hcju/isabelleps huggingface dataset:
class_add_shconf: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P,h \<turnstile>\<^sub>s sh \<surd>"   and "\<not> is_class P C"   and "\<And>C sobj D'. \<lbrakk>sh C = \<lfloor>sobj\<rfloor>; P \<turnstile> C \<preceq>\<^sup>* D'\<rbrakk> \<Longrightarrow> D' \<noteq> C" shows "class_add P (C, cdec),h \<turnstile>\<^sub>s sh \<surd>"
type_aexpr: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'AExp" shows "\<Gamma> \<turnstile>\<^sub>a e \<in> max_dom (to_total \<Gamma> ` aexp_vars e)"
nat_pow_closed: fixes x :: "'a list"   and n :: "nat" assumes "x \<in> carrier (K [X])" shows "x [^]\<^bsub>K [X]\<^esub> n \<in> carrier (K [X])"
pathsConvergeI: fixes g :: "'g"   and x :: "'node"   and xs :: "'node list"   and z :: "'node"   and y :: "'node"   and ys :: "'node list" assumes "g \<turnstile> x-xs\<rightarrow>z"   and "g \<turnstile> y-ys\<rightarrow>z"   and "1 < length xs"   and "1 < length ys"   and "set (butlast xs) \<inter> set (butlast ys) = {}" shows "old.pathsConverge g x xs y ys z"
case_2: fixes f1 :: "'a"   and f2 :: "enat \<Rightarrow> 'a"   and x2 :: "enat" shows "(case eSuc x2 of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = f2 x2"
prefix_length_le: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le> ys" shows "length xs \<le> length ys"
cp0: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f self a1 \<tau> = f (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) \<tau>"
I_lift_nnf_qe: fixes qe :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and xs :: "real list" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)"   and "\<And>xs \<phi>. nqfree \<phi> \<Longrightarrow> R.I (qe \<phi>) xs = (\<exists>x. R.I \<phi> (x # xs))" shows "R.I (R.lift_nnf_qe qe \<phi>) xs = R.I \<phi> xs"
C: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], x # h) v = ([y, x], rev v @ x # h)"
f_shrink_one: fixes k :: "nat"   and xs :: "'b message_af list" assumes "0 < k"   and "length xs = k" shows "xs \<div> k = [last_message xs]"
contour_integrable_continuous_part_circlepath: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on (path_image (part_circlepath z r s t)) f" shows "f contour_integrable_on part_circlepath z r s t"
single_eq_single: fixes a :: "'a"   and b :: "'a" shows "(\<LM> a  \<RM> = \<LM> b  \<RM>) = (a = b)"
sublistD_bounds_D: fixes PROB :: "(('b, bool) fmap \<times> ('b, bool) fmap) set" assumes "finite PROB" shows "problem_plan_bound_charles PROB \<le> problem_plan_bound PROB"
rep_eq: fixes x :: "'c Filters.filter"   and xa :: "'c Filters.filter" shows "(x \<le> xa) = (Filters.filter.Rep_filter x \<subseteq> Filters.filter.Rep_filter xa)"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (trg\<^sub>B f) = B.arr f"
bisim1AAccFail: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr expr1"   and i :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val"   and v' :: "'addr val" shows "P,a\<lfloor>i\<rceil>,h \<turnstile> (Throw ad, xs) \<leftrightarrow> ([v, v'], xs, length (compE2 a) + length (compE2 i), \<lfloor>ad\<rfloor>)"
split_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and t' :: "('a, 'b, 'c) event list"   and t'' :: "('a, 'b, 'c) event list" assumes "trace c t c'"   and "t = t' @ t''" shows "\<exists>c''. trace c t' c'' \<and> trace c'' t'' c'"
tendsto_real_sqrt: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. sqrt (f x)) \<longlongrightarrow> sqrt x) F"
condition1_part4: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k a (Suc k) A"   and "(if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) = nrows A" shows "is_zero_row_upt_k b (Suc k) A"
finite_measure_axioms: shows "finite_measure (Pi\<^sub>M I M)"
hyps: fixes s :: "'a"   and \<alpha> :: "'a"   and t :: "'a" assumes "(s, (\<alpha>, t) # xs) \<in> seq lrs" shows "set (labels (t, xs)) \<subseteq> (ars\<^sup>+)\<inverse> \<down>s {\<alpha>}"
butlast_map: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "butlast (map f xs) = map f (butlast xs)"
tB: shows "trace B = (\<Sum>i = 0..<n. B $$ (i, i))"
res_def: fixes \<eta> :: "nat" shows "MAC.res \<eta> = 1\<^sub>C |\<^sub>= MAC.enm \<eta> |\<^sub>= MAC.dem \<eta> \<rhd> 1\<^sub>C |\<^sub>= parallel_wiring \<rhd> parallel_resource1_wiring \<rhd> MAC.RO.res \<eta> \<parallel> MAC.INSEC.res"
open_vimage_fst: fixes S :: "'b set" assumes "open S" shows "open (fst -` S)"
nz: fixes z :: "complex" assumes "z \<in> S" shows "g__ z + g__ a \<noteq> 0"
map_disc_iff: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d tree" shows "(map_tree f a = \<langle>\<rangle>) = (a = \<langle>\<rangle>)"
strand_subst_fv_bounded_if_img_bounded: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list" assumes "range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S" shows "fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> fv\<^sub>s\<^sub>t S"
l2product_lsum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i square_integrable S"   and "g square_integrable S" shows "l2product S (\<lambda>x. \<Sum>i\<in>I. f i x) g = (\<Sum>i\<in>I. l2product S (f i) g)"
IH_2: fixes n :: "'node"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "n \<notin> set ns"   and "set ns \<subseteq> set (\<alpha>n g)"   and "distinct ns"   and "foldr (aux_2 g) ns (Mapping.empty, Mapping.empty) = (u', p')" shows "Mapping.lookup u' n = None"
chine_simps_3: shows "C.cod \<mu>\<nu>\<pi>.chine = \<mu>\<nu>\<pi>.cod.apex"
subsetI: fixes A :: "'c set"   and B :: "'c set" assumes "\<And>x. s2p A x \<Longrightarrow> s2p B x" shows "A \<subseteq> B"
maybe_arith_int_never_string: fixes f :: "int \<Rightarrow> int \<Rightarrow> int"   and a :: "value option"   and b :: "value option"   and x :: "String.literal" shows "maybe_arith_int f a b \<noteq> Some (Str x)"
cod_src_eq_dom_src: shows "\<epsilon>.cod.src = \<epsilon>.dsrc"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and attach_shadow_root :: "(_) element_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr) prog"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog" assumes "known_ptr = ShadowRootClass.known_ptr"   and "attach_shadow_root = l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_attach_shadow_root set_shadow_root set_mode get_tag_name get_shadow_root" shows "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr set_shadow_root set_mode attach_shadow_root get_tag_name get_shadow_root"
fully_faithful_functor_axioms: shows "fully_faithful_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R"
assms_1: shows "\<forall>r'\<ge>r. v_hist r' = Map.empty"
permutations_of_set_image_permutes: fixes \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "\<sigma> permutes A" shows "map \<sigma> ` permutations_of_set A = permutations_of_set A"
LastPoint: fixes msg :: "('p, 'v) message" assumes "enabled (last exec) msg" obtains n :: "nat" where "n < length exec" and "enabled (execConf n) msg" and "\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> execMsg n'" and "\<forall>n0. n0 < length exec \<and> enabled (execConf n0) msg \<and> (\<forall>n'\<ge>n0. n' < length trace \<longrightarrow> msg \<noteq> execMsg n') \<longrightarrow> n \<le> n0"
add_def: fixes lp1 :: "(nat, rat) fmap"   and lp2 :: "(nat, rat) fmap" shows "add lp1 lp2 = foldl (\<lambda>lp v. add_monom (get_var_coeff lp1 v) v lp) lp2 (ordered_keys lp1)"
steps_Fault: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and f :: "'f" shows "\<Gamma> \<turnstile> (c, Fault f) \<rightarrow>\<^sup>* (Skip, Fault f)"
hom_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "f (inv\<^bsub>G\<^esub> x) = inv\<^bsub>C\<^esub> f x"
vcard_finite_set: fixes A :: "V set" assumes "finite A" shows "vcard (ZFC_in_HOL.set A) = ord_of_nat (card A)"
poincare_mapsto_imp_flowsto: fixes P :: "'e set"   and X0 :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and S :: "'e set"   and CX :: "'e set"   and X1 :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set" assumes "poincare_mapsto P X0 S CX X1"   and "closed P" shows "flowsto X0 {0<..} (CX \<times> UNIV) (fst ` X1 \<times> UNIV)"
simps_17: fixes M :: "char list"   and n :: "nat"   and P :: "'b prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Invoke M n, P, pc, mxs, T\<^sub>r, ST, LT) = (n < length ST \<and> (ST ! n \<noteq> NT \<longrightarrow> (\<exists>C D Ts T m. ST ! n = Class C \<and> P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = m in D \<and> P \<turnstile> rev (take n ST) [\<le>] Ts)))"
assms_1: shows "x \<cdot> z = z \<cdot> y"
meet_cong_l: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and x' :: "'c"   and y :: "'c" assumes "weak_lower_semilattice L"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "y \<in> carrier L"   and "x .=\<^bsub>L\<^esub> x'" shows "x \<sqinter>\<^bsub>L\<^esub> y .=\<^bsub>L\<^esub> x' \<sqinter>\<^bsub>L\<^esub> y"
smc_1_components_1: fixes \<aa> :: "V"   and \<ff> :: "V" shows "smc_1 \<aa> \<ff>\<lparr>Obj\<rparr> = ZFC_in_HOL.set {\<aa>}"
complete_treeD_del: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c" assumes "complete t" shows "complete (treeD (Tree23_Map.del x t))"
ccINF_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` {}) = top"
AE_restrict_space_iff: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "\<Omega> \<inter> space M \<in> sets M" shows "almost_everywhere (restrict_space M \<Omega>) P = set_almost_everywhere \<Omega> M P"
from_nat_into_def: fixes S :: "'c set"   and n :: "nat" shows "from_nat_into S n = (if n \<in> to_nat_on S ` S then inv_into S (to_nat_on S) n else SOME s. s \<in> S)"
fin_ParT_noWhileL: fixes cl :: "('a, 'b, 'c) cmd list" shows "noWhile (ParT cl) = noWhileL cl"
nI: shows "n = pickFT cl \<longrightarrow> 0 \<notin> I"
tranclp_trans_induct: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x y"   and "\<And>x y. r x y \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>r\<^sup>+\<^sup>+ x y; P x y; r\<^sup>+\<^sup>+ y z; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x y"
msb_int_def: fixes x :: "int" shows "msb x = (x < 0)"
axioms_2: fixes f :: "'d \<Rightarrow> 'e" assumes "antilinear f" shows "antilinear_axioms f"
ext_inject: fixes mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d"   and mult' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one' :: "'c"   and more' :: "'d" shows "(\<lparr>mult = mult, one = one, \<dots> = more\<rparr> = \<lparr>mult = mult', one = one', \<dots> = more'\<rparr>) = (mult = mult' \<and> one = one' \<and> more = more')"
unit_unique_upto_unique_iso: shows "\<exists>!f. S.in_hom f M.unity M\<^sub>\<omega>.unity \<and> S.iso f \<and> S.comp f \<i>[a] = S.comp \<omega> (M\<^sub>\<omega>.tensor f f)"
yun_erel_mult: fixes F :: "int poly"   and f :: "rat poly"   and G :: "int poly"   and g :: "rat poly" assumes "yun_erel F f"   and "yun_erel G g" shows "yun_erel (F * G) (f * g)"
not_v: shows "eval r_univ [i, x] \<noteq> Some v"
ord_equiv_isom: fixes E :: "('g, 'h) Order_scheme" assumes "Order E"   and "ord_equiv D E" shows "\<exists>f. ord_isom D E f"
stutter_sampler_def: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'f" shows "stutter_sampler f \<sigma> \<equiv> f 0 = 0 \<and> strict_mono f \<and> (\<forall>k n. f k < n \<and> n < f (Suc k) \<longrightarrow> \<sigma> n = \<sigma> (f k))"
wide_subdigraph_axioms: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<CC>"
eq_swap_indoc: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a list"   and v :: "'a list"   and c :: "'a" assumes "traces ind"   and "traces.eq_swap ind u v"   and "traces.indoc ind c u" shows "traces.indoc ind c v"
some_worst_in_worst: shows "local.some_worst \<in> local.worst"
assms_2: assumes "sqn (rt \<xi>) dip \<le> sqn (rt \<xi>') dip" shows "R dip \<xi> \<xi>'"
nullhomotopic_through_contractible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and U :: "'c set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "continuous_on T g"   and "g ` T \<subseteq> U"   and "contractible T" obtains c :: "'c" where "homotopic_with_canon (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
strategy_winning_by_Even_def: fixes \<sigma>\<^sub>e :: "'a list \<Rightarrow> 'a"   and p :: "'a list" shows "strategy_winning_by_Even \<sigma>\<^sub>e p \<equiv> \<forall>\<sigma>\<^sub>o. induced_play (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) p \<in> A"
mset_quicksort: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "mset (quicksort R xs) = mset xs"
list_all_conv_same_lab: fixes ls :: "t list list"   and ts :: "t list"   and "is" :: "nat list"   and i :: "nat" assumes "list_all (\<lambda>i. i < length ls \<and> ls ! i = ts) (is @ [i])" shows "same_lab (is @ [i]) ls = Some ts"
exec_gpv_map_gpv_id: fixes "oracle" :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and f :: "'e \<Rightarrow> 'a"   and gpv :: "('e, 'c, 'd) gpv"   and \<sigma> :: "'b" shows "exec_gpv oracle (map_gpv f id gpv) \<sigma> = map_spmf (apfst f) (exec_gpv oracle gpv \<sigma>)"
polyfun_eq_0: fixes c :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<forall>z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)) = (\<forall>k\<le>n. c k = (0::'a))"
induct: fixes P :: "('d, 'c) stateful_strand_step list \<Rightarrow> ('c \<Rightarrow> ('d, 'c) Term.term) \<Rightarrow> (('d, 'c) Term.term \<times> ('d, 'c) Term.term) set \<Rightarrow> bool"   and a0 :: "('d, 'c) stateful_strand_step list"   and a1 :: "'c \<Rightarrow> ('d, 'c) Term.term"   and a2 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" assumes "\<And>I D. P [] I D"   and "\<And>t s A I D. P A I (insert (t \<cdot> I, s \<cdot> I) D) \<Longrightarrow> P (insert\<langle>t,s\<rangle> # A) I D"   and "\<And>t s A I D. P A I (D - {(t \<cdot> I, s \<cdot> I)}) \<Longrightarrow> P (delete\<langle>t,s\<rangle> # A) I D"   and "\<And>v A I D. P A I D \<Longrightarrow> P (send\<langle>v\<rangle> # A) I D"   and "\<And>v A I D. P A I D \<Longrightarrow> P (receive\<langle>v\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<langle>v: va \<doteq> vb\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<langle>v: va \<in> vb\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va \<or>\<notin>: vb\<rangle> # A) I D" shows "P a0 a1 a2"
axioms_1: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and \<psi> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c" assumes "meta_adjunction C D F G \<phi> \<psi>" shows "category C"
fbox_cond: fixes p :: "'a"   and x :: "'a"   and y :: "'a"   and q :: "'a" shows " |if p then x else y] q = (ad p +  |x] q) \<cdot> (d p +  |y] q)"
le_simps_3: fixes m :: "nat"   and n :: "nat" shows "(Suc m \<le> n) = (m < n)"
openin_prod_Times_iff: fixes X :: "'d topology"   and Y :: "'e topology"   and S :: "'d set"   and T :: "'e set" shows "openin (prod_topology X Y) (S \<times> T) = (S = {} \<or> T = {} \<or> openin X S \<and> openin Y T)"
rtrancl_on_into_rtrancl_on: fixes a :: "'e"   and b :: "'e"   and F :: "'e set"   and r :: "('e \<times> 'e) set"   and c :: "'e" assumes "(a, b) \<in> rtrancl_on F r"   and "(b, c) \<in> r"   and "c \<in> F" shows "(a, c) \<in> rtrancl_on F r"
hcomp_reassoc_1: fixes \<tau> :: "'b"   and \<mu> :: "'b"   and \<nu> :: "'b" assumes "D.arr \<tau>"   and "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<tau> = trg\<^sub>D \<mu>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>" shows "(\<tau> \<star>\<^sub>D \<mu>) \<star>\<^sub>D \<nu> = \<a>\<^sub>D\<^sup>-\<^sup>1[D.cod \<tau>, D.cod \<mu>, D.cod \<nu>] \<cdot>\<^sub>D (\<tau> \<star>\<^sub>D \<mu> \<star>\<^sub>D \<nu>) \<cdot>\<^sub>D \<a>\<^sub>D[D.dom \<tau>, D.dom \<mu>, D.dom \<nu>]"
check_mult_alt_def: fixes p :: "nat"   and A :: "(nat, int mpoly) fmap"   and i :: "nat"   and q :: "int mpoly"   and \<V> :: "nat set"   and r :: "int mpoly" shows "SPEC (\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars q \<subseteq> \<V> \<and> vars r \<subseteq> \<V>) \<bind> (\<lambda>b. if \<not> b then RETURN False else ASSERT (p \<in># dom_m A) \<bind> (\<lambda>_. let p = the (fmlookup A p) in mult_poly_spec p q \<bind> (\<lambda>pq. weak_equality pq r \<bind> RETURN))) \<le> check_mult A \<V> p q i r"
ms_closure_embed: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "P xs f basis" shows "ms_closure P xs f basis"
witness_list_append: fixes ps :: "('c, 'b) fm list"   and qs :: "('c, 'b) fm list"   and used :: "'b set" shows "\<exists>extra. witness_list (ps @ qs) used = witness_list ps used @ witness_list qs (extra \<union> used)"
lnth_llist_of: fixes xs :: "'b list" shows "($) (llist_of xs) = (!) xs"
Interleaves_nil: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and ys :: "'b list"   and zs :: "'b list" assumes "[] \<cong> {ys, zs, P}" shows "ys = [] \<and> zs = []"
quasi_inverse_simps_1: fixes f :: "'b" assumes "D.equivalence_map f" shows "D.equivalence_map f\<^sup>~\<^sup>D"
rreq_rrep_fresh_def: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and m :: "msg" shows "rreq_rrep_fresh crt m \<equiv> case m of Rreq hopsc x xa xb oipc osnc ipcc \<Rightarrow> ipcc \<noteq> oipc \<longrightarrow> oipc \<in> kD crt \<and> (osnc < sqn crt oipc \<or> sqn crt oipc = osnc \<and> the (dhops crt oipc) \<le> hopsc \<and> the (flag crt oipc) = val) | Rrep hopsc dipc dsnc x ipcc \<Rightarrow> ipcc \<noteq> dipc \<longrightarrow> dipc \<in> kD crt \<and> sqn crt dipc = dsnc \<and> the (dhops crt dipc) = hopsc \<and> the (flag crt dipc) = val | _ \<Rightarrow> True"
simps_3: fixes fresh :: "'a set \<Rightarrow> 'a"   and S :: "'b formula \<Rightarrow> 'b formula"   and F :: "'b formula" assumes "freshstuff fresh" shows "freshnats.tseytin_tran1 S (\<^bold>\<not> F) = [S (\<^bold>\<not> F) \<^bold>\<leftrightarrow> \<^bold>\<not> (S F)] @ freshnats.tseytin_tran1 S F"
pair_subst_ident: fixes t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "(fv t \<union> fv t') \<inter> subst_domain \<theta> = {}" shows "(t, t') \<cdot>\<^sub>p \<theta> = (t, t')"
induct: fixes x1 :: "exp"   and x2 :: "exp"   and P :: "exp \<Rightarrow> exp \<Rightarrow> bool" assumes "x1 \<longrightarrow> x2"   and "\<And>v x e. is_val v \<Longrightarrow> P (EApp (ELam x e) v) (subst x v e)"   and "\<And>e1 e1' e2. \<lbrakk>e1 \<longrightarrow> e1'; P e1 e1'\<rbrakk> \<Longrightarrow> P (EApp e1 e2) (EApp e1' e2)"   and "\<And>e2 e2' e1. \<lbrakk>e2 \<longrightarrow> e2'; P e2 e2'\<rbrakk> \<Longrightarrow> P (EApp e1 e2) (EApp e1 e2')"   and "\<And>f n1 n2. P (EPrim f (ENat n1) (ENat n2)) (ENat (f n1 n2))"   and "\<And>e1 e1' f e2. \<lbrakk>e1 \<longrightarrow> e1'; P e1 e1'\<rbrakk> \<Longrightarrow> P (EPrim f e1 e2) (EPrim f e1' e2)"   and "\<And>e2 e2' f e1. \<lbrakk>e2 \<longrightarrow> e2'; P e2 e2'\<rbrakk> \<Longrightarrow> P (EPrim f e1 e2) (EPrim f e1 e2')"   and "\<And>thn els. P (EIf (ENat 0) thn els) els"   and "\<And>n thn els. n \<noteq> 0 \<Longrightarrow> P (EIf (ENat n) thn els) thn"   and "\<And>cond cond' thn els. \<lbrakk>cond \<longrightarrow> cond'; P cond cond'\<rbrakk> \<Longrightarrow> P (EIf cond thn els) (EIf cond' thn els)" shows "P x1 x2"
simps_1: fixes uu :: "'b pmf" shows "replicate_pmf 0 uu = return_pmf []"
suc_not_zero: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep a + 1 \<noteq> CARD('a)" shows "plus a one \<noteq> zero"
coboundedI1: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<le> a" shows "c \<le> a + b"
geq0: shows "\<forall>v. dim_vec v = dim_col A \<and> vec_norm v = 1 \<longrightarrow> 0 \<le> inner_prod v (A *\<^sub>v v)"
frontier_of_subset_eq: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "(X frontier_of S \<subseteq> S) = closedin X S"
le2: shows "tensor_mat (1\<^sub>m d1) m2 \<le>\<^sub>L 1\<^sub>m d"
Rep_circline_mat_cases: fixes y :: "complex \<times> complex \<times> complex \<times> complex" assumes "y \<in> hermitean_nonzero" obtains x :: "circline_mat" where "y = Rep_circline_mat x"
LI_preserves_trm_wf: fixes S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list"   and \<theta>' :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S, \<theta>) \<leadsto>\<^sup>* (S', \<theta>')"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S')"
strong_sum_character: fixes x :: "i"   and y :: "i" shows "(P x (x \<oplus> y) \<and> P y (x \<oplus> y)) \<and> (\<forall>w. P w (x \<oplus> y) \<longrightarrow> O w x \<or> O w y)"