hcju/isabelleps huggingface dataset:
pivotandupdate_bounds_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<B>\<^sub>i (pivot_and_update x\<^sub>i x\<^sub>j c s) = \<B>\<^sub>i s"
dim_row_tensor_mat: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" shows "dim_row (A \<Otimes> B) = dim_row A * dim_row B"
imult_infinity: fixes n :: "eint" assumes "0 < n" shows "\<infinity> * n = \<infinity>"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>src\<^sub>C f : src\<^sub>C a \<Rightarrow>\<^sub>C src\<^sub>C b\<guillemotright>"
rtrancl_path_Range: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and z :: "'b" assumes "rtrancl_path R x xs y"   and "z \<in> set xs" shows "Rangep R z"
heap_write_total: fixes h :: "'heap"   and a :: "'addr"   and al :: "addr_loc"   and T :: "ty"   and v :: "'addr val" assumes "hconf h"   and "P,h \<turnstile> a@al : T"   and "P,h \<turnstile> v :\<le> T" shows "\<exists>h'. heap_write h a al v h'"
cod_cod: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.cod (CC.cod f) = CC.cod f"
disjunctive_def: fixes sup_b :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup_c :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and times_abc :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "disjunctive.disjunctive sup_b sup_c times_abc = {x. \<forall>y z. times_abc x (sup_b y z) = sup_c (times_abc x y) (times_abc x z)}"
submonoid_axioms: shows "submonoid N M (\<cdot>) \<one>"
parallel_reduction_varE: fixes x :: "'a"   and A :: "'a trm" assumes "Var x >> A" shows "A = Var x"
DBM_reset'_complete': fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and cs :: "'c list"   and d :: "'d" assumes "u \<turnstile>\<^bsub>v,n\<^esub> M"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n" shows "\<exists>u'. u' \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d"
linear_less: fixes var :: "nat"   and L :: "real list"   and C :: "real"   and b :: "real mpoly"   and c :: "real mpoly"   and B :: "real"   and p :: "real mpoly"   and v :: "real" assumes "var < length L"   and "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 (L[var := B / C])) b = B"   and "insertion (nth_default 0 (L[var := B / C])) c = C" shows "aEval (Less p) (L[var := B / C]) = aEval (linear_substitution var b c (Less p)) (L[var := v])"
snapshot_step_cs_preservation_p: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and p :: "'d"   and cid :: "nat"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.next channel trans send recv c ev c'"   and "\<not> regular_event ev"   and "occurs_on ev = p"   and "channel cid = Some (p, q)" shows "map Msg (fst (cs c cid)) @ takeWhile ((\<noteq>) Marker) (msgs c cid) = map Msg (fst (cs c' cid)) @ takeWhile ((\<noteq>) Marker) (msgs c' cid) \<and> snd (cs c cid) = snd (cs c' cid)"
\<beta>_dim_2: shows "dim_vec \<beta> = N"
that: assumes "x \<in> F"   and "y = f x" shows "thesis"
conseq: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2 {P} c {Q}"   and "\<And>s. P s \<le> P' s"   and "\<And>s. Q' s \<le> Q s" shows "\<turnstile>\<^sub>2 {P'} c {Q'}"
awalk_app_isoD: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'e"   and p :: "'f list"   and v :: "'e" assumes "pre_digraph.awalk (app_iso hom G) u p v"   and "digraph_isomorphism hom" shows "awalk (iso_verts (inv_iso hom) u) (map (iso_arcs (inv_iso hom)) p) (iso_verts (inv_iso hom) v)"
intro: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "cring R"   and "cring S"   and "ring_hom_cring_axioms R S h" shows "ring_hom_cring R S h"
\<psi>_1: fixes p :: "'a" assumes "p \<in> carrier" shows "codomain (\<psi> p) = ball (0::'b) 3"
c_of_merge: fixes env2 :: "v sem_env"   and env1 :: "v sem_env" shows "c (extend_dec_env env2 env1) = nsAppend (c env2) (c env1)"
IH: fixes z :: "complex" assumes "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
sum_map_sq_matrix: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and A :: "('e, 'c) sq_matrix"   and I :: "'d set" shows "(\<Sum>i\<in>I. map_sq_matrix (f i) A) = map_sq_matrix (\<lambda>x. \<Sum>i\<in>I. f i x) A"
advantage_def: fixes \<A> :: "('c, 'd) distinguisher"   and res1 :: "('c, 'd) resource"   and res2 :: "('c, 'd) resource" shows "advantage \<A> res1 res2 = \<bar>spmf (connect \<A> res1) True - spmf (connect \<A> res2) True\<bar>"
assms_4: shows "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"
simps_9: fixes f :: "'e \<Rightarrow> 'd"   and x1 :: "'e" shows "map_decision f (allow x1) = allow (f x1)"
adjoint_transpose_left_6: fixes u :: "'a"   and v :: "'a" assumes "ide u"   and "ide v"   and "src k = trg v"   and "src k\<^sup>* = trg u" shows "bij_betw (k.trnl\<^sub>\<epsilon> u) (hom v (k\<^sup>* \<star> u)) (hom (k \<star> v) u)"
rotate1_rotate_swap: fixes n :: "nat"   and xs :: "'b list" shows "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
lcmof_exists: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "\<exists>c. c \<in> carrier G \<and> c lcmof a b"
ln_le_zero_iff: fixes x :: "real" assumes "0 < x" shows "(ln x \<le> 0) = (x \<le> 1)"
negligible_eq_zero_density: fixes S :: "'a set" shows "negligible S = (\<forall>x\<in>S. \<forall>r>0. \<forall>e>0. \<exists>d>0. d \<le> r \<and> (\<exists>U. S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue U < e * Sigma_Algebra.measure lebesgue (ball x d)))"
U_consts_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n true\<^sub>n"
vcomp_can: fixes t :: "'a E.term"   and u :: "'a E.term"   and v :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "E.Ide v"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"   and "\<^bold>\<lfloor>u\<^bold>\<rfloor> = \<^bold>\<lfloor>v\<^bold>\<rfloor>" shows "B.can v u \<cdot>\<^sub>B B.can u t = B.can v t"
accp_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "Wellfounded.accp r a"   and "\<And>x. \<lbrakk>Wellfounded.accp r x; \<forall>y. r y x \<longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
simps_1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'b"   and y :: "'c" shows "tZip f (tInit x) (tInit y) = tInit (f x y)"
nprv_ptrmE_exi: fixes \<sigma> :: "'fmla"   and n :: "nat"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "\<sigma> \<in> ptrm n"   and "nprv (insert \<sigma> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "out \<notin> Fvars \<psi>"   and "out \<notin> \<Union> (Fvars ` F)" shows "nprv F \<psi>"
fgs_sub_carrier: fixes a :: "'a set" assumes "a \<subseteq> carrier M" shows "fgs R M a \<subseteq> carrier M"
simps_1: fixes x11 :: "agent"   and x12 :: "agent"   and x13 :: "msg"   and y11 :: "agent"   and y12 :: "agent"   and y13 :: "msg" shows "(Says x11 x12 x13 = Says y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
contains_star: fixes s :: "'a"   and r :: "'b \<times> 'c"   and rs :: "('b \<times> 'c) list" assumes "contains s r" shows "contains (s \<star> rs) r"
bind_split_asm: fixes P :: "'b option \<Rightarrow> bool"   and m :: "'c option"   and f :: "'c \<Rightarrow> 'b option" shows "P (m \<bind> f) = (\<not> (m = None \<and> \<not> P None \<or> (\<exists>x. m = \<lfloor>x\<rfloor> \<and> \<not> P (f x))))"
nonzero_neg_divide_eq_eq2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(c = uminus (divide a b)) = (times c b = uminus a)"
ReZ_of_int: fixes n :: "int" shows "ReZ (of_int n) = n"
set_drop_auxlist_cong: fixes x :: "nat \<times> event_data option list set \<times> event_data option list set"   and t :: "nat"   and a1 :: "event_data option list set"   and a2 :: "event_data option list set" assumes "x \<in> set (drop (length done) auxlist)"   and "x = (t, a1, a2)" shows "mem (nt - t) I = (left I \<le> nt - t)"
open_ball: fixes x :: "'c"   and e :: "real" shows "open (ball x e)"
assms_4: fixes s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "P as s"   and "index_valid as s" shows "P (insert a as) (assert a s)"
compile_s_correct': fixes n :: "nat"   and anything :: "nat list" shows "{\<lambda>nl. nl = n # 0 \<up> 2 @ anything} addition 0 (Suc 0) 2 [+] [Inc (Suc 0)] {\<lambda>nl. nl = n # Suc n # 0 # anything}"
cons3: shows "(evaluate ck env s1 e (s2, Rval v1) \<and> (\<forall>x count count' s' r'. clock s1 = count + x \<and> (s2, Rval v1) = (s', r') \<and> clock s' = count' + x \<and> ck = True \<longrightarrow> evaluate True env (update_clock (\<lambda>_. count) s1) e (update_clock (\<lambda>_. count') s', r'))) \<and> evaluate_list ck env s2 es (s3, Rerr err) \<and> (\<forall>x count count' s' r'. clock s2 = count + x \<and> (s3, Rerr err) = (s', r') \<and> clock s' = count' + x \<and> ck = True \<longrightarrow> evaluate_list True env (update_clock (\<lambda>_. count) s2) es (update_clock (\<lambda>_. count') s', r'))"
PiE: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set"   and x :: "'e" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
iT_Minus_Plus_empty: fixes k :: "nat"   and n :: "nat"   and I :: "nat set" assumes "k < n" shows "k \<ominus> (I \<oplus> n) = {}"
exp_ln_iff: fixes x :: "real" shows "(exp (real_ln x) = x) = (0 < x)"
Abs_mod_ring_def: fixes x :: "int" shows "Abs x = of_int_mod_ring x"
finite_lprefix_nitpick_simps_3: fixes xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "Coinductive_List.finite_lprefix xs (LCons y ys) = (xs = LNil \<or> (\<exists>xs'. xs = LCons y xs' \<and> Coinductive_List.finite_lprefix xs' ys))"
count_list_snoc: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" shows "count_list (xs @ [x]) y = (if y = x then Suc (count_list xs y) else count_list xs y)"
assms: shows "\<not> liftAll (\<lambda>eA. eA \<noteq> ERR) ebinp"
mzero_def: shows "mzero = COERCE(udom\<cdot>'d, 'c\<cdot>'d)\<cdot>zeroU"
urel32_eq: fixes x :: "uint32"   and y :: "int"   and x' :: "uint32"   and y' :: "int" assumes "urel32 x y"   and "urel32 x' y'" shows "(x = x') = (y = y')"
surfpS: fixes x :: "'a" assumes "norm x = 1"   and "x \<in> affine hull S" shows "surf (proj x) \<in> S"
intros_2: fixes u :: "'e \<Rightarrow> 'd"   and c :: "'e"   and d :: "'d" assumes "u c < d" shows "u \<turnstile> LT c d"
set_intros_2: fixes y :: "'c \<times> ('c, 'd) rose_tree\<^sub>m list"   and x :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m"   and yb :: "('c, 'd) rose_tree\<^sub>m list"   and yc :: "('c, 'd) rose_tree\<^sub>m"   and yd :: "'c" assumes "y \<in> set1_blindable\<^sub>m x"   and "yb \<in> snds y"   and "yc \<in> set yb"   and "yd \<in> set1_rose_tree\<^sub>m yc" shows "yd \<in> set1_rose_tree\<^sub>m (Tree\<^sub>m x)"
limit1: shows "LIM x at_left 4. sqrt (4 / x - 1) / (x - 4) :> at_bot"
sets_restrict_space_cong: fixes M :: "'c measure"   and N :: "'c measure"   and \<Omega> :: "'c set" assumes "sets M = sets N" shows "sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)"
return_transfer: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and dp :: "'a \<Rightarrow> 'c" assumes "dp_consistency lookup update P" shows "rel_fun R (dp_consistency.crel_vs lookup P dp R) Wrap State_Monad.return"
bound_deg_sum': fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "g \<in> carrier P"   and "deg R f < n"   and "deg R g < n" shows "deg R (f \<oplus>\<^bsub>P\<^esub> g) < n"
path_def: shows "gp2.path \<equiv> \<lambda>g. lfp (\<lambda>p x. (\<exists>n. x = [n] \<and> n \<in> set (\<alpha>n2 g) \<and> invar2 g) \<or> (\<exists>ns n'. x = n' # ns \<and> p ns \<and> n' \<in> set (gp2.predecessors g (hd ns))))"
Rep_cfun_strict1: fixes x :: "'s" shows "\<bottom>\<cdot>x = \<bottom>"
expand_aimpl_refine: fixes n_ns' :: "'a node \<times> 'a node set"   and n_ns :: "'a node \<times> 'a node set" assumes "(n_ns', n_ns) \<in> Id \<inter> {(uu_, n, ns). \<forall>n'\<in>ns. name n' < name n}" shows "expand_aimpl n_ns' \<le> \<Down> (Id \<inter> {(uu_, n, ns). \<forall>n'\<in>ns. name n' < n}) (expand\<^sub>T n_ns)"
extreme_point_of_def: fixes x :: "'b"   and S :: "'b set" shows "(x extreme_point_of S) = (x \<in> S \<and> (\<forall>a\<in>S. \<forall>b\<in>S. x \<notin> open_segment a b))"
wf_dbfm_iff_is_fm: fixes x :: "dbfm" shows "wf_dbfm x = (\<exists>A. x = trans_fm [] A)"
UNIF1_range_vars_subset: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "range_vars \<sigma> \<subseteq> vars_mset E"
choose_2: fixes S :: "'b set" assumes "card S = 2" shows "\<exists>x y. S = {x, y}"
FreeGroup_def: fixes S :: "'c set" shows "FreeGroup S \<equiv> {x. order.greater_eq S (fst ` set (freeword x))}"
commonArefl_subset_commonDomain: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('a \<times> 'a) set" shows "Arefl R1 \<inter> Arefl R2 \<subseteq> Domain R1 \<inter> Domain R2"
chain_mono_less: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "class.po below"   and "po.chain below Y"   and "i < j" shows "below (Y i) (Y j)"
class_ex_def: fixes rel :: "('a \<times> 'a) set"   and c :: "'a" shows "class_ex rel c = (c \<in> Field rel)"
derivable_clauses_lemma: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "derivable C P S \<sigma> k C'" shows "cl_ecl C = subst_cl C' \<sigma>"
subgroup_def: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" shows "subgroup H G \<equiv> (H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H)) \<and> \<one>\<^bsub>G\<^esub> \<in> H \<and> (\<forall>x. x \<in> H \<longrightarrow> inv\<^bsub>G\<^esub> x \<in> H)"
adds_triv_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a (plus a b)"
integral_subalgebra: fixes f :: "'b \<Rightarrow> 'c"   and N :: "'b measure"   and M :: "'b measure" assumes "f \<in> borel_measurable N"   and "sets N \<subseteq> sets M"   and "space N = space M"   and "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" shows "integral\<^sup>L N f = integral\<^sup>L M f"
assms_1: fixes x :: "nat" shows "B x \<in> sets M"
neg_0_less_iff_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (uminus a) = less a zero"
word_ao_absorbs_4: fixes y :: "'a word"   and x :: "'a word" shows "y AND x OR x = x"
assms: fixes v :: "nat" assumes "v \<in> vars p" shows "f v = g v"
sum_le_included: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'b set"   and t :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and i :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite s"   and "finite t"   and "\<forall>y\<in>t. less_eq zero (g y)"   and "\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> less_eq (f x) (g y)" shows "less_eq (comm_monoid_add.sum plus zero f s) (comm_monoid_add.sum plus zero g t)"
empty_neutral_1: fixes x :: "'a multiset" shows "\<Empt> + x = x"
empty: shows "\<Sum>\<^sub># {#} = (0::'k)"
positive_ux: assumes "float_plus_up prec ux (- floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) * 2 * (if floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) < 0 then float_round_up prec (ub_pi prec) else float_round_down prec (lb_pi prec))) \<le> float_round_down prec (lb_pi prec)"   and "0 \<le> x - real_of_int k * (2 * pi)" shows "real_of_float (lb_cos prec (float_plus_up prec ux (- floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) * 2 * (if floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) < 0 then float_round_up prec (ub_pi prec) else float_round_down prec (lb_pi prec))))) \<le> cos x"
cancel_pseudo_hoop_algebra_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.cancel_pseudo_hoop_algebra times inf imp_l less_eq less one imp_r \<equiv> class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r \<and> class.cancel_pseudo_hoop_algebra_axioms times"
3: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. linear_orderable_6 x)"
list_update_id: fixes xs :: "'a list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
br'_pref_br: fixes \<delta> :: "('c, 'd) ta_rule set" shows "wa_precise_refine (br'_algo \<delta>) (br_algo \<delta>) br'_\<alpha>"
Comp_Hom_Id: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a"   and f :: "'b"   and B :: "'a" assumes "concrete_category Obj Hom Id Comp"   and "A \<in> Obj"   and "f \<in> Hom A B" shows "Comp B A A f (Id A) = f"
set_1: fixes x1 :: "'c" shows "set1_blindable\<^sub>m (Unblinded x1) = {x1}"
invar_def: fixes empty :: "'b"   and insert :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and delete :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and isin :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and inorder :: "'b \<Rightarrow> 'c list"   and inv :: "'b \<Rightarrow> bool"   and t :: "'b" assumes "Set_by_Ordered empty insert delete isin inorder inv" shows "Set_by_Ordered.invar inorder inv t = (inv t \<and> sorted (inorder t))"
mat_row_carrierI: fixes nr :: "nat"   and nc :: "nat"   and r :: "nat \<Rightarrow> 'b Matrix.vec" shows "mat\<^sub>r nr nc r \<in> carrier_mat nr nc"
bangUnfoldRight: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "Id \<subseteq> Rel" shows "!P \<leadsto>[Rel] P \<parallel> !P"
frontier_indist_le: fixes x :: "'a"   and S :: "'a set"   and r :: "real" assumes "x \<in> frontier {y. infdist y S \<le> r}" shows "infdist x S = r"
empty: shows "GEN_OP emp op_map_empty (\<langle>A, nat_rel\<rangle>Rm)"
prod_in_hom': fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and d :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "local.dom f = a"   and "cod f = c"   and "arr g"   and "local.dom g = b"   and "cod g = d" shows "\<guillemotleft>HF'.prod f g : HF'.prod a b \<rightarrow> HF'.prod c d\<guillemotright>"
mult: fixes a :: "nat"   and b :: "nat" shows "\<chi> (a * b) = \<chi> a * \<chi> b"
div_rounding_down_2: fixes q :: "nat" shows "Suc (2 * q) div 2 = q"
ch2ch_Rep_cfunL: fixes F :: "nat \<Rightarrow> 'b \<rightarrow> 'c"   and x :: "'b" assumes "chain F" shows "chain (\<lambda>i. F i\<cdot>x)"
l_minus: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> x \<otimes>\<^bsub>S\<^esub> y = \<ominus>\<^bsub>S\<^esub> (x \<otimes>\<^bsub>S\<^esub> y)"
ub_pi_def: fixes prec :: "nat" shows "ub_pi prec = (let A = rapprox_rat prec 1 5; B = lapprox_rat prec 1 239 in Float 1 2 * float_plus_up prec (Float 1 2 * float_round_up prec (A * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (A * A)))) (- float_round_down prec (B * lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (float_round_up (Suc prec) (B * B)))))"
weakInput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F \<langle>\<epsilon>, \<Psi> \<otimes> \<one>\<rangle>" shows "\<Psi> : Q \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<Longrightarrow>K\<lparr>N[xvec::=Tvec]\<rparr> \<prec> P[xvec::=Tvec]"
loose_bvar1_decr'''': fixes t :: "Core.term"   and lev' :: "nat"   and lev :: "nat" assumes "\<not> loose_bvar1 t lev'"   and "lev' \<le> lev"   and "\<not> loose_bvar1 t (Suc lev)" shows "\<not> loose_bvar1 (decr lev' t) lev"
norm_triangle_lt: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y < e" shows "norm (plus x y) < e"
assms: shows "\<forall>n. (E n \<and> X) = (E n \<and> Y)"