hcju/isabelleps huggingface dataset:
simps_3: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and X :: "char list set" shows "L (c\<^sub>1;; c\<^sub>2) X = L c\<^sub>1 (L c\<^sub>2 X)"
zero_dim: fixes K :: "'a set" shows "dimension 0 K {\<zero>}"
Z_last: shows "\<forall>x\<in>Z. x $ n = (0::'a)"
bal_list: fixes n :: "nat"   and xs :: "'b list" shows "Root_Balanced_Tree.bal_list n xs = Balance.bal_list n xs"
SUP_upper: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
connectedin_eq_not_separated: fixes X :: "'c topology"   and S :: "'c set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>C1 C2. C1 \<union> C2 = S \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> separatedin X C1 C2))"
disc_eq_case_2: fixes option :: "'b option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | Some uu_ \<Rightarrow> True)"
Skip: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile>\<^sub>1 {P} SKIP { \<lambda>s. Suc 0 \<Down> P}"
trimono_spec_defs_3: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "f \<circ> g = (\<lambda>x. f (g x))"
eq_esing_up: fixes x :: "Vars.var"   and a :: "Arity"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" shows "eq\<rho> (esing x\<cdot>(up\<cdot>a)) \<rho>1 \<rho>2 = eq a (\<rho>1 x) (\<rho>2 x)"
l3_inv7_derived: fixes implem :: "chan \<Rightarrow> msg" assumes "dhlvl3 implem" shows "dhlvl3.l3_inv3 implem \<inter> l3_inv4 \<subseteq> dhlvl3.l3_inv7 implem"
get_def: fixes h :: "heap" shows "Ref.get h = (from_nat \<circ>\<circ>\<circ> refs) h TYPEREP('d) \<circ> addr_of_ref"
is_faithful: fixes f :: "'d"   and f' :: "'d" assumes "D.par f f'"   and "G f = G f'" shows "f = f'"
carrier_HOM_M_Z: fixes M :: "('f, 'a, 'g) Module_scheme"   and e :: "'h" assumes "R module M" shows "carrier (HOM\<^bsub>R\<^esub> M Zm R e) = {mzeromap M (Zm R e)}"
assms_3: shows "gauss_int_norm z = p\<^sup>2"
angle_actrans_disj: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "|~ \<langle>F \<or> G\<rangle>_v = (\<langle>F\<rangle>_v \<or> \<langle>G\<rangle>_v)"
residue_prime_has_primroot: fixes p :: "nat" assumes "prime p" shows "\<exists>a\<in>totatives p. ord p a = p - 1"
a: shows "((\<lambda>s. True) \<and>* P) h"
redT_preserves_final_thread: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "s -t'\<triangleright>ta\<rightarrow> s'"   and "final_thread s t" shows "final_thread s' t"
bd_cts_tr_def: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "bd_cts_tr T = (\<forall>M. (\<forall>i. le_trans (M i) (M (Suc i)) \<and> feasible (M i)) \<longrightarrow> equiv_trans (T (Sup_trans (range M))) (Sup_trans (range (T \<circ> M))))"
gcd_diff2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "'a"   and m :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd (minus n m) n = gcd m n"
frameEq_2: fixes \<Psi>' :: "'e"   and A\<^sub>F :: "name list"   and \<Psi> :: "'e" shows "(\<langle>\<epsilon>, \<Psi>'\<rangle> = \<langle>A\<^sub>F, \<Psi>\<rangle>) = (A\<^sub>F = [] \<and> \<Psi> = \<Psi>')"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "Right_a.ide (Right_a.comp g f)" obtains "Right_a.seq g f" and "Right_a.seq f g" and "Right_a.comp g f = Right_a.dom f" and "Right_a.comp g f = Right_a.cod g"
put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) document_ptr"   and setter :: "('w \<Rightarrow> 'w) \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme"   and v :: "'w" assumes "l_put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
sublist_MEM: fixes h :: "'a"   and l1 :: "'a list"   and l2 :: "'a list" assumes "subseq (h # l1) l2" shows "ListMem h l2"
assms: shows "\<forall>i\<in>FO. \<exists>!p. p < length w \<and> snd (w ! p) ! i"
Spy_analz_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> kerbIV" shows "(Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
Class_widen: fixes P :: "'a prog"   and C :: "char list"   and T :: "ty" assumes "P \<turnstile> Class C \<le> T" shows "\<exists>D. T = Class D"
assms_3: shows "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
plus_eint_simps_2: fixes q :: "eint" shows "\<infinity> + q = \<infinity>"
polynomial_pow_not_zero: fixes p :: "'a list"   and n :: "nat" assumes "p \<in> carrier (poly_ring R)"   and "p \<noteq> []" shows "p [^]\<^bsub>poly_ring R\<^esub> n \<noteq> []"
abs_eq: fixes x :: "'b multiset \<times> 'b multiset" shows "zcount (abs_zmultiset x) = (case x of (Mp, Mn) \<Rightarrow> \<lambda>x. int (count Mp x) - int (count Mn x))"
squarefree_of_int_polyI: fixes p :: "int poly" assumes "squarefree p"   and "Polynomial.content p = 1" shows "squarefree (Ring_Hom_Poly.of_int_poly p)"
strict_prefix_Cons_elim: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix (x # xs) ys" obtains ys' :: "'b list" where "ys = x # ys'" and "strict_prefix xs ys'"
Assign: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and a :: "aexp" shows "\<turnstile>\<^sub>t {\<lambda>l s. P l (s[a/x])} x ::= a {P}"
lam_def: shows "lam = [:0::'a, 1::'a:] \<cdot>\<^sub>m 1\<^sub>m n"
f_Suc: fixes x :: "'s"   and y :: "'s"   and n :: "nat" assumes "x \<noteq> y" shows "f x y (Suc n) = u x y n"
group_left_neutral: fixes a :: "'c" shows "(0::'c) + a = a"
sdrop_length_shift: fixes xs :: "'d list"   and s :: "'d stream" shows "sdrop (length xs) (xs @- s) = s"
trans_not_delay_mono: fixes u :: "'c \<Rightarrow> 'a"   and u' :: "'c \<Rightarrow> 'a"   and x :: "'c"   and c :: "'c" assumes "local.trans u u'"   and "u \<in> V"   and "x \<in> \<X>"   and "u' x = (0::'a)"   and "c \<in> \<X>" shows "u' c \<le> u c"
Enabled_disj2: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> Enabled F \<longrightarrow> Enabled (G \<or> F)"
from_list_def: fixes l :: "(('v \<times> 'v) \<times> nat) list" shows "from_list l = foldr (\<lambda>(e, d). add_edge e d) l empty"
theorem_5_invI: fixes fp :: "'a set"   and ds :: "'b set" assumes "\<And>z X. \<lbrakk>z \<in> Rf CH fp; z \<in> X; stable_on ds X\<rbrakk> \<Longrightarrow> False" shows "theorem_5_inv ds fp"
card_length: fixes xs :: "'c list" shows "card (set xs) \<le> length xs"
ys: shows "linorder.sorted (\<sqsubseteq>\<^sub>b) (map fst ys)"
Un_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B = _Collect x (x \<in> A \<or> x \<in> B)"
prebicategory_axioms: shows "prebicategory (\<cdot>) (\<star>) \<a>"
RED_props_3: fixes \<tau> :: "ty" shows "CR3 \<tau>"
sup: fixes f :: "'b \<Rightarrow> 'c"   and F1 :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and F2 :: "'b filter" assumes "f \<in> \<Omega>[F1](g)"   and "f \<in> \<Omega>[F2](g)" shows "f \<in> \<Omega>[sup F1 F2](g)"
set_val_ok: fixes h :: "(_) heap"   and character_data_ptr :: "(_) character_data_ptr"   and tag :: "char list" assumes "ShadowRootClass.type_wf h"   and "character_data_ptr |\<in>| character_data_ptr_kinds h" shows "h \<turnstile> ok set_val character_data_ptr tag"
set_less_eq_aux_Compl_fusion_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and proper_interval :: "'c option \<Rightarrow> 'c option \<Rightarrow> bool"   and g1 :: "('c, 'd) generator"   and g2 :: "('c, 'e) generator"   and ao :: "'c option"   and s1 :: "'d"   and s2 :: "'e" shows "proper_intrvl.set_less_eq_aux_Compl_fusion less proper_interval g1 g2 ao s1 s2 = proper_intrvl.set_less_eq_aux_Compl less proper_interval ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
intros: fixes W :: "('a \<times> 'b) list"   and Q10 :: "'a set"   and Q20 :: "'b set" assumes "distinct W"   and "set W = Q10 \<times> Q20" shows "(Q10 \<times> Q20, W, {}) \<in> frp_initial Q10 Q20"
ListSum_disj_union: fixes A :: "'a list"   and B :: "'a list"   and C :: "'a list"   and f :: "'a \<Rightarrow> nat" assumes "distinct A"   and "distinct B"   and "distinct C"   and "set C = set A \<union> set B"   and "set A \<inter> set B = {}" shows "ListSum C f = ListSum A f + ListSum B f"
False: shows "insort_key (\<lambda>q. dist q p) q ps \<noteq> ps @ [q]"
atLeastLessThan_Suc: fixes a :: "nat"   and b :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "a \<le> b" shows "sum g {a..<Suc b} = sum g {a..<b} + g b"
borel_measurable_lebesgue_integral: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow> 'e"   and N :: "'d measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M restrict_space (measure_pmf A) (set_pmf A))" shows "(\<lambda>x. A.expectation (f x)) \<in> borel_measurable N"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>A. g x = (1::'d)" shows "prod g A = (1::'d)"
alw_ev_coinduct: fixes R :: "'b stream \<Rightarrow> bool"   and w :: "'b stream"   and \<phi> :: "'b stream \<Rightarrow> bool" assumes "R w"   and "\<And>w. R w \<Longrightarrow> ev \<phi> w \<and> ev R (stl w)" shows "alw (ev \<phi>) w"
row_add_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f"   and b :: "'f"   and q :: "'d" shows "row_add A a b q = (\<chi>i j. if i = a then A $ a $ j + q * A $ b $ j else A $ i $ j)"
errMOD_igSubstIGOpSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igWlsAbsIsInBar MOD"   and "igVarIPresIGWls MOD"   and "igOpIPresIGWls MOD"   and "igSubstIPresIGWls MOD"   and "igSubstAbsIPresIGWlsAbs MOD"   and "igWlsDisj MOD"   and "igWlsAbsDisj MOD"   and "igSubstIGOp MOD" shows "igSubstIGOpSTR (errMOD MOD)"
distinct_23: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> x41 ;; x42"
unit_char_4: fixes a :: "'b" assumes "B.obj a" shows "\<exists>!\<psi>. \<guillemotleft>\<psi> : GoH.map\<^sub>0 a \<Rightarrow>\<^sub>D GoH.map a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[GoH.map\<^sub>0 a] = (GoH.map \<i>\<^sub>B[a] \<cdot>\<^sub>D GoH.cmp (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
h'_nonneg: fixes n :: "nat"   and xy :: "nat \<times> nat" shows "0 \<le> h' n xy"
bij_betw_implies_surj_on_and_card_eq: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "finite A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "bij_betw f A B = (f ` A = B \<and> card A = card B)"
join_self_below_6: fixes y :: "'a"   and x :: "'a" shows "(y \<squnion> x \<sqsubseteq> x) = (y \<sqsubseteq> x)"
mono_write_FD: fixes P :: "'d process"   and Q :: "'d process"   and c :: "'e \<Rightarrow> 'd"   and x :: "'e" assumes "P \<le> Q" shows "(c`!`x \<rightarrow> P) \<le> (c`!`x \<rightarrow> Q)"
repc'_def: fixes cfg' :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "repc' cfg' \<equiv> repcs (THE s. s \<in> set_pmf (rept (reps (state cfg)) (action cfg)) \<and> abss s = state cfg') cfg'"
ir_to_Skip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_hoare P c c' Q = (ir_hoare P c SKIP (decomp P c c' Q) \<and> ir_hoare (decomp P c c' Q) SKIP c' Q)"
f: shows "f \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M"
nn_integral_empty: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "space M = {}" shows "integral\<^sup>N M f = 0"
monoI: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "\<And>x y. less_eq_prod x y \<Longrightarrow> f x \<le> f y" shows "prod.mono f"
borel_measurable_enn2real: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. enn2real (f x)) \<in> borel_measurable M"
is_increasingE: fixes f :: "nat \<Rightarrow> nat"   and m :: "nat"   and n :: "nat" assumes "is_increasing f"   and "m < n" shows "f m < f n"
bernoulli_fps_altdef: shows "bernoulli_fps = Abs_fps (\<lambda>n. of_real (bernoulli n) / fact n)"
binomial_pmf_altdef': fixes A :: "'b set"   and n :: "nat"   and p :: "real"   and dflt :: "bool" assumes "finite A"   and "card A = n"   and "p \<in> {0..1}" shows "binomial_pmf n p = map_pmf (\<lambda>f. card {x \<in> A. f x}) (Pi_pmf A dflt (\<lambda>_. bernoulli_pmf p))"
q_minimal: fixes q' :: "nat" assumes "q' \<in># prime_factorization (2 ^ p - 1)" shows "q \<le> q'"
p_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "y \<le> ego.t_stop" shows "ego.p x \<le> ego.p y"
convex_affine_rel_interior_Int: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "affine T"   and "rel_interior S \<inter> T \<noteq> {}" shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
Farkas_Lemma: fixes A :: "'b mat"   and n :: "nat"   and nr :: "nat"   and b :: "'b vec" assumes "A \<in> carrier_mat n nr"   and "b \<in> carrier_vec n" shows "(\<exists>x\<ge>0\<^sub>v nr. A *\<^sub>v x = b) = (\<forall>y. y \<in> carrier_vec n \<longrightarrow> 0\<^sub>v nr \<le> A\<^sup>T *\<^sub>v y \<longrightarrow> (0::'b) \<le> y \<bullet> b)"
subst_del_id: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "id -\<^sub>s x = id"
sign_simps_42: fixes am :: "'a"   and bm :: "'a"   and ck :: "'a" shows "am - bm + ck = am + ck - bm"
tw_def: shows "tw \<equiv> takeWhile ((\<noteq>) u) (tj_stack s)"
holomorphic_nonconstant: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S"   and "open S"   and "\<xi> \<in> S"   and "deriv f \<xi> \<noteq> 0" shows "\<not> f constant_on S"
distinct_1: shows "sort_dir.Left \<noteq> sort_dir.Right"
mcont_Pair: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and lubb :: "'e set \<Rightarrow> 'e"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'e" assumes "mcont lub ord luba orda f"   and "mcont lub ord lubb ordb g" shows "mcont lub ord (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
image_sub0: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
d_2: fixes x :: "'b"   and u :: "'b"   and v :: "'b" assumes "x \<in> cbox a b"   and "x \<in> cbox u v"   and "cbox u v \<subseteq> ball x (d x)"   and "cbox u v \<subseteq> cbox a b" shows "f integrable_on cbox u v"
fv\<^sub>r\<^sub>c\<^sub>v_def: fixes x :: "('d, 'c) strand_step" shows "fv\<^sub>r\<^sub>c\<^sub>v x \<equiv> case x of receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> fv x | _ \<Rightarrow> {}"
degree_smult_not_zero_divisor_left: fixes c :: "'b"   and p :: "'b poly" assumes "\<not> zero_divisor c" shows "degree (smult c p) = degree p"
Rep_dense: fixes x :: "'c dense" shows "Rep_dense x \<in> dense_elements"
comp_arr_dom: fixes f :: "'c"   and a :: "'c" assumes "C.arr f"   and "C.dom f = a" shows "f \<cdot> a = f"
splitA': shows "split_block A' 1 1 = (A1, A2, A0, A3)"
step_Nil': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and asscss :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and ass :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile> (cs, asscss, s) \<rightarrow> (cs', css', t)"   and "cs = []"   and "asscss = ass @ css"   and "ass \<noteq> []" shows "css' = tl ass @ css \<and> (case s of Abrupt s' \<Rightarrow> cs' = snd (hd ass) \<and> t = Normal s' | _ \<Rightarrow> cs' = fst (hd ass) \<and> t = s)"
GE_arOf_def: shows "GE_arOf \<equiv> ProblemIkTpartG.GE_arOf arOf"
fps_divide_uminus': fixes f :: "'a fps"   and g :: "'a fps" shows "f / - g = - (f / g)"
mertens_summable: obtains c :: "real" where "summable (\<lambda>x. (\<MM> (real x) - ln (real x) + c) / real x)"
2: shows "fold_graph Or_ltln false\<^sub>n y (Or\<^sub>n y)"
tranclp_idemp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "P\<^sup>+\<^sup>+\<^sup>+\<^sup>+ = P\<^sup>+\<^sup>+"
has_vector_derivative: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "'b" assumes "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at x within s)" shows "((\<lambda>x. f x ** g x) has_vector_derivative f x ** g' + f' ** g x) (at x within s)"
scaleRat_real_def: fixes x :: "rat"   and y :: "real" shows "x *R y = real_of_rat x * y"
parse_nodes_def: shows "parse_nodes \<equiv> parse_nodes_sumC"
step_5: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "f \<noteq> 0"   and "lt f adds\<^sub>t t" shows "thesis"
ass_5: shows "line_integral_exists F basis (rec_join (a # l))"
minus_u_def: fixes x :: "'a\<^sub>\<bottom>"   and y :: "'a\<^sub>\<bottom>" shows "x - y = (case x of up\<cdot>a \<Rightarrow> \<Lambda> (up\<cdot>b). up\<cdot>(a - b))\<cdot>y"
ba_def: fixes \<phi> :: "'cmp \<Rightarrow> bool" shows "[\<phi>]\<^sub>b \<equiv> \<lambda>t n. \<phi> (t n)"
zero_carrier_mat: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<in> carrier_mat nr nc"
aminus_add_distrib: fixes z :: "ant"   and w :: "ant" shows "- (z + w) = - z + - w"
sndI: fixes x :: "'d \<times> 'e"   and y :: "'d"   and z :: "'e" assumes "x = (y, z)" shows "snd x = z"
has_integral_contour_integral_subpath: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and u :: "real"   and v :: "real" assumes "f contour_integrable_on g"   and "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "u \<le> v" shows "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral contour_integral (subpath u v g) f) {u..v}"
simps: fixes x :: "'a"   and xs :: "'a list" shows "last (x # xs) = (if xs = [] then x else last xs)"
sets_P: fixes J :: "'i set" assumes "finite J"   and "J \<subseteq> I" shows "sets (P J) = sets (Pi\<^sub>M J (\<lambda>_. borel))"
lookup_Some_imp_find_Some: fixes P :: "'b \<Rightarrow> bool"   and xs :: "('b \<times> 'a) list"   and v :: "'a" assumes "lookup P xs = Some v" shows "\<exists>x. List.find (\<lambda>(k, uu_). P k) xs = Some (x, v)"
subdivFace_subdivFace'_eq: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and vol :: "nat option list" assumes "pre_subdivFace g f v vol" shows "subdivFace g f vol = subdivFace' g f v 0 (tl vol)"
psi_inner_alpha: shows "inner_prod \<psi> \<alpha> = complex_of_real (cos (\<theta> / 2))"
case_rews_1: fixes f1 :: "'b"   and f2 :: "'c \<rightarrow> 'b" shows "Maybe_case\<cdot>f1\<cdot>f2\<cdot>\<bottom> = \<bottom>"
projr_def: fixes sum :: "'f + 'e" shows "projr sum = (case sum of Inr x2 \<Rightarrow> x2)"
is_scc_ex1: fixes E :: "('a \<times> 'a) set"   and v :: "'a" shows "\<exists>!scc. is_scc E scc \<and> v \<in> scc"
sel_4: fixes x :: "quat" shows "Im3 (- x) = - Im3 x"
iso_verts_inv_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'a" assumes "digraph_isomorphism hom"   and "u \<in> verts G" shows "iso_verts (inv_iso hom) (iso_verts hom u) = u"
R3_H2_commute2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R3 (H2 P) = H2 (R3 P)"
add_comm: fixes x :: "'a list set"   and y :: "'a list set" shows "x \<union> y = y \<union> x"
simple_syntactic_and_axioms: fixes n :: "nat" shows "simple_syntactic_and (wellformed' n)"
times_eq_iff: fixes A :: "'b set"   and B :: "'c set"   and C :: "'b set"   and D :: "'c set" shows "(A \<times> B = C \<times> D) = (A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
merge_Cons: fixes c :: "'a list"   and pc :: "nat"   and l :: "nat \<times> 'a"   and ls :: "(nat \<times> 'a) list"   and x :: "'a" shows "merge c pc (l # ls) x = merge c pc ls (if fst l = pc + 1 then snd l \<squnion>\<^bsub>f\<^esub> x else if snd l \<sqsubseteq>\<^bsub>r\<^esub> c ! fst l then x else \<top>)"
check_precond_r2_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "check_precond_r2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> = (s\<^sub>e < s\<^sub>o \<and> 0 \<le> v\<^sub>e \<and> 0 \<le> v\<^sub>o \<and> a\<^sub>e < 0 \<and> a\<^sub>o < 0 \<and> 0 < \<delta> \<and> - v\<^sub>o / a\<^sub>o < \<delta>)"
sign_changes_inf_def: fixes ps :: "'a poly list" shows "sign_changes_inf ps = length (remdups_adj (filter (\<lambda>x. x \<noteq> (0::'a)) (map poly_inf ps))) - 1"
order_iff_strict: fixes b :: "'a list"   and a :: "'a list" shows "(b \<le>lex a) = (b <lex a \<or> a = b)"
leg0_lax: shows "\<exists>\<theta>. \<guillemotleft>\<theta> : s\<^sub>0 \<star> w \<Rightarrow> r\<^sub>0\<guillemotright>"
iterate_ball_def: fixes it :: "(bool \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool \<Rightarrow> bool) \<Rightarrow> bool \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" shows "iterate_ball it P = it id (\<lambda>x \<sigma>. P x) True"
simps_1: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d set" assumes "class.wellorder less_eq less" shows "wellorder.enumerate less_eq S 0 = ord.Least less_eq (\<lambda>n. n \<in> S)"
D3: fixes \<psi> :: "\<nu> \<Rightarrow> \<o>" shows "Semantics.d\<^sub>\<kappa> (\<^bold>\<iota>x. \<psi> x) = (if \<exists>x. [\<psi> x in Semantics.w\<^sub>0] \<and> (\<forall>y. [\<psi> y in Semantics.w\<^sub>0] \<longrightarrow> y = x) then Some (THE x. [\<psi> x in Semantics.w\<^sub>0]) else None)"
simps_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C" shows "monoidal_language.Can C (t \<^bold>\<otimes> u) = (monoidal_language.Can C t \<and> monoidal_language.Can C u)"
Fun_\<Psi>: fixes f' :: "'t"   and x' :: "'t" assumes "S'.arr f'"   and "x' \<in> S'.set (S'.dom f')" shows "S.Fun (\<Psi> f') (\<psi> x') = \<Psi>a f' (\<psi> x')"