hcju/isabelleps huggingface dataset:
integrable: fixes c :: "real" shows "(\<lambda>t. c / (ln t)\<^sup>2) integrable_on {2..x}"
pw_ASSUME_1: fixes \<Phi> :: "bool" shows "nofail (ASSUME \<Phi>)"
take_bit_0: fixes a :: "'a" shows "take_bit 0 a = (0::'a)"
hd_prog_append_Read\<^sub>s\<^sub>b: fixes p :: "'b"   and sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "hd_prog p (sb @ [Read\<^sub>s\<^sub>b volatile a t v]) = hd_prog p sb"
fold_plus_sum_list_rev: fixes xs :: "((predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option) list" shows "fold compose_action xs = compose_action (local.sum_list (rev xs))"
simps_4: fixes v\<^sub>1 :: "ref"   and l :: "dag"   and v\<^sub>2 :: "ref"   and r :: "dag" shows "isLeaf (Node Tip v\<^sub>1 (Node l v\<^sub>2 r)) = False"
remove: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> bool" assumes "finite A"   and "x \<in> A" shows "bool.F g A = (g x \<and> bool.F g (A - {x}))"
isPath_head: fixes u :: "nat"   and e :: "nat \<times> nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "isPath u (e # p) v = (fst e = u \<and> e \<in> E \<and> isPath (snd e) p v)"
components_are_iso: fixes a :: "'c" assumes "C.ide a" shows "local.iso (E a)"
set_intros_1: fixes x1 :: "'b" shows "(x1::'b) \<in> atoms (Atom x1)"
1: shows "\<forall>m<length s. \<forall>x. m = length x \<longrightarrow> finite {(t, u). x setinterleaves ((t, u), A)}"
th2_4: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2"   and d :: "hyp2" assumes "a b \<congruent>\<^sub>K c d" shows "b a \<congruent>\<^sub>K c d"
mult_left_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "b \<cdot> (a \<cdot> c) = a \<cdot> (b \<cdot> c)"
not_iTILL_iMOD_subset: fixes n' :: "nat"   and m :: "nat"   and r :: "nat" assumes "0 < n'"   and "m \<noteq> Suc 0" shows "\<not> [\<dots>n'] \<subseteq> [ r, mod m ]"
sp_generalize: fixes F :: "'n \<Rightarrow> 'o fundef option"   and xs :: "('c, 'p, 'n, 'e, 'f, 'g, 'h, 'i) instr list"   and \<Sigma> :: "Unboxed.type option list"   and \<Sigma>' :: "Unboxed.type option list" assumes "sp F xs \<Sigma> = Ok \<Sigma>'" shows "sp F (map generalize_instr xs) (map Map.empty \<Sigma>) = Ok (map Map.empty \<Sigma>')"
algebraic_times_i_iff: fixes x :: "complex" shows "algebraic (\<i> * x) = algebraic x"
da_elim_cases_46: fixes Env :: "env"   and B :: "lname set"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and e :: "expr"   and mn :: "mname"   and pTs :: "ty list"   and args :: "expr list"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l ({accC,statT,mode}e\<cdot>mn( {pTs}args))\<guillemotright> A" obtains E :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "Env\<turnstile> nrm E \<guillemotright>\<langle>args\<rangle>\<guillemotright> A"
sign_simps_3: fixes bf :: "'c"   and af :: "'c"   and cf :: "'c" shows "bf + (af + cf) = af + (bf + cf)"
bind_writer_def: fixes m :: "('w, 'a, 'm) writerT"   and f :: "'a \<Rightarrow> ('w, 'a, 'm) writerT" shows "local.bind_writer m f = WriterT (bind (run_writer m) (\<lambda>(a, ws). bind (run_writer (f a)) (\<lambda>(b, ws'). return (b, ws @ ws'))))"
uminus_Pair: fixes a :: "'c"   and b :: "'d" shows "- (a, b) = (- a, - b)"
cong_pow_2n: fixes m :: "nat" assumes "0 < m"   and "m < p" shows "[m ^ (2 * n) = m ^ n'] (mod p)"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a3 :: "expr list"   and a4 :: "ty list" assumes "P,a1,a2 \<turnstile> a3 [:] a4" obtains E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" where "a1 = Ea__" and "a2 = h__" and "a3 = []" and "a4 = []"   | E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and e :: "expr" and T :: "ty" and es :: "expr list" and Ts :: "ty list" where "a1 = E" and "a2 = h" and "a3 = e # es" and "a4 = T # Ts" and "P,E,h \<turnstile> e : T" and "P,E,h \<turnstile> es [:] Ts"
forall_pos_mono: fixes P :: "real \<Rightarrow> bool"   and e :: "real" assumes "\<And>d e. \<lbrakk>d < e; P d\<rbrakk> \<Longrightarrow> P e"   and "\<And>n. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))"   and "0 < e" shows "P e"
remove1_tail_invar: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tail_invar bq" shows "SkewBinomialHeapStruc.tail_invar (remove1 t bq)"
delete_def: fixes x :: "'a"   and t :: "('a, 'b) rbt" shows "delete x t = paint B (del x t)"
restrict_complement_singleton_eq: fixes f :: "'b \<Rightarrow> 'c option"   and x :: "'b" shows "f |` (- {x}) = f(x := None)"
exists_prerecording_violation_when_card_greater_0: 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 i :: "nat"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "0 < card {j \<in> {i + 1..<length t}. computation.prerecording_event channel trans send recv init t j}" shows "\<exists>j\<in>{i + 1..<length t}. computation.prerecording_event channel trans send recv init t j"
Le_def: fixes v :: "'a aexp"   and va :: "'a aexp" shows "Le v va \<equiv> gNot (Gt v va)"
simps_2: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option" shows "rm_MT_rules C [] = []"
MRList3: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and xa :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<notin> dom (Cp xa)" shows "applied_rule_rev Cp x (a @ b # xs @ [xa]) = applied_rule_rev Cp x (a @ b # xs)"
maxsimp_def: fixes x :: "'b set" shows "codomain.chamber x \<equiv> x \<in> Y \<and> (\<forall>z\<in>Y. order.greater_eq z x \<longrightarrow> z = x)"
vertical_chop_restriction_res_consec_or_empty: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=v1--v2 \<and> restrict v1 (res ts) c \<noteq> bot \<and> consec (lan v1) (lan v2) \<and> \<not> consec (restrict v1 (res ts) c) (restrict v2 (res ts) c) \<longrightarrow> restrict v2 (res ts) c = bot"
Pd'_d'_range: shows "Field (Pd' (Xd x)) \<subseteq> {y. Xd y = Xd x}"
H_fcs_eq_G_fcs: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d"   and b :: "'d" assumes "bidel_arc G M a"   and "b \<in> arcs G"   and "{b, pre_digraph_map.face_cycle_succ M b} \<inter> {a, edge_rev M a} = {}" shows "pre_digraph_map.face_cycle_succ (bidel_arc.HM G M a) b = pre_digraph_map.face_cycle_succ M b"
derivative_is_smooth: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "smooth_on S f" shows "smooth_on S (\<lambda>x. frechet_derivative f (at x) v)"
assms_2: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "HT \<pi> (\<lambda>s'. (f s', f s) \<in> R \<and> P s') (PCall p) Q"   and "P s" shows "wp \<pi> (PCall p) (Q s) s"
AE_I: fixes M :: "'d measure"   and P :: "'d \<Rightarrow> bool"   and N :: "'d set" assumes "{x \<in> space M. \<not> P x} \<subseteq> N"   and "emeasure M N = 0"   and "N \<in> sets M" shows "almost_everywhere M P"
in_set_dropD: fixes x :: "'c"   and n :: "nat"   and xs :: "'c list" assumes "x \<in> set (drop n xs)" shows "x \<in> set xs"
conc_trans: fixes C :: "'a nres"   and R :: "('a \<times> 'b) set"   and B :: "'b nres"   and R' :: "('b \<times> 'c) set"   and A :: "'c nres" assumes "C \<le> \<Down> R B"   and "B \<le> \<Down> R' A" shows "C \<le> \<Down> R (\<Down> R' A)"
normalize_quot_eq_iff: fixes b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<noteq> (0::'b)"   and "d \<noteq> (0::'b)" shows "(normalize_quot (a, b) = normalize_quot (c, d)) = (a * d = b * c)"
f3: shows "3 \<le> rk {A, B, A', P}"
re_def: shows "\<forall>c. re c = Abs_nat_int {1}"
1: fixes i :: "nat" shows "\<exists>a\<in>alphabet A. r (Suc i) \<in> transition A a (r i)"
intros_24: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and v0s :: "v list"   and n :: "nat"   and j :: "nat" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>" shows "\<lparr>s;v0s;[Local n i vs es]\<rparr> \<leadsto>_ j \<lparr>s';v0s;[Local n i vs' es']\<rparr>"
PDG_BS_valid_node: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and control_dependence :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g"   and S :: "'g set" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "n \<in> PDG.PDG_BS sourcenode targetnode valid_edge Def Use control_dependence S" shows "CFG.valid_node sourcenode targetnode valid_edge n"
infpath_model: fixes f :: "nat \<Rightarrow> bool list"   and Cs :: "fterm literal set set" assumes "wf_infpath f"   and "\<forall>n. \<not> falsifies\<^sub>c\<^sub>s (f n) Cs"   and "finite Cs"   and "\<forall>C\<in>Cs. finite C" shows "eval\<^sub>c\<^sub>s HFun (extend f) Cs"
Ints_scalar_prod: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n"   and "\<And>i. i < n \<Longrightarrow> v $ i \<in> \<int>"   and "\<And>i. i < n \<Longrightarrow> w $ i \<in> \<int>" shows "v \<bullet> w \<in> \<int>"
h2: shows "\<not> t mod cycleLength (nC k) mem schedule (nC k)"
perm_append1: fixes xs :: "'b list"   and ys :: "'b list"   and l :: "'b list" assumes "xs <~~> ys" shows "l @ xs <~~> l @ ys"
recip_succ_recip: shows "recip \<circ> succ \<circ> recip = (\<lambda>(x, y). (x, x + y))"
le_supI: fixes a :: "'b upred"   and x :: "'b upred"   and b :: "'b upred" assumes "x \<sqsubseteq> a"   and "x \<sqsubseteq> b" shows "x \<sqsubseteq> (a \<or> b)"
monom_finsum_decomp: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" shows "p = (\<Oplus>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub>t\<in>set (UP.dense_repr p). UP.monom (fst t) (snd t))"
map_id: fixes t :: "('c, 'd) resource" shows "map_resource' id t = t"
totalize_transient_iff: fixes F :: "'c program"   and A :: "'c set" shows "(totalize F \<in> transient A) = (\<exists>act\<in>Acts F. A \<subseteq> Domain act \<and> act `` A \<subseteq> - A)"
simps: fixes abort :: "bool"   and x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist"   and n :: "nat" shows "power_ms_aux abort (MSLCons x xs) n = scale_shift_ms_aux (power_expansion abort (fst x) n, snd x * real n) (powser_ms_aux (gbinomial_series abort (real n)) (scale_shift_ms_aux (inverse (fst x), - snd x) xs))"
abs01: fixes x :: "real"   and y :: "real" assumes "0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1" shows "(x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0) = (\<bar>x - y\<bar> \<in> {0, 1})"
shift_replicate_sconst: fixes n :: "nat"   and x :: "'b" shows "replicate n x @- sconst x = sconst x"
restr_simple_state_rel_def: fixes R :: "'d set"   and ES :: "('e \<times> 'f) set" shows "\<langle>ES\<rangle>restr_simple_state_rel R \<equiv> {(s, s'). (ss_stack s, map (\<lambda>u. (u, pending s' `` {u})) (stack s')) \<in> \<langle>Id \<times>\<^sub>r {(U, U'). U - R = U'}\<rangle>list_rel \<and> on_stack s = set (stack s') \<and> visited s = dom (discovered s') \<union> R \<and> dom (discovered s') \<inter> R = {} \<and> dom (finished s') = dom (discovered s') - set (stack s') \<and> set (stack s') \<subseteq> dom (discovered s') \<and> (simple_state.more s, state.more s') \<in> ES}"
heap_heap_upd: fixes g :: "oref \<Rightarrow> obj option"   and a :: "loc"   and obj :: "obj"   and l :: "lname \<Rightarrow> val option" shows "heap (st (g(Inl a \<mapsto> obj)) l) = heap (st g l)(a \<mapsto> obj)"
Suc: shows "(1::'a) + of_nat n * x \<le> ((1::'a) + x) ^ n"
eventually_gt_at_top: shows "eventually ((<) (0::'b)) at_top"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.map (g \<cdot> f) = C.map g \<cdot> C.map f"
blinding_of_prod_mono: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R1a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_prod R1 R2 \<le> rel_prod R1a R2a"
vfsequence_vcard_vcons: fixes x :: "V" shows "vcard (xs\<^sub>1 #\<^sub>\<circ> x) = ZFC_in_HOL.succ (vcard xs\<^sub>1)"
semiring_simprules_6: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<oplus> x = x"
sup_absorb1: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "sup x y = x"
WTrtVar: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "E V = \<lfloor>T\<rfloor>" shows "P,E,h \<turnstile> Var V : T"
characteristic_iso_3: fixes f :: "'a" assumes "Left.ide f" shows "Left.in_hom (Left.comp (\<i>[trg f] \<star> f) (local.inv \<a>[trg f, trg f, f])) (Left.L (Left.L f)) (Left.L f)"
countable_iff_bij: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "countable A = countable B"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "R f = S.null"
uminus_def: fixes x :: "'a" shows "- x = ! (d x)"
rep_eq: fixes x :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and xa :: "('e, 'c) uexpr"   and xb :: "('f, 'c) uexpr" shows "\<lbrakk>bop x xa xb\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b))"
in_gamma_meet: fixes x :: "int"   and a1 :: "ivl"   and a2 :: "ivl" assumes "x \<in> \<gamma>_ivl a1"   and "x \<in> \<gamma>_ivl a2" shows "x \<in> \<gamma>_ivl (a1 \<sqinter> a2)"
l_cancel_one: fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier H"   and "a \<in> carrier H" shows "(x \<otimes>\<^bsub>H\<^esub> a = x) = (a = \<one>\<^bsub>H\<^esub>)"
P2: shows "Bet B A' A0 \<and> Cong A' A0 E D'"
leI: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
ide_char: fixes a :: "'a \<times> 'a" shows "B.VV.ide a = (B.VV.arr a \<and> B.VxV.ide a)"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Left.in_hom f a b = (Left.arr a \<and> Left.arr b \<and> Left.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"
one_free_Abelian_group: fixes S :: "'b set" shows "\<one>\<^bsub>free_Abelian_group S\<^esub> = 0"
simps_1: fixes x :: "pdf_type" shows "op_type Add x = (case x of PRODUCT INTEG INTEG \<Rightarrow> Some INTEG | PRODUCT INTEG (PRODUCT pdf_type1 x) \<Rightarrow> Map.empty x | PRODUCT INTEG _ \<Rightarrow> None | PRODUCT REAL REAL \<Rightarrow> Some REAL | PRODUCT REAL (PRODUCT pdf_type1 x) \<Rightarrow> Map.empty x | PRODUCT REAL _ \<Rightarrow> None | PRODUCT (PRODUCT pdf_type1a x) pdf_type2 \<Rightarrow> Map.empty x | PRODUCT _ pdf_type2 \<Rightarrow> None | _ \<Rightarrow> None)"
instance_of\<^sub>l\<^sub>s_self: fixes L :: "fterm literal set" shows "instance_of\<^sub>l\<^sub>s L L"
add_diff_cancel_right': fixes a :: "'a"   and b :: "'a" shows "a + b - b = a"
inv_into_inv_into_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and a :: "'b" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A' (inv_into A f) a = f a"
map: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'b hterm list" shows "map_hterm f (HApp x1 x2) = HApp (f x1) (map (map_hterm f) x2)"
is_auto_reduced_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "is_auto_reduced B \<equiv> \<forall>b\<in>B. \<not> is_red (B - {b}) b"
Sigma_Int_distrib1: fixes I :: "'a set"   and J :: "'a set"   and C :: "'a \<Rightarrow> 'b set" shows "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C"
is_faithful: fixes f :: "'d"   and f' :: "'d" assumes "D.par f f'"   and "D.L f = D.L f'" shows "f = f'"
enabled_opr'_def: shows "enabled_opr' \<equiv> \<lambda>(name, pres, effs, cost) s. match_pres pres s \<and> match_implicit_pres effs s"
exp_minus_Euler: fixes z :: "complex" shows "exp (- (\<i> * z)) = cos z - \<i> * sin z"
idom_tOp_nonzeros: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<noteq> \<zero>"   and "y \<noteq> \<zero>" shows "x \<cdot>\<^sub>r y \<noteq> \<zero>"
semIntAll_prAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prAbs (semInt SEM) (semIntAbs SEM) SEM"
seq_simps_2: fixes x :: "'b" shows "seq\<cdot>x\<cdot>\<bottom> = \<bottom>"
single_valued_def: fixes r :: "('a \<times> 'c) set" shows "single_valued r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
cases: fixes x :: "typ \<times> typ \<times> (variable \<times> String.literal set \<Rightarrow> typ option)" obtains v :: "variable" and S :: "String.literal set" and T :: "typ" and subs :: "variable \<times> String.literal set \<Rightarrow> typ option" where "x = (Tv v__ S, T, subs__)"   | a :: "String.literal" and Ts :: "typ list" and b :: "String.literal" and Us :: "typ list" and subs :: "variable \<times> String.literal set \<Rightarrow> typ option" where "x = (Ty a Ts, Ty b Us, subs)"   | v :: "String.literal" and va :: "typ list" and vb :: "variable" and vc :: "String.literal set" and uw_ :: "variable \<times> String.literal set \<Rightarrow> typ option" where "x = (Ty v va, Tv vb vc, uw_)"
unitary_comp: fixes P :: "'c \<Rightarrow> real"   and f :: "'d \<Rightarrow> 'c" assumes "unitary P" shows "unitary (P \<circ> f)"
hyps: fixes x :: "'a" shows "lnth (iterates f x) n = (f ^^ n) x"
is_lubD1: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b" assumes "class.po below"   and "po.is_lub below S x" shows "po.is_ub below S x"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P (NOT p); P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q; P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (Eq a)) \<Longrightarrow> P (Dvd i a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (NEq a)) \<Longrightarrow> P (NDvd i a)"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (And p q))"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Or p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Imp p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q); P (NOT p); P q\<rbrakk> \<Longrightarrow> P (NOT (Iff p q))"   and "\<And>p. P p \<Longrightarrow> P (NOT (NOT p))"   and "P (NOT T)"   and "P (NOT F)"   and "\<And>a. P (Ge a) \<Longrightarrow> P (NOT (Lt a))"   and "\<And>a. P (Gt a) \<Longrightarrow> P (NOT (Le a))"   and "\<And>a. P (Le a) \<Longrightarrow> P (NOT (Gt a))"   and "\<And>a. P (Lt a) \<Longrightarrow> P (NOT (Ge a))"   and "\<And>a. P (NEq a) \<Longrightarrow> P (NOT (Eq a))"   and "\<And>a. P (Eq a) \<Longrightarrow> P (NOT (NEq a))"   and "\<And>i a. P (NDvd i a) \<Longrightarrow> P (NOT (Dvd i a))"   and "\<And>i a. P (Dvd i a) \<Longrightarrow> P (NOT (NDvd i a))"   and "P T"   and "P F"   and "\<And>va. P (NOT (E va))"   and "\<And>va. P (NOT (A va))"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
subst_apply_extstrand_singleton_4: fixes t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "[Decomp t] \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta> = [Decomp (t \<cdot> \<theta>)]"
sum_list_0: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x = (0::'a)" shows "sum_list xs = (0::'a)"
"3_4": fixes i :: "nat" assumes "i < n" shows "(0::'a) < s i"
char_conv: fixes t :: "real" shows "(\<lambda>n. char (M n) t) \<longlonglongrightarrow> char M' t"
prob_space_convolution_density: fixes f :: "real \<Rightarrow> ennreal"   and g :: "real \<Rightarrow> ennreal" assumes "f \<in> borel_measurable borel"   and "g \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "\<And>x. 0 \<le> g x"   and "prob_space (density lborel f)"   and "prob_space (density lborel g)" shows "prob_space (density lborel (\<lambda>x. \<integral>\<^sup>+ y. f (x - y) * g y \<partial>lborel))"
mult_less_cancel_left: 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times c a) (times c b) = ((less_eq zero c \<longrightarrow> less a b) \<and> (less_eq c zero \<longrightarrow> less b a))"
digraph_map_def: fixes G :: "('e, 'f) pre_digraph"   and M :: "'f pre_map" shows "digraph_map G M \<equiv> fin_digraph G \<and> bidirected_digraph G (edge_rev M) \<and> digraph_map_axioms G M"
elims: fixes x :: "nat"   and y :: "'a bro set" assumes "Um x = y" obtains "x = 0" and "y = {}"   | h :: "nat" where "x = Suc h" and "y = N1 ` T h"
s_antimono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "t_stop \<le> x" shows "s y \<le> s x"
Par1B_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P' \<parallel> Q"