec R addition multiplication zero unit) f \<phi>\<^sub>f" shows "affine_scheme R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str f \<phi>\<^sub>f"
prime_odd_int: fixes p :: "int" assumes "prime p"   and "2 < p" shows "odd p"
ffun_minus_zero: fixes f :: "('a, 'b) ffun" shows "{}\<^sub>f - f = {}\<^sub>f"
rel_alt: shows "rel = (\<Union>(t, rel)\<in>set auxlist'. if left I \<le> \<tau> \<sigma> ne - t then rel else empty_table)"
F2_def: shows "F2 = (\<lambda>x. F x + \<bar>u s x / real s\<bar>)"
rtranclp_eq_rtranclp': fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<lambda>x y. P x y \<and> x \<noteq> y)\<^sup>*\<^sup>* = P\<^sup>*\<^sup>*"
float_defs_7: fixes x :: "('a, 'b) IEEE.float" shows "valof x = (if IEEE.exponent x = 0 then (- 1) ^ sign x * (2 / 2 ^ bias TYPE(('a, 'b) IEEE.float)) * (real (fraction x) / 2 ^ LENGTH('b)) else (- 1) ^ sign x * (2 ^ IEEE.exponent x / 2 ^ bias TYPE(('a, 'b) IEEE.float)) * (1 + real (fraction x) / 2 ^ LENGTH('b)))"
constant_rep_designI: fixes \<r> :: "int" assumes "\<And>x. x \<in> \<V> \<Longrightarrow> \<B> rep x = \<r>" shows "constant_rep_design \<V> \<B> \<r>"
P_least: shows "P (LEAST y. \<exists>x\<in>A. y = f x)"
tern_rel_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "heap_sep_algebra.tern_rel plus sep_disj a b c \<equiv> sep_disj a b \<and> plus a b = c"
assms_1: fixes i :: "'a" assumes "i \<in> Basis" shows "a \<bullet> i < b \<bullet> i"
halfspace_simps_3: fixes sctn :: "'c sctn" shows "below_halfspace sctn = Collect (le_halfspace sctn)"
nprod_memTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>i\<le>n. f i \<in> carrier R) \<longrightarrow> (\<forall>l\<le>n. e\<Pi>\<^bsub>R,l\<^esub> f \<in> carrier R)"
simps: fixes p :: "linear_poly" shows "lpoly_to_vec p = vec (dim_poly p) (Abstract_Linear_Poly.coeff p)"
CallRedsFinal: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es :: "char list exp list"   and vs :: "val list"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and l\<^sub>2' :: "char list \<Rightarrow> val option"   and ef :: "char list exp"   and h\<^sub>3 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>3 :: "char list \<Rightarrow> val option" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "l\<^sub>2' = [this \<mapsto> Addr a, pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2')\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3)\<rangle>"   and "final ef" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2)\<rangle>"
pseudonatural_transformation_whisker_right_def: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>F :: "'d \<times> 'd \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'d \<Rightarrow> 'e"   and \<tau>\<^sub>1 :: "'d \<Rightarrow> 'e" shows "pseudonatural_transformation_whisker_right V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B 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>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<and> bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> (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>\<^sub>F \<and> 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 G \<Phi>\<^sub>G) \<and> pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C H \<Phi>\<^sub>H \<and> pseudonatural_transformation 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>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
parallel2_wiring_simps: fixes f :: "'h \<Rightarrow> 'j"   and g :: "'l \<Rightarrow> 'n"   and f' :: "'i \<Rightarrow> 'k"   and g' :: "'m \<Rightarrow> 'o" shows "(f, g) |\<^sub>w (f', g') = (map_sum f f', map_sum g g')"
preserves_seq: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "A.seq (F g) (F f)"
image_bind: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'd set" shows "f ` (A \<bind> g) = A \<bind> (`) f \<circ> g"
finsum_Un_disjoint: fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier V"   and "g \<in> B \<rightarrow> carrier V" shows "finsum V g (A \<union> B) = finsum V g A \<oplus>\<^bsub>V\<^esub> finsum V g B"
U: shows "splits_at \<alpha> i a1 U a2 \<and> splits_at \<gamma> j b1 U b2 \<and> n \<le> length D \<and> LeftDerivation a1 (take n D) b1 \<and> derivation_ge (drop n D) (Suc (length b1)) \<and> LeftDerivation a2 (derivation_shift (drop n D) (Suc (length b1)) 0) b2 \<and> (n = length D \<or> n < length D \<and> is_word (b1 @ [U]))"
brc_invar_final: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist" assumes "rqrm_invar rqrm"   and "rqrm_prop (ls.\<alpha> \<delta>) rqrm"   and "ls.invar \<delta>" shows "\<forall>sc. sc \<in> wa_invar (det_wa_wa (brc_det_algo rqrm \<delta>)) \<and> sc \<notin> wa_cond (det_wa_wa (brc_det_algo rqrm \<delta>)) \<longrightarrow> hs.\<alpha> (fst sc) = b_accessible (ls.\<alpha> \<delta>)"
assms_2: shows "\<forall>rsg\<in>ran \<Gamma> \<union> {[Rule m a]}. wf_chain \<Gamma> rsg"
fps_divide_def: fixes f :: "'a fps"   and g :: "'a fps" shows "f / g = fps_shift (subdegree g) (f * inverse (unit_factor g))"
int_pow_neg: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier G" shows "x [^] - i = inv (x [^] i)"
stateful_strand_step_subst_comp: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and x :: "('d, 'c) stateful_strand_step"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "range_vars \<delta> \<inter> set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p x) = {}" shows "x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<circ>\<^sub>s \<theta> = (x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>"
distinct_1: fixes x1 :: "'a"   and x21 :: "'a"   and x22 :: "('a, 'b) Expression"   and x23 :: "'a" shows "\<turnstile> x1 Type \<noteq> Vx : x21 \<turnstile> x22 : x23"
E: shows "Min {k' \<in> {k..<length cfs}. \<not> (size P \<le> fst (cfs ! k') \<and> fst (cfs ! k') - size P < size (P' x))} \<in> {k' \<in> {k..<length cfs}. \<not> (size P \<le> fst (cfs ! k') \<and> fst (cfs ! k') - size P < size (P' x))}"
full_component: fixes x :: "'a"   and i :: "nat"   and t :: "(nat, nat) pp" assumes "snd (rep_nat_term x) = i" shows "\<exists>y. rep_nat_term y = (t, i)"
in_w_valuesD: fixes v :: "'a val"   and P :: "'c prog"   and vs0 :: "'a \<times> addr_loc \<Rightarrow> 'a val set"   and obs :: "('a, 'b) obs_event action list"   and ad :: "'a"   and al :: "addr_loc" assumes "v \<in> w_values P vs0 obs (ad, al)"   and "v \<notin> vs0 (ad, al)" shows "\<exists>obs' wa obs''. obs = obs' @ wa # obs'' \<and> is_write_action wa \<and> (ad, al) \<in> action_loc_aux P wa \<and> value_written_aux P wa al = v"
row_one: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "row (1\<^sub>m n) i = unit_vec n i"
prems: shows "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
lset_ltake: fixes n :: "nat"   and xs :: "'c llist"   and A :: "'c set" assumes "\<And>m. m < n \<Longrightarrow> xs $ m \<in> A" shows "lset (ltake (enat n) xs) \<subseteq> A"
iT_Plus_neg_Plus_inverse: fixes I :: "nat set"   and k :: "nat" shows "I \<oplus>- k \<oplus> k = I \<down>\<ge> k"
swap_msgs_Recv_Trans: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isTrans ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
1: shows "det_fold_set ordR c' f' \<sigma>' result"
le_SucE: fixes m :: "nat"   and n :: "nat" assumes "m \<le> Suc n" obtains "m \<le> n"   | "m = Suc n"
insert_remove: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
full_pmdl_closed_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "'k set"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<in> full_pmdl K"   and "q \<in> full_pmdl K" shows "p + q \<in> full_pmdl K"
G: shows "(\<Union> (range (relations M)))\<^sup>* `` {t} \<subseteq> worlds M"
separation_t0: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.t0_space open" shows "(x \<noteq> y) = (\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U))"
two_special: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set"   and b :: "'a"   and e :: "real" assumes "closed A"   and "A \<subseteq> S"   and "a \<in> A"   and "closed B"   and "B \<subseteq> S"   and "b \<in> B"   and "A \<inter> B = {}"   and "0 < e"   and "e < 1" shows "\<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>x\<in>A. f x < e) \<and> (\<forall>x\<in>B. 1 - e < f x)"
xxsys_1: shows "altnst @ cs = xs @ [x] @ ys"
get_shadow_root_shadow_root_ptr_in_heap: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
finprod_def: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'a" shows "finsum I f \<equiv> if finite I then foldD R ((+) \<circ> f) \<zero> I else \<zero>"
measurable_translation: fixes S :: "'b set"   and a :: "'b" assumes "S \<in> lmeasurable" shows "(+) a ` S \<in> lmeasurable"
attach_CNV_RES: fixes callee :: "'v \<Rightarrow> 'r \<Rightarrow> ('s \<times> 'v, 't, 'u) gpv"   and s :: "'v"   and res :: "'w \<Rightarrow> 't \<Rightarrow> ('u \<times> 'w) spmf"   and s' :: "'w" shows "CNV callee s \<rhd> RES res s' = RES (attach_callee callee res) (s, s')"
res_function_Complete_set_residues: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "res_function f" shows "Complete_set_residues (\<lambda>c. range (f c))"
k': shows "(clock_simAbs (fst k'), snd k') \<in> (\<Union>a. Clock.sim_equiv_class a ` Clock.jkbpC) \<times> UNIV"
vlesspollD: fixes A :: "V"   and B :: "V" assumes "A \<prec>\<^sub>\<circ> B" shows "A \<lesssim>\<^sub>\<circ> B \<and> \<not> A \<approx>\<^sub>\<circ> B"
Fix_lsu_iff: fixes x :: "'b" shows "(x \<in> Fix \<nu>\<^sup>\<natural>) = (\<nu>\<^sup>\<natural> x = x)"
trivial_reduced_homology_group_empty: fixes X :: "'a topology"   and p :: "int" assumes "topspace X = {}" shows "trivial_group (reduced_homology_group p X)"
fmapU_lifted_def: shows "fmapU \<equiv> lifted_map"
mod_mult_self1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a + c * b) mod b = a mod b"
axioms_2: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" assumes "sylow G p a m" shows "sylow_axioms G p a m"
Torder_Sadjunction: fixes D :: "'c Order"   and X :: "'c set"   and a :: "'c" assumes "Order D"   and "X \<subseteq> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>X. x \<preceq>\<^bsub>D\<^esub> a"   and "Torder (SIod D X)" shows "Torder (SIod D (X \<union> {a}))"
MWLfSteps_det_cases_2: fixes x :: "'id"   and e :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and W :: "('exp, 'id) MWLfCom list"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (x:=e) m W m'" obtains "W = []" and "m' = m(x := E e m)"
GSpan_single_nonzero: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and m :: "'c" assumes "FinGroupRepresentation G smult V"   and "m \<in> V"   and "m \<noteq> (0::'c)" shows "R_scalar_mult.RSpan (aezfun_setspan G) smult [m] \<noteq> 0"
ortho_def: fixes f :: "'b \<Rightarrow> 'b option"   and g :: "'b \<Rightarrow> 'b option" shows "ortho f g \<equiv> dom f \<inter> dom g = {}"
next_not_member: fixes "next" :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and arb :: "'b"   and s :: "'b" assumes "fresh next"   and "finite S"   and "fresh.Next next arb S \<le> s" shows "next s \<notin> S"
length_list_update: fixes xs :: "'g list"   and i :: "nat"   and x :: "'g" shows "length (xs[i := x]) = length xs"
inverse2: fixes f :: "'b \<Rightarrow> 'a option" shows "U (C f) = f"
X_variance: fixes n :: "nat" shows "expectation (\<lambda>x. (X n x - expectation (X n))\<^sup>2) = \<sigma>\<^sup>2"
that: fixes x :: "nat" assumes "x \<le> (n - 1) div 2"   and "[x\<^sup>2 + 1 = 0] (mod n)" shows "thesis"
transcendental_sinh: fixes z :: "complex" assumes "algebraic z"   and "z \<noteq> 0" shows "\<not> algebraic (sinh z)"
minimalize_offending_overapprox_subset: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and ff :: "('c \<times> 'c) list"   and keeps :: "('c \<times> 'c) list"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "set (SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox sinvar ff keeps G nP) \<subseteq> set ff \<union> set keeps"
partition_on_mset_distinct: fixes A :: "'a multiset"   and P :: "'a multiset multiset" assumes "partition_on_mset A P"   and "distinct_mset A" shows "distinct_mset P"
induct: fixes P :: "('c, 'd) bintree option list \<Rightarrow> ('c, 'd) bintree option list \<Rightarrow> bool"   and a0 :: "('c, 'd) bintree option list"   and a1 :: "('c, 'd) bintree option list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>xs y ys. P xs ys \<Longrightarrow> P (None # xs) (y # ys)"   and "\<And>x xs ys. P xs ys \<Longrightarrow> P (x # xs) (None # ys)"   and "\<And>t xs r ys. P xs ys \<Longrightarrow> P (Some t # xs) (Some r # ys)" shows "P a0 a1"
cases: fixes a1 :: "Player"   and a2 :: "'a llist"   and a3 :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy a1 a2 a3" obtains p :: "Player" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = pa__" and "a2 = LNil" and "a3 = \<sigma>'__"   | p :: "Player" and v :: "'a" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = pb__" and "a2 = LCons va__ LNil" and "a3 = \<sigma>''__"   | v :: "'a" and p :: "Player" and w :: "'a" and \<sigma> :: "'a \<Rightarrow> 'a" and Ps :: "'a llist" where "a1 = pc__" and "a2 = LCons vb__ (LCons w Ps__)" and "a3 = \<sigma>'''__" and "vb__ \<in> VV pc__" and "w = \<sigma>'''__ vb__" and "path_conforms_with_strategy pc__ (LCons w Ps__) \<sigma>'''__"   | v :: "'a" and p :: "Player" and Ps :: "'a llist" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = p" and "a2 = LCons v Ps" and "a3 = \<sigma>" and "v \<notin> VV p" and "path_conforms_with_strategy p Ps \<sigma>"
sel: fixes x2 :: "'l" shows "the (Some x2) = x2"
east_west_tangents_incident_far_north_1: shows "proj2_incident far_north (polar east)"
cobounded2: fixes b :: "'b"   and a :: "'b" shows "(b::'b) \<le> sup (a::'b) b"
dvdm_imp_div_mod: fixes u :: "int poly"   and g :: "int poly" assumes "pl.dvdm u g" shows "\<exists>q r. g = q * u + smult (p ^ l) r"
rtranclp_normalize_poly_p_poly_of_mset: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset" assumes "normalize_poly_p\<^sup>*\<^sup>* p q" shows "polynomial_of_mset p = polynomial_of_mset q"
diff_zero: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
mertens_bound_real': assumes "1 \<le> x" shows "\<MM> x - ln x \<in> {- 2<..25 / 18}"
sint_lt: fixes x :: "'a word" shows "sint x < 2 ^ (LENGTH('a) - 1)"
f1: shows "((\<lambda>x. flow0 x t) \<circ> (flow0 x \<circ> s)) \<longlonglongrightarrow> flow0 p t"
submultiset_implies_subset: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "M \<subseteq># N" shows "set_mset M \<subseteq> set_mset N"
igSubstCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstCls MOD \<equiv> igSubstIGVar1 MOD \<and> igSubstIGVar2 MOD \<and> igSubstIGAbs MOD \<and> igSubstIGOp MOD"
integrable_spike_set_eq: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible (S - T \<union> (T - S))" shows "(f integrable_on S) = (f integrable_on T)"
span_breakdown: fixes b :: "'b"   and S :: "'b set"   and a :: "'b" assumes "b \<in> S"   and "a \<in> span S" shows "\<exists>k. a - k *s b \<in> span (S - {b})"
m_comm: fixes x :: "'a vec"   and y :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "x + y = y + x"
continuous_funpower: fixes f :: "'b set \<Rightarrow> 'b set"   and n :: "nat" assumes "continuous f" shows "continuous (funpower f n)"
map_def: shows "dnelist.map \<equiv> \<lambda>f. (\<lambda>x. Abs_dnelist (list_of_dlist x)) \<circ> map_dlist f \<circ> (\<lambda>x. Dlist (list_of_dnelist x))"