hcju/isabelleps huggingface dataset:
emb_step_at_remove_arg: fixes p :: "dir list"   and t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "emb_step_at p dir.Left t = s"   and "position_of t (p @ [dir.Left])"   and "list_all (\<lambda>x. x = dir.Left) p" shows "let i = num_args t - Suc (length p) in head t = head s \<and> i < num_args t \<and> args s = take i (args t) @ drop (Suc i) (args t)"
typeof_NullPointer: fixes h :: "heap" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt NullPointer)) = \<lfloor>Class NullPointer\<rfloor>"
mono_rtranclp: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "\<And>a b. x a b \<longrightarrow> y a b" shows "x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
iterate_funpow_step: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "f x \<noteq> y"   and "y \<in> orbit f x" shows "iterate 0 (funpow_dist1 f x y) f x = x # iterate 0 (funpow_dist1 f (f x) y) f (f x)"
hyps_3: shows "pi \<bullet> (\<parallel>(B)\<parallel>) = \<parallel>(B)\<parallel>"
inversion_unit_circle: fixes x :: "complex_homo" assumes "on_circline unit_circle x" shows "inversion x = x"
m_nz: fixes A :: "'a" assumes "A \<in> set As" shows "Aj_of A \<noteq> {#}"
is_sur_def: fixes x :: "'a" shows "is_sur x \<equiv> reflexive (x\<^sup>T ; x)"
Units_m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Units \<G>"   and "y \<in> Units \<G>" shows "x \<otimes> y \<in> Units \<G>"
fold_graph_closed_lemma: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and A :: "'b set"   and x :: "'c"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "fold_graph g z A x"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> f a b = g a b"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> B"   and "z \<in> B" shows "fold_graph f z A x \<and> x \<in> B"
domo_join: fixes S :: "'a Abs_State.st option"   and T :: "'a Abs_State.st option" shows "domo (S \<squnion> T) \<subseteq> domo S \<union> domo T"
h3f: assumes "MPoly_Type.degree p2 var = 0" shows "p2 = A + B * Var var"
the1_equality': fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "\<exists>\<^sub>\<le>\<^sub>1 x. P x"   and "P a" shows "(THE x. P x) = a"
succ_pred_no_3: fixes w :: "num" shows "word_succ (- numeral w) = - numeral w + 1"
evContainsAtom1: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and z :: "signs"   and P :: "predicate"   and vs :: "vbl list" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f (i, FAtom z P vs))" shows "\<exists>n. (z, P, vs) \<in> set (fst (f n))"
onorm_sphere: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "onorm f = \<Squnion> {norm (f x) |x. norm x = 1}"
eventually_INF1: fixes i :: "'a"   and I :: "'a set"   and P :: "'b \<Rightarrow> bool"   and F :: "'a \<Rightarrow> 'b filter" assumes "i \<in> I"   and "eventually P (F i)" shows "eventually P (Inf (F ` I))"
p_div_gcd_p_pderiv: shows "p div gcd p (pderiv p) = (\<Prod>(a, i)\<in>as. a)"
expandLeft: fixes R :: "pi"   and P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q"   and "Id \<subseteq> Rel" shows "P \<parallel> Q \<leadsto>[Rel] R"
simps: fixes ws :: "('a list \<times> 's \<times> 's) list"   and ps :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set" shows "step_invariant (ws, ps, N) = (let (w, r, s) = hd ws; ps' = (r, s) # ps; succs = map (\<lambda>a. let r' = delta a r; s' = delta a s in ((a # w, r', s'), post r', post s')) (\<sigma> n); new = remdups' snd (filter (\<lambda>(uu_, rs). rs \<notin> N) succs); ws' = tl ws @ map fst new; N' = set (map snd new) \<union> N in (ws', ps', N'))"
iso_char: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b"   and \<phi> :: "'b" assumes "right_hom V H \<omega>"   and "partial_magma.arr (subcategory.comp V (weak_composition.right V H \<omega>)) \<phi>" shows "category.iso (subcategory.comp V (weak_composition.right V H \<omega>)) \<phi> = category.iso V \<phi>"
updVal_simp: fixes val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'g"   and x :: "'var"   and sX :: "'g"   and xs :: "'varSort"   and ys :: "'varSort"   and y :: "'var" shows "(val (x := sX)_xs) ys y = (if ys = xs \<and> y = x then sX else val ys y)"
bind_eq_None_conv: fixes a :: "'n option"   and f :: "'n \<Rightarrow> 'm option" shows "(Option.bind a f = None) = (a = None \<or> f (the a) = None)"
dsf': shows "matrix_sum d f' n \<in> carrier_mat d d"
repcs_def: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "repcs s cfg = cfg_corec s (\<lambda>(s, cfg). rept s (action cfg)) (\<lambda>(s, cfg) s'. (s', cont cfg (abss s'))) (s, cfg)"
I_l_closed: fixes a :: "'a"   and x :: "'a" assumes "a \<in> J"   and "x \<in> carrier R" shows "x \<otimes> a \<in> J"
D_Mprefix: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "D (Mprefix A P) = {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))}"
shiftl_shiftr2: fixes b :: "nat"   and c :: "nat"   and a :: "'b word" assumes "b < c" shows "a << b >> c = (a >> c - b) AND mask (size a - c)"
proj_stoch_eq_pseudo_proj_True: fixes m :: "nat"   and x :: "bool stream"   and y :: "bool stream" assumes "proj_stoch_proc geom_proc m x = proj_stoch_proc geom_proc m y" shows "pseudo_proj_True m x = pseudo_proj_True m y"
vconverse_vid_on: fixes A :: "V" shows "(vid_on A)\<inverse>\<^sub>\<circ> = vid_on A"
homeomorphic_empty_1: fixes S :: "'b set" shows "S homeomorphic {} = (S = {})"
axioms_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b"   and \<eta> :: "'b \<Rightarrow> 'b"   and \<epsilon> :: "'a \<Rightarrow> 'a" assumes "equivalence_of_categories C D F G \<eta> \<epsilon>" shows "category D"
transfer: shows "pcr_pmf (=) (\<lambda>x. indicat_real S x / real (card S)) local.pmf_of_set"
awhd_if_cas: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "cas u p v" shows "awhd u p = u"
to_tree_order_same_root: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root_ptr"   and "h \<turnstile> to_tree_order root_ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to" shows "h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root_ptr"
l_sorted_app: fixes xs :: "'b list"   and ys :: "'b list" assumes "l_sorted xs"   and "l_sorted ys"   and "last xs \<le> hd ys" shows "l_sorted (xs @ ys)"
right_totalI: fixes A :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>y. \<exists>x. A x y" shows "right_total A"
right_def: shows "right \<equiv> map_fun Rep_\<I> id snd"
indH: shows "\<forall>p'. (p', p) \<in> sb \<longrightarrow> (\<forall>n. (\<exists>P. sub P \<in> bbrp (\<sigma>\<^bsub>the_bb\<^esub>t n) \<and> p' \<in> P) \<longrightarrow> (\<exists>m\<ge>n. (p', solve p') = bbcs (\<sigma>\<^bsub>the_bb\<^esub>t m)))"
optimum_splitLeaf: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat" assumes "consistent t"   and "optimum t"   and "a \<in> alphabet t"   and "b \<notin> alphabet t"   and "freq t a = w\<^sub>a + w\<^sub>b"   and "\<forall>c\<in>alphabet t. w\<^sub>a \<le> freq t c \<and> w\<^sub>b \<le> freq t c" shows "optimum (splitLeaf t w\<^sub>a a w\<^sub>b b)"
f: shows "(\<Sum>v\<in>R - {a}. f v *s v) = a"
OUTfromChCorrect_data21: shows "OUTfromChCorrect data21"
rank_invar_hd_cons: fixes bq :: "('c, 'd) BinomialTree list"   and t :: "('c, 'd) BinomialTree" assumes "rank_invar bq"   and "rank t < rank (hd bq)" shows "rank_invar (t # bq)"
reflclp_tranclp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "r\<^sup>+\<^sup>+\<^sup>=\<^sup>= = r\<^sup>*\<^sup>*"
mgp_edge_face_ex: fixes g :: "graph"   and v :: "nat"   and f :: "face"   and u :: "nat" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)"   and "(u, v) \<in> \<E> f" shows "\<exists>f'\<in>set (facesAt g v). (v, u) \<in> \<E> f'"
divisor_set_not_empty: fixes m :: "nat" assumes "1 \<le> m" shows "m \<in> divisor_set m"
measure_mono_fmeasurable: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure" assumes "A \<subseteq> B"   and "A \<in> sets M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M B"
s_stop_def: fixes s :: "real"   and v :: "real"   and a :: "real" shows "s_stop s v a \<equiv> s + rel_dist_to_stop v a"
elims_3: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real"   and l0 :: "nat"   and x :: "nat \<times> nat" assumes "TableInv f1 f2 f1' f2' e c l0"   and "\<not> TableInv.invar f1 f2 c l0 x" obtains n :: "nat" and l :: "nat" where "x = (n, l)" and "\<not> (l0 \<le> l \<and> (int l0 \<le> \<lfloor>real l / c\<rfloor> \<longrightarrow> f1 * real l \<le> real n) \<and> real n \<le> f2 * real l)"
t_App_elim: fixes \<Gamma> :: "(name \<times> ty) list"   and t1 :: "trm"   and t2 :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> App t1 t2 : T" obtains T' :: "ty" where "\<Gamma> \<turnstile> t1 : T'\<rightarrow>T" and "\<Gamma> \<turnstile> t2 : T'"
stopping_timeI: fixes T :: "'d \<Rightarrow> 'c"   and F :: "'c \<Rightarrow> 'd measure" assumes "\<And>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t)" shows "stopping_time F T"
m_2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
variance_eq: fixes X :: "'a \<Rightarrow> real" assumes "integrable M X"   and "integrable M (\<lambda>x. (X x)\<^sup>2)" shows "expectation (\<lambda>x. (X x - expectation X)\<^sup>2) = expectation (\<lambda>x. (X x)\<^sup>2) - (expectation X)\<^sup>2"
star_inductl_one: fixes x :: "'a"   and y :: "'a" assumes "1' + x ; y \<le> y" shows "x\<^sup>\<star> \<le> y"
UES: fixes \<Gamma> :: "'b formula list"   and \<AA> :: "'b formula"   and \<Theta> :: "'b formula list"   and \<BB> :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<AA> # \<Theta>"   and "\<Gamma> \<Rightarrow> \<BB> # \<Theta>" shows "\<Gamma> \<Rightarrow> \<AA> \<^bold>\<and> \<BB> # \<Theta>"
dg_Cod_vsubset_Vset: shows "\<CC>\<lparr>Cod\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
sel_3: fixes x1 :: "'b"   and x2 :: "'c"   and x3 :: "'a list" shows "callstack (State x1 x2 x3) = x3"
time_mergesort_conv_mergesort_recurrence: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "real (time (mergesort_tm f xs)) \<le> mergesort_recurrence (length xs)"
normal_oVeblen: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal" assumes "normal F" shows "normal (oVeblen F x)"
mulex1_add: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and N :: "'b multiset"   and M0 :: "'b multiset"   and a :: "'b" assumes "mulex1 P N (M0 + {#a#})" shows "(\<exists>M. mulex1 P M M0 \<and> N = M + {#a#}) \<or> (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> P b a) \<and> N = M0 + K)"
Cons_9: shows "getSATFlag stateA = getSATFlag stateB"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "r1.redT a1 a2 a3" obtains t :: "'t" and x :: "'x1" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action" and x' :: "'x1" and m' :: "'m1" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "a1 = s__" and "a2 = (ta__, ta)" and "a3 = s'__" and "r1.r_syntax ta__ x__ (shr s__) ta x' m'" and "thr s__ ta__ = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "r1.actions_ok s__ ta__ ta" and "redT_upd s__ ta__ ta x' m' s'__"   | s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x1" and n :: "'l" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ln :: "'l \<Rightarrow>f nat" where "a1 = s" and "a2 = (t, K$ [], [], [], [], [], convert_RA ln)" and "a3 = s'" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
noipdj: shows "\<forall>l\<in>{j..k}. \<pi> l \<noteq> ipd (\<pi> j)"
possible_steps_select_wrong_arity: fixes a :: "String.literal"   and b :: "value list" assumes "a = STR ''select''"   and "length b \<noteq> 1" shows "possible_steps drinks 0 <> a b = {||}"
exec_d_eq_Normal: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list" shows "(exec_d P t \<sigma> = Normal (exec P t \<sigma>)) = check P \<sigma>"
greatest_Lower_cong_r: fixes A :: "'a set"   and A' :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier L"   and "A' \<subseteq> carrier L"   and "A {.=} A'" shows "is_glb L x A = is_glb L x A'"
cases: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym a1 a2" obtains t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = t__" and "a2 = s__" and "tkbo_coefs.gt_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t__) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s__)"   | t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = ta__" and "a2 = sa__" and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym ta__) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym sa__)" and "\<not> ground_heads.comp_hd gt_sym ground_heads_var (head ta__) (head sa__)" and "num_args ta__ = 1" and "\<exists>f\<in>ground_heads.ground_heads ground_heads_var (head ta__). arity_sym f = 1 \<and> wt_sym f = 0" and "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym (arg ta__) sa__ \<or> arg ta__ = sa__"   | t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = tb__" and "a2 = sb__" and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym tb__) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym sb__)" and "ground_heads.gt_hd gt_sym ground_heads_var (head tb__) (head sb__)"   | t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = t" and "a2 = s" and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s)" and "head t = head s" and "\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym) (args t) (args s)"
has_sgnx_divide: fixes f :: "real \<Rightarrow> real"   and c1 :: "real"   and F :: "real filter"   and g :: "real \<Rightarrow> real"   and c2 :: "real" assumes "(f has_sgnx c1) F"   and "(g has_sgnx c2) F" shows "((\<lambda>x. f x / g x) has_sgnx c1 / c2) F"
heap_eq_List_eq: fixes ps :: "ref list"   and h :: "ref \<Rightarrow> ref"   and g :: "ref \<Rightarrow> ref"   and p :: "ref" assumes "\<forall>x\<in>set ps. h x = g x" shows "List p h ps = List p g ps"
of_nat_less_0_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (semiring_1.of_nat one plus zero m) zero"
vsv_ex1: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> [a, b]\<^sub>\<circ>" shows "\<exists>!ba. \<langle>a, ba\<rangle> \<in>\<^sub>\<circ> [a, b]\<^sub>\<circ>"
roofed_mono: fixes S :: "'k set"   and S' :: "'k set"   and G :: "('k, 'l) graph_scheme"   and B :: "'k set" assumes "S \<subseteq> S'" shows "roofed_gen G B S \<subseteq> roofed_gen G B S'"
ht_distinct_replicate: fixes n :: "nat" shows "ht_distinct (replicate n [])"
path_connectedin_topspace: fixes X :: "'b topology" shows "path_connectedin X (topspace X) = path_connected_space X"
unit_imp_no_irreducible_divisors: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and p :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times x one"   and "comm_semiring_1.irreducible times one zero p" shows "\<not> dvd.dvd times p x"
shiftl_bl: fixes w :: "'b word"   and n :: "nat" shows "w << n = of_bl (to_bl w @ replicate n False)"
coeff_degree_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "lead_coeff (p * q) = lead_coeff q * lead_coeff p"
tensor_Tangle_def: fixes x :: "Tangle_Diagram"   and y :: "Tangle_Diagram" shows "x \<otimes> y = Abs_Tangle_Diagram (Rep_Tangle_Diagram x \<otimes> Rep_Tangle_Diagram y)"
comp_assoc: fixes f :: "'j \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'j"   and h :: "'g \<Rightarrow> 'i" shows "f \<circ> g \<circ> h = (f \<circ>\<circ>\<circ> (\<circ>)) g h"
simps_1: fixes ts :: "t list"   and cons :: "ct list" shows "consume (Type ts) cons = (if ct_suffix cons (to_ct_list ts) then Type (take (length ts - length cons) ts) else Bot)"
absolutely_integrable_reflect: fixes b :: "'c"   and a :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "((\<lambda>x. f (- x)) absolutely_integrable_on cbox (- b) (- a)) = (f absolutely_integrable_on cbox a b)"
xs: shows "\<forall>j\<in>set xs. j < min CARD('nr) CARD('nc)"
induct: fixes P :: "'b list \<Rightarrow> bool"   and a0 :: "'b list" assumes "P []"   and "\<And>uu_. P [uu_]"   and "\<And>x1 x2 xs. \<lbrakk>x1 * x2 < (0::'b) \<Longrightarrow> P (x2 # xs); \<lbrakk>\<not> x1 * x2 < (0::'b); x2 = (0::'b)\<rbrakk> \<Longrightarrow> P (x1 # xs); \<lbrakk>\<not> x1 * x2 < (0::'b); x2 \<noteq> (0::'b)\<rbrakk> \<Longrightarrow> P (x2 # xs)\<rbrakk> \<Longrightarrow> P (x1 # x2 # xs)" shows "P a0"
lincomb_list_alt: fixes c :: "nat \<Rightarrow> 'a"   and s :: "'a vec list" shows "lincomb_list c s = M.sumlist (map2 (\<lambda>x y. x \<cdot>\<^sub>v s ! y) (map c [0..<length s]) [0..<length s])"
elims_2: fixes x :: "'a"   and xa :: "'a interval" assumes "x \<in>\<^sub>i xa" obtains y :: "'a" and lx :: "'a" and ux :: "'a" where "x = y" and "xa = Interval lx ux" and "lx \<le> y \<and> y \<le> ux"
1: fixes xb :: "real" assumes "poly (fss ! i) xb = 0" shows "insertAt i 0 (signs_at (removeAt i fss) xb) \<in> range (signs_at fss)"
zmult_pos_neg: fixes i :: "int"   and j :: "int" assumes "0 < i"   and "j < 0" shows "i * j < 0"
mem_list2: shows "\<forall>r\<in>set (characterize_root_list_p p). consistent_sign_vec_copr qs2 r \<in> set (map (consistent_sign_vec_copr qs2) (characterize_root_list_p p))"
elims_2: fixes x :: "'a ifex" assumes "ifex_ordered x" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "Ball (ifex_var_set t \<union> ifex_var_set e) ((<) v) \<and> ifex_ordered t \<and> ifex_ordered e"   | "x = Trueif"   | "x = Falseif"
inject_1: fixes x11 :: "char list"   and x12 :: "'a list"   and y11 :: "char list"   and y12 :: "'a list" shows "(Pos x11 x12 = Pos y11 y12) = (x11 = y11 \<and> x12 = y12)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "P x"
cInf_lower: fixes x :: "'b multiset"   and X :: "'b multiset set" assumes "x \<in> X"   and "subset_mset.bdd_below X" shows "\<Sqinter> X \<subseteq># x"
c: fixes x :: "trm"   and z :: "trm" assumes "\<And>y. x \<mapsto> y \<Longrightarrow> P y z"   and "\<And>u. z \<mapsto> u \<Longrightarrow> P x u" shows "P x z"
simps_1: fixes f :: "'b \<Rightarrow> 'a option" shows "traverse f [] = Some []"
icheck_ss_exp: fixes P :: "'a prog"   and C :: "char list"   and e :: "'b exp" assumes "icheck P C e" shows "ss_exp e = e"
U\<^sub>f_mult_without_empty_summands_odd: fixes i :: "nat"   and j :: "nat"   and A :: "complex Matrix.mat" assumes "i < dim_row U\<^sub>f"   and "j < dim_col A"   and "odd i"   and "dim_col U\<^sub>f = dim_row A" shows "(U\<^sub>f * A) $$ (i, j) = (\<Sum>k\<in>{i - 1, i}. U\<^sub>f $$ (i, k) * A $$ (k, j))"
filter_mset_image_mset: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d multiset" shows "filter_mset P (image_mset f A) = image_mset f {#x \<in># A. P (f x)#}"
C1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q"   and "(\<Psi>, Q, R) \<in> Rel" shows "(\<Psi>, P, R) \<in> Rel"
e2i_f: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2i (encode_config ((f, xs, ls) # ss, rv)) = encode f"
expectation_gpv_bind: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'c \<Rightarrow> ennreal"   and gpv :: "('d, 'a, 'b) gpv"   and g :: "'d \<Rightarrow> ('c, 'a, 'b) gpv" shows "expectation_gpv fail \<I> f (gpv \<bind> g) = expectation_gpv fail \<I> (expectation_gpv fail \<I> f \<circ> g) gpv"
cobounded2: fixes a :: "'d"   and b :: "'d" shows "inf_class.inf a b \<le> b"
real_interval_uminus: fixes b :: "float interval" shows "real_interval (- b) = - real_interval b"
p_max: fixes x :: "real" shows "other.p x \<le> other.p_max"
wcode_prepare_le_def: shows "wcode_prepare_le \<equiv> inv_image Recursive.lex_triple wcode_prepare_measure"
Can_red_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a" shows "monoidal_language.Can C (monoidal_language.red C a)"
rep_eq: fixes x :: "Vars.var"   and xa :: "CoCalls" shows "ccNeighbors x xa = {y. (y, x) \<in> Rep_CoCall xa \<or> (x, y) \<in> Rep_CoCall xa}"