hcju/isabelleps huggingface dataset:
GE_wtPsym_def: shows "GE_wtPsym \<equiv> ProblemIkTpartG.GE_wtPsym wtPsym tpD tpFD"
integral_restrict: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<subseteq> T"   and "S \<in> sets lebesgue"   and "T \<in> sets lebesgue" shows "LINT x|lebesgue_on T. (if x \<in> S then f x else (0::'b)) = integral\<^sup>L (lebesgue_on S) f"
is_weak_ranking_map_inj: fixes xs :: "'b set list"   and f :: "'b \<Rightarrow> 'c" assumes "is_weak_ranking xs"   and "inj_on f (\<Union> (set xs))" shows "is_weak_ranking (map ((`) f) xs)"
prefix_length_less: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix xs ys" shows "length xs < length ys"
vid_on_vfield: fixes A :: "V" shows "\<F>\<^sub>\<circ> (vid_on A) = A"
angle_property_axioms: shows "angle_property bullet R R\<^sub>d"
assms: shows "(3 ^ 3 * 5 * 11 * 17 * 227) Amic (3 ^ 3 * 5 * 23 * 37 * 53)"
complex_cnj_inverse: fixes x :: "complex" shows "cnj (inverse x) = inverse (cnj x)"
has_derivative_at_withinI: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set" assumes "(f has_derivative f') (at x)" shows "(f has_derivative f') (at x within s)"
list_comb'_def: shows "list_comb' = foldl ($)"
finite_closed_segment: fixes a :: "'a"   and b :: "'a" shows "finite (closed_segment a b) = (a = b)"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b \<squnion> (a \<squnion> c) = a \<squnion> (b \<squnion> c)"
dollar_def: fixes q :: "nat" shows "$ q = (\<lambda>(s, c). s = 0 \<and> c = q)"
supp_7: fixes x2 :: "pi"   and x1 :: "pi" shows "supp (x2 \<oplus> x1) = supp x2 \<union> supp x1"
concat_cuts: fixes xs :: "int list" shows "concat (cuts xs) = xs"
prefix_length_le: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>p ys" shows "\<^bold>|xs\<^bold>| \<le> \<^bold>|ys\<^bold>|"
collapse: fixes state :: "('b, 'c) state" shows "State (run_state state) = state"
mark_out_aux_def: fixes n :: "nat"   and m :: "nat"   and bs :: "bool list" shows "mark_out_aux n m bs = map (\<lambda>(q, b). b \<and> (q < m + n \<or> \<not> Suc n dvd Suc (Suc q) + (n - m mod Suc n))) (enumerate n bs)"
assms_1: shows "backward_simulation step1 step2 final1 final2 order1 match1"
sign_idempotent: fixes p :: "'a \<Rightarrow> 'a" shows "sign p * sign p = 1"
rep_eq: fixes x :: "('c, 'd) fmap"   and xa :: "('c, 'd) fmap" shows "fmlookup (x ++\<^sub>f xa) = fmlookup x ++ fmlookup xa"
DBM_reset'_int_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry"   and d :: "'e"   and v :: "'f \<Rightarrow> nat"   and cs :: "'f list" assumes "dbm_int M n"   and "d \<in> \<int>"   and "\<forall>c. 0 < v c"   and "\<forall>c\<in>set cs. v c \<le> n" shows "dbm_int (reset' M n cs v d) n"
min: fixes z :: "real" assumes "a \<le> z"   and "z \<le> b"   and "y \<le> f z" shows "x \<le> z"
normalize_right_idem: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "bounded_quasi_semilattice f top bot normalize" shows "f a (normalize b) = f a b"
hrabs_add_Infinitesimal_cancel: fixes e :: "real star"   and e' :: "real star"   and x :: "real star"   and y :: "real star" assumes "e \<in> Infinitesimal"   and "e' \<in> Infinitesimal"   and "\<bar>x + e\<bar> = \<bar>y + e'\<bar>" shows "\<bar>x\<bar> \<approx> \<bar>y\<bar>"
cases: fixes a :: "'a list" assumes "steps a" obtains x :: "'a" where "a = [x__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "a = x # y # xs" and "x \<rightarrow> y" and "steps (y # xs)"
hom_0_iff: fixes x :: "'a mod_ring" shows "(to_int_mod_ring x = 0) = (x = 0)"
l: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "a \<in> UNIV \<rightarrow> carrier R"   and "b \<in> UNIV \<rightarrow> carrier R"   and "k \<le> n" shows "(\<Oplus>i\<in>{..k}. a i \<otimes> b (n - i)) = (\<Oplus>i\<in>{..k}. a (k - i) \<otimes> b (i + n - k))"
a_2: shows "{1..p - 1} = {a ^ i mod p |i. i \<in> UNIV}"
set_of_candidate_values_def: fixes S :: "'a eclause set"   and t :: "'a trm" shows "set_of_candidate_values S t = {pair. \<exists>z CC C' C s L L' \<sigma> t' s'. pair = (z, s) \<and> candidate_values z CC C' C s L L' \<sigma> t' s' t S}"
isomTr1: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme" assumes "Group F"   and "Group G"   and "F \<cong> G" shows "G \<cong> F"
s_def: shows "s \<equiv> Cn 1 r_ifz [Cn 1 r_consist_phi [t, recf.Id 1 0], t, r_auxhyp]"
Cons_5: shows "list_all (\<lambda>Q. index_free Q i) Qs"
tseyting_update_keep_subformula_sema: fixes G :: "'a formula"   and F :: "'a formula"   and \<A> :: "'a \<Rightarrow> bool" assumes "G \<in> set (subformulae F)" shows "tseytin_update \<A> F \<Turnstile> G = \<A> \<Turnstile> G"
HEndPhase1_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4c s q"   and "Inv2b s" shows "HInv4c s' q"
backward_terminating_path_root: fixes r :: "'a"   and x :: "'a" assumes "backward_terminating_path_root r x" shows "backward_terminating_path x"
InitConf_status: fixes A :: "('d, 'e, 'f) hierauto" shows "(A, InitConf A, {}, HAInitValue A) \<in> status"
no_loop: fixes x :: "'v option" shows "\<not> edge \<Psi> x x"
inverse_distrib_swap: fixes a :: "'a perm"   and b :: "'a perm" shows "inverse (a * b) = inverse b * inverse a"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C H \<Phi>\<^sub>H"
to_conjugate_space_inject: fixes x :: "'c"   and y :: "'c" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(to_conjugate_space x = to_conjugate_space y) = (x = y)"
GT: fixes x :: "'b"   and y :: "'b" shows "(cmp x y = GT) = (y < x)"
face_cycle_succ_inI: fixes x :: "'b"   and y :: "'b" assumes "x \<in> H.face_cycle_set y" shows "H.face_cycle_succ x \<in> H.face_cycle_set y"
AbstrLevels_A4_A42: fixes i :: "AbstrLevelsID" assumes "sA4 \<in> AbstrLevel i" shows "sA42 \<notin> AbstrLevel i"
elims_2: fixes x :: "Core.term" assumes "is_all x" obtains alla :: "String.literal" and S :: "typ" and T :: "typ" and t :: "Core.term" where "x = Ct alla S $ Abs T t" and "alla = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT"
Polys_alt: fixes X :: "'c set" shows "P[X] = {p. indets p \<subseteq> X}"
not_equals_and_not_equals_not_in: fixes y :: "'d"   and x :: "'d"   and x' :: "'d"   and phi :: "bool" shows "(y \<noteq> x \<and> y \<noteq> x' \<and> phi) = (y \<notin> {x, x'} \<and> phi)"
induct: fixes P :: "('c, 'd) Node list \<Rightarrow> ('c, 'd) FingerTreeStruc \<Rightarrow> ('c, 'd) Digit \<Rightarrow> bool"   and a0 :: "('c, 'd) Node list"   and a1 :: "('c, 'd) FingerTreeStruc"   and a2 :: "('c, 'd) Digit" assumes "\<And>m sf. P [] m sf"   and "\<And>v va m sf. P (v # va) m sf" shows "P a0 a1 a2"
repeatable_both_both: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "repeatable t" shows "t \<otimes>\<otimes> t' \<otimes>\<otimes> t = t \<otimes>\<otimes> t'"
insert_is_Un: fixes a :: "'a"   and A :: "'a set" shows "insert a A = \<eta> a \<union> A"
strict_implies_order: fixes a :: "'a"   and b :: "'a" assumes "a \<sqsubset> b" shows "a \<sqsubseteq> b"
Abs_num1_inverse: fixes y :: "unit" assumes "y \<in> UNIV" shows "Rep_num1 (Abs_num1 y) = y"
crel_vs_intro: fixes v\<^sub>T :: "('mem, 'd) state"   and R :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and v :: "'e" assumes "\<And>M v' M'. \<lbrakk>cmem M; P M; run_state v\<^sub>T M = (v', M')\<rbrakk> \<Longrightarrow> R v v' \<and> cmem M' \<and> P M'" shows "crel_vs R v v\<^sub>T"
full_chain_imp_chain: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "full_chain R xs" shows "chain R xs"
monad_duplicate_axioms_def: fixes bind :: "'d \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd" shows "monad_duplicate_axioms bind \<equiv> \<forall>m f. bind m (\<lambda>x. bind m (f x)) = bind m (\<lambda>x. f x x)"
isPath_fwd_cases: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "Graph.isPath c s p t" obtains "p = []" and "t = s"   | p' :: "(nat \<times> nat) list" and u :: "nat" where "p = (s, u) # p'" and "(s, u) \<in> Graph.E c" and "Graph.isPath c u p' t"
has_sort_code: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and T :: "typ"   and S :: "String.literal set" assumes "exe_osig_conds oss" shows "has_sort (translate_osig oss) T S = exehas_sort oss T S"
shift_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "shift_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; flagi = get_operand_flag (op_list ! 0); rs1 = get_operand_w5 (op_list ! 1); rs2_shcnt = get_operand_w5 (op_list ! 2); rd = get_operand_w5 (op_list ! 3) in do curr_win \<leftarrow> get_curr_win (); shift_count \<leftarrow> gets (\<lambda>s. if flagi = 0 then ucast (user_reg_val curr_win rs2_shcnt s) else rs2_shcnt); rs1_val \<leftarrow> gets (user_reg_val curr_win rs1); if instr_name = shift_type SLL \<and> rd \<noteq> 0 then do rd_val \<leftarrow> gets (\<lambda>s. rs1_val << unat shift_count); write_reg rd_val curr_win rd; return () od else if instr_name = shift_type SRL \<and> rd \<noteq> 0 then do rd_val \<leftarrow> gets (\<lambda>s. rs1_val >> unat shift_count); write_reg rd_val curr_win rd; return () od else if instr_name = shift_type SRA \<and> rd \<noteq> 0 then do rd_val \<leftarrow> gets (\<lambda>s. rs1_val >>> unat shift_count); write_reg rd_val curr_win rd; return () od else return () od"
degree_fls_prpart: fixes f :: "'a fls" shows "degree (fls_prpart f) = nat (- fls_subdegree f)"
injective_comp_right_dist_inf: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<sqinter> id z \<le> top" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> z \<sqinter> (y \<sqinter> z)"
orient_lit_inst_vars: fixes L :: "'a literal"   and t :: "'a trm"   and s :: "'a trm"   and p :: "sign"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "orient_lit_inst L t s p \<sigma>" shows "vars_of t \<subseteq> vars_of_lit L \<and> vars_of s \<subseteq> vars_of_lit L"
pell_valuation_eq_iff: fixes D :: "nat"   and z1 :: "int \<times> int"   and z2 :: "int \<times> int" assumes "pell D" shows "(pell.pell_valuation D z1 = pell.pell_valuation D z2) = (z1 = z2)"
domI: fixes m :: "'h \<Rightarrow> 'g option"   and a :: "'h"   and b :: "'g" assumes "m a = Some b" shows "a \<in> dom m"
appNew: fixes C :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (New C, P, pc, mxs, T\<^sub>r, s) = (\<exists>ST LT. s = (ST, LT) \<and> is_class P C \<and> length ST < mxs)"
noStuck_start: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "t \<noteq> Stuck" shows "s \<noteq> Stuck"
rel_G_mono: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co1' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'x \<Rightarrow> 'y \<Rightarrow> bool" assumes "L1 \<le> L1'"   and "L2 \<le> L2'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2" shows "rel_G L1 L2 Co1 Co2 Contra1 Contra2 \<le> rel_G L1' L2' Co1' Co2' Contra1' Contra2'"
BB_Suc: fixes j :: "nat"   and j0 :: "nat"   and K :: "nat set" assumes "j \<le> j0"   and "K \<in> \<K> j0 (Suc j)" shows "BB j0 (Suc j) K = usplit (\<lambda>L k. BB j0 j L @ list_of (b k (j0, j))) K"
del_UnEdge_even': fixes v :: "'v"   and w :: "'w"   and v' :: "'v" assumes "(v, w, v') \<in> E"   and "finite E" shows "(v' \<in> odd_nodes_set (del_unEdge v w v' G)) = even (degree v' G)"
map_semantics_5: shows "\<phi>\<^sub>i = map_ltlc ((the \<circ>\<circ> idx) (IArray.list_of APs)) \<phi>"
strict_order_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subset># b) = (a = a \<inter># b \<and> a \<noteq> b)"
arr_iff_in_hom: fixes f :: "'a" shows "arr f = \<guillemotleft>f : local.dom f \<rightarrow> cod f\<guillemotright>"
downclosed_set_def: fixes X :: "'a set" shows "downclosed_set X = (X \<in> downsets)"
strict_iff_order: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b \<subset># a) = (b \<subseteq># a \<and> a \<noteq> b)"
commutative_square_def: fixes f :: "(bool, bool list) concrete_category.arr"   and g :: "(bool, bool list) concrete_category.arr"   and h :: "(bool, bool list) concrete_category.arr"   and k :: "(bool, bool list) concrete_category.arr" shows "J.commutative_square f g h k \<equiv> J.cospan f g \<and> J.span h k \<and> J.dom f = J.cod h \<and> f \<cdot>\<^sub>J h = g \<cdot>\<^sub>J k"
IH: assumes "lookup \<rho> x = None" shows "x \<notin> sdom \<rho>"
append_Nil: fixes ys :: "'d list" shows "[] @ ys = ys"
bernoulli_expl_cond_expect_adapt: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat"   and f :: "bool stream \<Rightarrow> real" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "expl_cond_expect N (proj_stoch_proc geom_proc n) f \<in> borel_measurable (G n)"
rec_ci_s_def: shows "rec_ci_s \<equiv> addition 0 1 2 [+] [Inc 1]"
that: fixes y :: "'a" assumes "limit_cycle y"   and "range (flow0 y) \<subseteq> K" shows "thesis"
map_sub_finer_than_commute: fixes f :: "'d \<Rightarrow> 'c"   and T :: "'d topology"   and X :: "'c set" shows "map_topology f (subtopology T (f -` X)) finer_than subtopology (map_topology f T) X"
sel_2: fixes x21 :: "'b"   and x22 :: "('b, 'a) tllist" shows "terminal (TCons x21 x22) = terminal0 x22"
sectionI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C (C g f)" shows "category.section C f"
valuation_models_encoding_cnf_formula_equals: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "\<A> \<Turnstile> \<Phi> \<Pi> t = cnf_semantics \<A> (cnf (\<Phi> \<Pi> t))"
ring_simprules_18: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> y) = y"
simps_30: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "nat" shows "(case Pkt x51 x52 x53 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f5 x51 x52 x53"
dom_char: fixes f :: "'a \<times> 'a" shows "VV.dom f = (if VV.arr f then VxV.dom f else VxV.null)"
lfp_greatest: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>u. less_eq (f u) u \<Longrightarrow> less_eq A u" shows "less_eq A (complete_lattice.lfp Inf less_eq f)"
init: shows "[this # pns [\<mapsto>] vs] \<subseteq>\<^sub>m [this # pns [\<mapsto>] vs @ rest]"
fixp_lowerbound: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "monotone leq1 leq1 f"   and "leq1 (f z) z" shows "leq1 (c1.fixp f) z"
STA_sub': fixes A :: "'d set"   and n :: "nat"   and branch :: "(('f, 'd) fm list \<times> 'd) list"   and f :: "'d \<Rightarrow> 'e" assumes "\<And>f i A. \<lbrakk>finite A; i \<notin> A\<rbrakk> \<Longrightarrow> \<exists>j. j \<notin> f ` A"   and "finite A"   and "A, n \<turnstile> branch" shows "f ` A \<turnstile> sub_branch f branch"
constructPerm: fixes xvec :: "name list"   and yvec :: "name list" assumes "length xvec = length yvec"   and "xvec \<sharp>* yvec"   and "distinct xvec"   and "distinct yvec" obtains p :: "(name \<times> name) list" where "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" and "distinctPerm p" and "yvec = p \<bullet> xvec"
array_idx_in_set: fixes pc :: "nat"   and ins :: "'a array"   and x :: "'a" assumes "pc < array_length ins"   and "array_get ins pc = x" shows "x \<in> set (list_of_array ins)"
p_incr_def: fixes bs :: "bool list" shows "\<Phi> bs = real (length (filter id bs))"
eq: shows "\<bar>standard_liouville p q - real_of_int (a n) / real_of_int (b n)\<bar> = (\<Sum>k. real_of_int (p (k + n + 1)) / real_of_int q ^ fact (Suc (k + n + 1)))"
butlast_cs_not_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "IFC prob"   and "butlast (IFC_def.cs prob \<pi> m) = butlast (IFC_def.cs prob \<pi> n)" shows "\<not> IFC_def.is_cdi prob m \<pi> n"
crossproduct_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and w :: "'e"   and y :: "'e"   and x :: "'e"   and z :: "'e" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(plus (times w y) (times x z) = plus (times w z) (times x y)) = (w = x \<or> y = z)"
of_nat_mod: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "semiring_1.of_nat one plus zero (m mod n) = modulo (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
Interp_imp_INTERP: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and C :: "'a literal multiset"   and D :: "'a literal multiset"   and N :: "'a literal multiset set" assumes "ground_resolution_with_selection S"   and "C \<le> D"   and "ground_resolution_with_selection.Interp S N D \<Turnstile> C" shows "ground_resolution_with_selection.INTERP S N \<Turnstile> C"
EWHILEIT_rule: fixes R :: "('a \<times> 'a) set"   and I :: "'a \<Rightarrow> bool"   and s\<^sub>0 :: "'a"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> ('b + 'a) nres"   and E :: "'b \<Rightarrow> bool"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "wf R"   and "I s\<^sub>0"   and "\<And>s. \<lbrakk>I s; b s; (s, s\<^sub>0) \<in> R\<^sup>*\<rbrakk> \<Longrightarrow> f s \<le> ESPEC E (\<lambda>s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s; (s, s\<^sub>0) \<in> R\<^sup>*\<rbrakk> \<Longrightarrow> \<Phi> s" shows "EWHILEIT I b f s\<^sub>0 \<le> ESPEC E \<Phi>"
completely_multiplicative_fds_deriv': fixes f :: "'a fds" assumes "completely_multiplicative_function (fds_nth f)" shows "fds_deriv f = - fds (\<lambda>n. fds_nth f n * mangoldt n) * f"
update_convs_3: fixes Doms' :: "'c set \<Rightarrow> 'c set"   and Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "Doms_update Doms' \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = \<lparr>Pol = Pol, Map = Map, Doms = Doms' Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr>"
f7: shows "s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') = s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> (s' \<otimes> r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> (s \<otimes> r' \<otimes> r'')"
assms: shows "poly (hd ps) x\<^sub>0 \<noteq> 0"
starfun_minus: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d star" shows "- (*f* f) x = (*f* (\<lambda>x. - f x)) x"
dim_basis: fixes A :: "'c set" assumes "finite A"   and "basis A" shows "local.dim = card A"
preserves_iso: fixes f :: "'a" assumes "S.iso f" shows "S.iso (\<rr> f)"
pp_ub2: fixes x :: "'a list"   and y :: "'a" assumes "set x \<subseteq> A"   and "y \<in> A" shows "y \<sqsubseteq>\<^bsub>r\<^esub> x \<Squnion>\<^bsub>f\<^esub> y"
AE_I2: fixes M :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x" shows "almost_everywhere M P"
composition_series_simple_group: shows "simple_group (G\<lparr>carrier := \<GG> ! 1\<rparr>) = (take 2 \<GG> = [{\<one>\<^bsub>G\<lparr>carrier := \<GG> ! 1\<rparr>\<^esub>}, carrier (G\<lparr>carrier := \<GG> ! 1\<rparr>)])"
lookup_union: fixes s :: "('d, 'e) RBT.rbt"   and t :: "('d, 'e) RBT.rbt" shows "RBT.lookup (RBT.union s t) = RBT.lookup s ++ RBT.lookup t"
transp_trans_eq: fixes r :: "('a \<times> 'a) set" shows "transp (\<lambda>x y. (x, y) \<in> r) = trans r"
prems_2: fixes j :: "nat" assumes "j < length (x # xs)"   and "j \<noteq> i" shows "(x # xs) ! j = (0::'a)"
strings_of_length_n: fixes n :: "nat" shows "set (List.n_lists n all_chars) = {s. length s = n}"
refl_onD: fixes A :: "'g set"   and r :: "('g \<times> 'g) set"   and a :: "'g" assumes "refl_on A r"   and "a \<in> A" shows "(a, a) \<in> r"
Abs_ExpList_Nil: shows "Abs_ExpList [] \<equiv> []"
assms: fixes s' :: "'state" shows "\<not> (c, s) \<rightarrow>*t s'"
Pi_I: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
S_verts_def: shows "S_verts \<equiv> {1..n}"
R28_1: shows "pmf (sds R28) a = 1 / 2"
fmeasurable_Union_bound: fixes \<D> :: "'a set set"   and M :: "'a measure"   and B :: "real" assumes "countable \<D>"   and "\<And>D. D \<in> \<D> \<Longrightarrow> D \<in> fmeasurable M"   and "\<And>\<E>. \<lbrakk>\<E> \<subseteq> \<D>; finite \<E>\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure M (\<Union> \<E>) \<le> B" shows "\<Union> \<D> \<in> fmeasurable M"
enumerate_Suc'': fixes S :: "'b set"   and n :: "nat" assumes "infinite S" shows "wellorder_class.enumerate S (Suc n) = (LEAST s. s \<in> S \<and> wellorder_class.enumerate S n < s)"
ord_isom_trans: fixes E :: "('e, 'f) Order_scheme"   and F :: "('g, 'h) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'g" assumes "Order E"   and "Order F"   and "ord_isom D E f"   and "ord_isom E F g" shows "ord_isom D F (compose (carrier D) g f)"
dirichlet_prod_moebius_commute: fixes f :: "nat \<Rightarrow> 'b" shows "f \<star> moebius_mu = moebius_mu \<star> f"
sum_list_closed: fixes f :: "'a \<Rightarrow> 'm"   and as :: "'a list" assumes "set (map f as) \<subseteq> M" shows "sum_list (map f as) \<in> M"
elts_0: shows "elts []\<^sub>\<circ> = {}"