hcju/isabelleps huggingface dataset:
whisker_can_left_0: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term"   and g :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u"   and "partial_magma.ide V g"   and "bicategorical_language.Trg trg t = bicategorical_language.Prim\<^sub>0 (src g)" shows "H g (bicategory.can V H \<a> \<i> src trg u t) = bicategory.can V H \<a> \<i> src trg (bicategorical_language.Hcomp (bicategorical_language.Prim g) u) (bicategorical_language.Hcomp (bicategorical_language.Prim g) t)"
measure_pmf_UNIV: fixes p :: "'b pmf" shows "measure_pmf.prob p UNIV = 1"
simps_11: fixes F :: "'a formula"   and G :: "'a formula" shows "nnf (F \<^bold>\<rightarrow> G) = nnf (\<^bold>\<not> F) \<^bold>\<or> nnf G"
blinfun_eq: fixes v :: "(real, 'sz) vec" shows "Blinfun (\<lambda>v'. sterm_sem I \<theta>\<^sub>1 v * frechet I \<theta>\<^sub>2 v v' + frechet I \<theta>\<^sub>1 v v' * sterm_sem I \<theta>\<^sub>2 v) = sterm_sem I \<theta>\<^sub>1 v *\<^sub>R blin_frechet (good_interp I) (simple_term \<theta>\<^sub>2) v + sterm_sem I \<theta>\<^sub>2 v *\<^sub>R blin_frechet (good_interp I) (simple_term \<theta>\<^sub>1) v"
simps_3: shows "op_sem Minus = lift_RealIntVal uminus uminus"
aeq: assumes "a = 0" shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + b * x + c < 0"
SIGT_times2: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<sigma> :: "('b, 'c, 'a, 'd) subst_scheme"   and t1 :: "('b, 'e) trm"   and t2 :: "('b, 'e) trm" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT (Times t1 t2). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT t2. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"
simps_21: fixes G :: "prog"   and t1 :: "ty"   and t2 :: "ty" shows "wt_binop G CondAnd t1 t2 = (t1 = PrimT Boolean \<and> t2 = PrimT Boolean)"
simps_3: fixes uw :: "'a" shows "lderiv uw One = Zero"
order_root: fixes p :: "'a list"   and a :: "'a" shows "(poly p a = (0::'a)) = (poly p = poly [] \<or> order a p \<noteq> 0)"
gcd: shows "gcd ((e mod order \<G> - e' mod order \<G>) mod order \<G>) (order \<G>) = 1"
assms_1: shows "RECOVER_PURE \<Gamma> \<Gamma>'"
sup_assoc: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
perm_append2: fixes xs :: "'b list"   and ys :: "'b list"   and l :: "'b list" assumes "xs <~~> ys" shows "xs @ l <~~> ys @ l"
0: shows "(HMA_Connect.map_matrix complex_of_real (HMA_Connect.map_matrix cmod F) - F) *v map_vector complex_of_real yp = 0"
apfst_def: fixes f :: "'r \<Rightarrow> 't" shows "apfst f = map_prod f id"
order_0_monom: fixes c :: "'b"   and n :: "nat" assumes "c \<noteq> (0::'b)" shows "order (0::'b) (monom c n) = n"
induct: fixes P :: "cld list \<Rightarrow> ctx \<Rightarrow> cl \<Rightarrow> (ctx \<times> cld) list \<Rightarrow> bool"   and a0 :: "cld list"   and a1 :: "ctx"   and a2 :: "cl"   and a3 :: "(ctx \<times> cld) list" assumes "\<And>Pa ctx path. P Pa ctx cl_object path"   and "\<And>Pa ctx fqn path. (\<And>x2 x y. \<lbrakk>\<not> \<not> acyclic_clds Pa; find_cld_f Pa ctx fqn = Some x2; (x, y) = x2\<rbrakk> \<Longrightarrow> P Pa x (superclass_name_f y) (path @ [(x, y)])) \<Longrightarrow> P Pa ctx (cl_fqn fqn) path" shows "P a0 a1 a2 a3"
list_Previous_gr0: fixes n :: "nat"   and xs :: "'d list"   and init :: "'d" assumes "0 < n" shows "xs\<^bsup>\<leftarrow>' init\<^esup> n = xs ! (n - Suc 0)"
valid_path_aux_valid_path: fixes cs :: "'edge list"   and as :: "'edge list" assumes "valid_path_aux cs as" shows "valid_path as"
atLeastLessThan_iff: fixes i :: "'b Word.word"   and l :: "'b Word.word"   and u :: "'b Word.word" shows "(i \<in> signed.atLeastLessThan l u) = (l \<le>s i \<and> i <s u)"
simps_5: fixes \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "not\<^sub>r (\<phi> and\<^sub>r \<psi>) = not\<^sub>r \<phi> or\<^sub>r not\<^sub>r \<psi>"
union_disjoint: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum g (A \<union> B) = sum g A + sum g B"
simps: fixes k :: "name"   and s :: "name"   and t :: "tm"   and vf :: "bool" assumes "atom k \<sharp> (s, t)"   and "atom s \<sharp> t" shows "CTermP vf t = SyntaxN.Ex s (SyntaxN.Ex k (SeqCTermP vf (Var s) (Var k) t))"
simps_3: fixes n :: "enat"   and xs :: "'c llist" assumes "n \<noteq> 0"   and "\<not> lnull xs" shows "lhd (ltake n xs) = lhd xs"
prems_2: shows "strict_sorted (concat (bs1 @ bs2))"
Retracts_def: fixes s :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and t :: "'c set"   and k :: "'c \<Rightarrow> 'b" shows "Retracts s h t k \<equiv> (continuous_on s h \<and> h ` s = t) \<and> continuous_on t k \<and> k ` t \<subseteq> s \<and> (\<forall>y. y \<in> t \<longrightarrow> h (k y) = y)"
simps_8: fixes c :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "flatten (DynCom c) = [DynCom c]"
assms_3: shows "t \<sqsubseteq> s \<cdot> \<delta>"
cut_ge_Int: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>\<ge> t = A \<down>\<ge> t \<inter> B \<down>\<ge> t"
stable_ordIso: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r =o r'" shows "stable r = stable r'"
c: shows "abc_steps_l (0, lm) A n = (s, lm')"
real_sqrt_abs: fixes x :: "real" shows "sqrt (x\<^sup>2) = \<bar>x\<bar>"
division_of_def: fixes s :: "'b set set"   and i :: "'b set" shows "(s division_of i) = (finite s \<and> (\<forall>K\<in>s. K \<subseteq> i \<and> K \<noteq> {} \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>K1\<in>s. \<forall>K2\<in>s. K1 \<noteq> K2 \<longrightarrow> interior K1 \<inter> interior K2 = {}) \<and> \<Union> s = i)"
that: fixes F :: "'a set \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>U. U \<in> \<C> \<Longrightarrow> continuous_on S (F U) \<and> (\<forall>x\<in>S. 0 \<le> F U x)"   and "\<And>x U. \<lbrakk>U \<in> \<C>; x \<in> S; x \<notin> U\<rbrakk> \<Longrightarrow> F U x = 0"   and "\<And>x. x \<in> S \<Longrightarrow> supp_sum (\<lambda>W. F W x) \<C> = 1"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. F U x \<noteq> 0}" shows "thesis"
snth_as_fct: fixes M :: "bool stream measure"   and p :: "real"   and i :: "nat"   and b :: "bool" assumes "M = bernoulli_stream p" shows "to_stream -` {w \<in> space M. w !! i = b} = {X. X i = b}"
Sign_getmsg: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "Sign X Y \<in> parts H" shows "X \<in> parts H"
ipresIGSwap_def: fixes h :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'w, 'ae, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ai, 'aj, 'ad, 'x, 'ak, 'al) model_scheme" shows "ipresIGSwap h MOD MOD' \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> h (igSwap MOD zs z1 z2 X) = igSwap MOD' zs z1 z2 (h X)"
min_int_poly_squarefree: fixes x :: "'a" shows "squarefree (min_int_poly x)"
distinct_26: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Stmt.If x51 x52 x53 \<noteq> x31@[x32]"
right_plus_circ: fixes x :: "'a" shows "(x\<^sup>\<circ> * x)\<^sup>\<circ> = x\<^sup>\<circ>"
selectlike_compose: fixes X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set"   and W :: "variable set" shows "selectlike (selectlike X \<nu> V) \<nu> W = selectlike X \<nu> (V \<union> W)"
set_sel_3: fixes a :: "('a, 'b) Term.term" assumes "is_Var a" shows "the_Var a \<in> fv a"
objE: fixes a :: "'a" assumes "C.obj a" obtains "C.ide a" and "src\<^sub>C a = a" and "trg\<^sub>C a = a"
useNodes_of_code: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and g :: "'g" shows "uninst_code.ssa.useNodes_of u g = fold (\<lambda>n. fold (\<lambda>v. Mapping.map_default v {} (insert n)) (sorted_list_of_set ((usesOf \<circ> u) g n))) (\<alpha>n g) Mapping.empty"
ring_distrib4: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "a \<cdot>\<^sub>r b \<plusminus> -\<^sub>a x \<cdot>\<^sub>r y = a \<cdot>\<^sub>r (b \<plusminus> -\<^sub>a y) \<plusminus> (a \<plusminus> -\<^sub>a x) \<cdot>\<^sub>r y"
vbrelation_vinsertD: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "vbrelation (vinsert \<langle>a, b\<rangle> r)" shows "vbrelation r"
RSpan_closed: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list" assumes "RModule R smult M"   and "set ms \<subseteq> M" shows "R_scalar_mult.RSpan R smult ms \<subseteq> M"
lens_override_def: fixes X :: "'e \<Longrightarrow> 'd"   and S\<^sub>1 :: "'d"   and S\<^sub>2 :: "'d" shows "S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 = put\<^bsub>X\<^esub> S\<^sub>1 (get\<^bsub>X\<^esub> S\<^sub>2)"
prv_cnj_imp_monoR: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<phi> \<chi>) (imp (imp \<phi> \<psi>) (imp \<phi> (cnj \<chi> \<psi>))))"
sum_my2: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>A. f x = g x" shows "sum f A = sum g A"
size_length: fixes t :: "('b \<times> 'c) tree" shows "size t = length (Tree2.inorder t)"
big_sound: shows "sound ([(close (q # \<Gamma>) (p \<leftrightarrow> q), \<Delta>), (close \<Gamma> (p \<leftrightarrow> q), p # \<Delta>)], \<Gamma>, \<Delta>)"
Domain_Id_on: fixes A :: "'a set" shows "Domain (Id_on A) = A"
f_def: shows "f = (\<lambda>t. (Im b - Im a) * t + (Im a - Im z))"
simps_3: fixes y :: "name"   and x :: "name"   and s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" assumes "atom y \<sharp> (x, s)" shows "subst x s (Abs y t) = Abs y (subst x s t)"
le: fixes i :: "nat" assumes "i < length \<alpha>" shows "\<alpha>' ! i \<le> \<alpha> ! i"
semantics_bigstep_state_vs_tagged: fixes stateful_matcher' :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma> :: "'b"   and p :: "'c"   and stateful_matcher_tagged' :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and packet_tagger' :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and rs :: "'a rule list"   and t :: "state" assumes "\<forall>m. stateful_matcher' \<sigma> m p = stateful_matcher_tagged' m (packet_tagger' \<sigma> p)" shows "\<Gamma>,stateful_matcher' \<sigma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t = \<Gamma>,stateful_matcher_tagged',packet_tagger' \<sigma> p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t"
after_rep: fixes xs :: "'e list"   and P :: "'e process" assumes "xs \<in> traces P" shows "Rep_process (P \<setminus> xs) = (futures P xs, future_divergences P xs)"
lang_quot: shows "lang n (samequot_exec (any, replicate n False) (Pr (rexp_of (Suc n) \<phi>))) = SAMEQUOT (any, replicate n False) (lang n (Pr (rexp_of (Suc n) \<phi>)))"
allUse_simpleUse: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and m :: "'node" assumes "v \<in> braun_ssa.allUses g m"   and "m \<in> set (\<alpha>n g)" obtains ms :: "'node list" and m' :: "'node" where "g \<turnstile> m-ms\<rightarrow>m'" and "var g v \<in> uses g m'" and "\<forall>x\<in>set (tl ms). var g v \<notin> defs g x"
maintain_upper_bound: fixes u :: "'v"   and v :: "'v" shows "\<delta> s u \<le> relax_outgoing v D u"
div_diff_mult_self1: fixes n :: "nat"   and k :: "nat"   and m :: "nat" shows "(n - k * m) div m = n div m - k"
set_mode_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_mode_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
contour_integrable_neg: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "f contour_integrable_on g" shows "(\<lambda>x. - f x) contour_integrable_on g"
run_asset_v_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_asset_v n s \<equiv> stage_asset_v n (run_owner_v n s)"
code: fixes ws :: "'d list stream" shows "flat ws = hd (shd ws) ## flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)"
invar_exit': fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat"   and u :: "nat"   and PRED' :: "nat \<Rightarrow> nat option" assumes "nf_invar c src dst PRED C N d"   and "u \<in> C"   and "(u, dst) \<in> Graph.E c"   and "dst \<in> dom PRED'"   and "PRED \<subseteq>\<^sub>m PRED'"   and "PRED' \<subseteq>\<^sub>m map_mmupd PRED (Graph.E c `` {u} - dom PRED) u" shows "f_invar c src dst PRED' (Suc d)"
sum_list_filter_le_nat: fixes f :: "'b \<Rightarrow> nat"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "sum_list (map f (filter P xs)) \<le> sum_list (map f xs)"
right_edge_smooth: shows "(\<lambda>y. twoC (1, y)) piecewise_C1_differentiable_on {0..1}"
z1: shows "\<forall>x. x \<sqsubset> z1 \<longrightarrow> P1 x = P1'"
closedin_subset: fixes U :: "'a topology"   and S :: "'a set" assumes "closedin U S" shows "S \<subseteq> topspace U"
lookup_pair_map_val_pair: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e"   and k :: "'a" assumes "oalist_inv_raw x xs"   and "f k (0::'d) = (0::'e)" shows "lookup_pair x (map_val_pair f xs) k = f k (lookup_pair x xs k)"
bot_filter_def: shows "bot \<equiv> Filters.filter.Abs_filter {top}"
assertedLiteralIsNotUnit: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantConsistent (getM state @ [(literal, decision)])"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)" shows "let state' = assertLiteral literal decision state in literal \<notin> set (getQ state') - set (getQ state)"
std\<^sub>2_renames: fixes C\<^sub>2 :: "fterm literal set" shows "var_renaming_of C\<^sub>2 (C\<^sub>2 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''2'' @ x)))"
Keys_mono: fixes A :: "('c \<Rightarrow>\<^sub>0 'd) set"   and B :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "A \<subseteq> B" shows "Keys A \<subseteq> Keys B"
onode_gcast: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and D :: "nat set"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and i :: "nat"   and R :: "nat set" assumes "((\<sigma>, s), groupcast D m, \<sigma>', s') \<in> S" shows "((\<sigma>, NodeS i s R), (R \<inter> D):*cast(m), \<sigma>', NodeS i s' R) \<in> onode_sos S"
de_Morgan_conj: fixes x :: "'b"   and y :: "'b" shows "NOT (x AND y) = NOT x OR NOT y"
induced_map_returns: fixes x :: "'a"   and A :: "'a set"   and n :: "nat" assumes "x \<in> A" shows "((T ^^ n) x \<in> A) = (\<exists>N\<le>n. n = (\<Sum>i<N. return_time_function A ((induced_map A ^^ i) x)))"
IH_1: shows "size1 l \<le> 2 ^ height l"
PartialMeetContraction_axioms: shows "PartialMeetContraction Cn \<gamma>\<^sub>T\<^sub>R"
Join_ac_3: fixes F :: "'b program"   and G :: "'b program" shows "F \<squnion> G = G \<squnion> F"
prj_simps_5: shows "Maps.dom SPN_fgh.Prj\<^sub>0\<^sub>1 = Chn ((SPN f \<circ> SPN g) \<circ> SPN h)"
wP: fixes s :: "'a" assumes "P s" shows "1 \<le> t \<guillemotleft> Q \<guillemotright> s"
attractive_axioms: shows "attractive Q (\<sqsubseteq>\<^sup>s)"
UNIV_sum: shows "UNIV = range Inl \<union> range Inr"
hpar: shows "src f = src f' \<and> trg f = trg f'"
prems: fixes a :: "'a" assumes "a \<in> insert x_ F_" shows "b dvd f a"
length_list_swap: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" shows "length (list_swap xs i j) = length xs"
add_def': fixes x :: "'a"   and y :: "'a" shows "x + y = Abs' (Rep x + Rep y)"
upd: shows "bij_betw u_s {..<n} {..<n}"
simps_1: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "sublist_at (x # xs) (y # ys) 0 = (x = y \<and> sublist_at xs ys 0)"
WT_lassocr\<^sub>C: fixes \<I>1 :: "('h, 'k) \<I>"   and \<I>2 :: "('i, 'l) \<I>"   and \<I>3 :: "('j, 'm) \<I>" shows "(\<I>1 \<oplus>\<^sub>\<I> \<I>2) \<oplus>\<^sub>\<I> \<I>3, \<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3) \<turnstile>\<^sub>C lassocr\<^sub>C \<surd>"
comp_arr_ide: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and f :: "'c" assumes "partial_magma C"   and "partial_magma.ide C a"   and "partial_magma.arr C (C f a)" shows "C f a = f"
dX_range': fixes x :: "'x"   and X :: "'x set"   and d :: "'d" assumes "x \<in> dX X d" shows "x \<in> X \<and> Xd x = d"
convex_on_mean_ineq: fixes A :: "real set"   and f :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real" assumes "convex_on A f"   and "x \<in> A"   and "y \<in> A" shows "f ((x + y) / 2) \<le> (f x + f y) / 2"
4: shows "int ((p ## trace u p) !! Max (lift (liset visible (llist_of_stream u)))) = int ((p ## trace u p) !! i)"
mann1: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> y) = - (- (- (- x \<squnion> y) \<squnion> x \<squnion> y) \<squnion> y)"
dom_subdist: fixes x :: "'a"   and y :: "'a" shows "ad (ad x) \<le> ad (ad (x + y))"
bisim1_xcp_pcD: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and a :: "'addr" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "pc < length (compE2 e)"
DisjointPaths_supergraph: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and v :: "'c"   and paths :: "'c list set" assumes "v0_v1_Digraph G v0 v1"   and "DisjointPaths (Digraph.remove_vertex G v) v0 v1 paths" shows "DisjointPaths G v0 v1 paths"
fst: fixes P :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and P' :: "'a trm" assumes "substitutes P x M P'" shows "substitutes (Fst P) x M (Fst P')"
assms_3: shows "factorize_rat_poly_monic p = (c, fs)"
hyps_1: fixes Z :: "'b" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s} c\<^sub>1 {t. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
card_ge_0_finite: fixes A :: "'d set" assumes "0 < card A" shows "finite A"
ab_event: shows "\<forall>\<^sub>F n in sequentially. 0 < b n \<and> a n \<le> a (n + 1)"
neg_diag_empty: fixes n :: "nat"   and v :: "'b \<Rightarrow> nat"   and i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "i \<le> n"   and "M i i < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
reindex_cong: fixes l :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'b \<Rightarrow> 'd" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "prod g A = prod h B"
mset_subset_of_empty: fixes A :: "'c multiset" shows "(A \<subset># {#}) = False"
S2Forward: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S2 rmhist p \<and> MClkFwd memCh crCh cst p \<and> unchanged (e p, r p, m p, rmhist!p) \<longrightarrow> S3 rmhist p$"
allDef_path_from_simpleDef: fixes v :: "'val"   and g :: "'g" assumes "v \<in> allVars g" obtains n :: "'node" and ns :: "'node list" where "g \<turnstile> n-ns\<rightarrow>defNode g v" and "old.EntryPath g ns" and "var g v \<in> oldDefs g n"
not\<pi>n: shows "\<forall>j\<le>n'. \<pi>' j \<noteq> \<pi> n"
simps_3: fixes uu :: "poscheckvariant"   and t :: "('c, 'd) Term.term"   and t' :: "('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>uu: t \<doteq> t'\<rangle> = {t, t'}"
Ints_subset_Rats: shows "\<int> \<subseteq> \<rat>"
word_upto_Cons_eq: fixes x :: "'b word"   and y :: "'b word" assumes "x < y" shows "[x .e. y] = x # [x + 1 .e. y]"
is_infinity_cases: fixes x :: "('a, 'b) IEEE.float" assumes "is_infinity x" obtains "x = \<infinity>"   | "x = - \<infinity>"
eff: shows "\<forall>(pc', \<tau>')\<in>set (eff i P pc xt \<lfloor>\<tau>\<rfloor>). pc' < mpc"
Rep_segment: fixes x :: "'b segment" shows "Rep_segment x \<in> {x. fst x \<le> snd x}"
inj_on_imp_bij_betw: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "bij_betw f A (\<P> f A)"
dvdm_imp_degree_le: fixes p :: "int poly"   and q :: "int poly" assumes "p dvdm q"   and "monic p"   and "Mp q \<noteq> 0"   and "1 < m" shows "degree p \<le> degree q"
weak_sup_post_fixed_point: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "f \<in> carrier L \<rightarrow> carrier L"   and "Mono\<^bsub>L\<^esub> f"   and "A \<subseteq> fps L f" shows "f (\<Sqinter>\<^bsub>L\<^esub>A) \<sqsubseteq>\<^bsub>L\<^esub> \<Sqinter>\<^bsub>L\<^esub>A"
sorted_smaller: fixes m :: "nat"   and t :: "nat \<Rightarrow> nat"   and j :: "nat"   and j' :: "nat" assumes "LoadBalancing m"   and "LoadBalancing.sorted t j"   and "j' \<le> j" shows "LoadBalancing.sorted t j'"
assms_5: shows "lf = (\<lambda>x y. k1 * x + k2 * y + n)"
poly_shift_s_mult: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "f \<in> carrier P"   and "s \<in> carrier R" shows "poly_shift (s \<odot>\<^bsub>P\<^esub> f) = s \<odot>\<^bsub>P\<^esub> poly_shift f"
nth_insert_nth_index_eq: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i \<le> length xs" shows "insert_nth i x xs ! i = x"
disc_eq_case_2: fixes list :: "'d list" shows "(list \<noteq> []) = (case list of [] \<Rightarrow> False | uu_ # uua_ \<Rightarrow> True)"
\<Delta>'''_def: shows "\<Delta>''' = \<lparr>edge = \<lambda>x y. edge \<Delta> x y \<and> 0 < capacity \<Delta> (x, y) \<and> y \<noteq> source \<Delta> \<and> x \<noteq> sink \<Delta>, capacity = \<lambda>(x, y). if x = sink \<Delta> \<or> y = source \<Delta> then 0 else capacity \<Delta> (x, y), source = source \<Delta>, sink = sink \<Delta>\<rparr>"