hcju/isabelleps huggingface dataset:
countable_UN: fixes A :: "'c \<Rightarrow> 'b set"   and X :: "'c set"   and M :: "'b measure" assumes "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
eq_iff: fixes a :: "'k"   and b :: "'k" shows "(a = b) = (a \<le> b \<and> b \<le> a)"
Texp_simp: fixes e :: "exp"   and a :: "Arity" shows "Texp e\<cdot>a = ccTTree (edom (\<A>\<^bsub>a\<^esub> e)) (\<G>\<^bsub>a\<^esub> e)"
fst_proj_def: fixes M :: "('c \<times> 'd) set" shows "fst_proj M \<equiv> {A. \<exists>B. (A, B) \<in> M}"
summable_of_complex: fixes X :: "nat \<Rightarrow> complex" assumes "summable X" shows "summable (\<lambda>n. of_complex (X n))"
euler_mascheroni_bounds': fixes n :: "nat"   and l :: "real"   and u :: "real" assumes "1 \<le> n"   and "ln (real (Suc n)) \<in> {l<..<u}" shows "euler_mascheroni \<in> {harm n - u + inverse (real (2 * (n + 1)))<..<harm n - l + inverse (real (2 * n))}"
adding_emptypair_keeps_SdlHPPB: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and E :: "'e \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and pp :: "'a \<Rightarrow> nat"   and DA :: "'b \<Rightarrow> 'f"   and lH :: "('f \<times> 'e \<times> nat) set"   and d :: "'f"   and PP :: "nat set"   and R :: "('a list \<times> 'a list) set" assumes "WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP R" shows "WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP (R \<union> {([], [])})"
distinct_1: fixes x1 :: "'a"   and x2 :: "'a" shows "allow x1 \<noteq> deny x2"
f_pos_iff: fixes x :: "real" shows "(0 < f x) = (0 < x)"
rsp: shows "fractrel (1::'a, 1::'a) (1::'a, 1::'a)"
action_params_match_def: fixes a :: "ast_action_schema"   and args :: "object list" shows "action_params_match a args \<equiv> list_all2 is_obj_of_type args (map snd (parameters a))"
mul_ext_cong: fixes xs1 :: "'a list"   and ys1 :: "'a list"   and xs2 :: "'a list"   and ys2 :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" assumes "mset xs1 = mset ys1"   and "mset xs2 = mset ys2"   and "\<And>x x'. \<lbrakk>x \<in> set ys1; x' \<in> set ys2\<rbrakk> \<Longrightarrow> f x x' = g x x'" shows "mul_ext f xs1 xs2 = mul_ext g ys1 ys2"
Lin_irrefl: fixes L :: "(alt \<times> alt) set"   and a :: "alt"   and b :: "alt" assumes "L \<in> Lin"   and "a <\<^bsub>L\<^esub> b"   and "b <\<^bsub>L\<^esub> a" shows "False"
cases: fixes x :: "nat \<times> C" obtains c :: "nat" where "x = (c__, C [])"   | c :: "nat" and ns :: "C list" where "x = (ca__, C (C [] # ns__))"   | c :: "nat" and n :: "C" and ns :: "C list" and ms :: "C list" where "x = (c, C (C (n # ns) # ms))"
A_2: shows "pderiv p div d = sturm_squarefree' p ! 1"
inj_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f A"   and "B \<subseteq> A" shows "inj_on f B"
constrainsD: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and act :: "('c \<times> 'c) set"   and s :: "'c"   and s' :: "'c" assumes "F \<in> A co A'"   and "act \<in> Acts F"   and "(s, s') \<in> act"   and "s \<in> A" shows "s' \<in> A'"
normal_subgroup_factorize: fixes N :: "'a set set"   and H :: "'a set set" assumes "N \<lhd> G Mod N"   and "N \<subseteq> H"   and "subgroup H (G Mod N)" shows "subgroup (rcosets\<^bsub>(G Mod N)\<lparr>carrier := H\<rparr>\<^esub> N) (G Mod N Mod N)"
point_injective: fixes x :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])" shows "coreflexive ((x[[top]]) * x)"
arg_div_2kpi: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0"   and "z2 \<noteq> 0" shows "\<exists>x. arg (z1 / z2) = arg z1 - arg z2 + 2 * real_of_int x * pi"
silent_moves_nonempty_nodestack_False: fixes S :: "'node SDG_node set"   and m :: "'node"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and nx :: "'node SDG_node" assumes "S,kind \<turnstile> ([m],[cf]) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   and "ms' \<noteq> []"   and "CFG_node m' \<in> sum_SDG_slice1 nx"   and "nx \<in> S" shows "False"
simps_2: fixes left :: "'c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf"   and right :: "'c \<Rightarrow> 'e \<Rightarrow> ('b \<times> 'c) spmf"   and s :: "'c"   and b :: "'e" shows "(left \<oplus>\<^sub>O right) s (Inr b) = map_spmf (apfst Inr) (right s b)"
eqBut_def: fixes val :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and val' :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and xs :: "'b"   and x :: "'c" shows "eqBut val val' xs x \<equiv> \<forall>ys y. ys = xs \<and> y = x \<or> val ys y = val' ys y"
pth_7_1: fixes x :: "'c" shows "(0::'c) + x \<equiv> x"
orbit_def: fixes f :: "'a perm"   and a :: "'a" shows "orbit f a = range (\<lambda>n. (f ^ n) \<langle>$\<rangle> a)"
summable_norm: fixes f :: "nat \<Rightarrow> 'b" assumes "summable (\<lambda>n. norm (f n))" shows "norm (suminf f) \<le> (\<Sum>n. norm (f n))"
take_bit_diff: fixes n :: "nat"   and k :: "int"   and l :: "int" shows "take_bit n (take_bit n k - take_bit n l) = take_bit n (k - l)"
da_true: shows "I \<Turnstile> DA \<cdot> \<sigma> \<cdot> \<eta>"
consistency_def: fixes C :: "('c, 'd) form set set" shows "consistency C = (\<forall>S. S \<in> C \<longrightarrow> (\<forall>p ts. \<not> (Pred p ts \<in> S \<and> Neg (Pred p ts) \<in> S)) \<and> \<bottom> \<notin> S \<and> Neg \<top> \<notin> S \<and> (\<forall>Z. Neg (Neg Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. And A B \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. Neg (Or A B) \<in> S \<longrightarrow> S \<union> {Neg A, Neg B} \<in> C) \<and> (\<forall>A B. Or A B \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (And A B) \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {Neg B} \<in> C) \<and> (\<forall>A B. Impl A B \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (Impl A B) \<in> S \<longrightarrow> S \<union> {A, Neg B} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> S \<longrightarrow> S \<union> {P[t/0]} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> S \<longrightarrow> S \<union> {Neg (P[t/0])} \<in> C) \<and> (\<forall>P. Exists P \<in> S \<longrightarrow> (\<exists>x. S \<union> {P[App x []/0]} \<in> C)) \<and> (\<forall>P. Neg (Forall P) \<in> S \<longrightarrow> (\<exists>x. S \<union> {Neg (P[App x []/0])} \<in> C)))"
Collect_subset: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "order.greater_eq A {x \<in> A. P x}"
unit_simps_4: fixes a :: "'a" assumes "obj a" shows "S.dom (UP.unit a) = UP.map\<^sub>0 a"
set_ubounds: fixes as :: "atom list" shows "set (ubounds as) = {(r / c, (- 1 / c) *\<^sub>s cs) |r c cs. Less r (c # cs) \<in> set as \<and> c < 0}"
Pre_element: fixes a :: "'a" assumes "a \<in> carrier D"   and "ExPre D a" shows "Pre D a \<in> carrier D \<and> Pre D a \<prec> a \<and> \<not> (\<exists>y\<in>carrier D. Pre D a \<prec> y \<and> y \<prec> a)"
nat_mult_le_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "(k * m \<le> k * n) = (m \<le> n)"
map_upd_triv: fixes t :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" assumes "t k = Some x" shows "t(k \<mapsto> x) = t"
Inf_eq_Sup: 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 A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf A = Sup {b. \<forall>a\<in>A. less_eq b a}"
wordinterval_UNIV_set_eq: shows "wordinterval_to_set wordinterval_UNIV = UNIV"
crename_fresh_interesting2: fixes x :: "name"   and M :: "trm"   and a :: "coname"   and b :: "coname" assumes "x \<sharp> M[a\<turnstile>c>b]" shows "x \<sharp> M"
span_def: fixes S :: "complex vec set" shows "local.span S = {lincomb a A |a A. finite A \<and> A \<subseteq> S}"
Inf_pres_Inf: fixes F :: "('b \<Rightarrow> 'c) set" assumes "\<forall>f\<in>F. Inf_pres f" shows "Inf_pres (\<Sqinter> F)"
PiE_def: fixes S :: "'f set"   and T :: "'f \<Rightarrow> 'g set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
cmp_simps_4: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "S.dom (S.cmp\<^sub>U\<^sub>P (f, g)) = S.UP f \<star>\<^sub>S S.UP g"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "page_t \<Rightarrow> 'b"   and ipc_stage_t :: "ipc_stage_t" shows "h (case ipc_stage_t of PREP \<Rightarrow> f1 | WAIT \<Rightarrow> f2 | BUF x \<Rightarrow> f3 x) = (case ipc_stage_t of PREP \<Rightarrow> h f1 | WAIT \<Rightarrow> h f2 | BUF x \<Rightarrow> h (f3 x))"
n_mult_add: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "n_mult (p \<oplus>\<^bsub>P\<^esub> q) = n_mult p \<oplus>\<^bsub>P\<^esub> n_mult q"
n: fixes \<epsilon> :: "'b"   and n :: "'b" assumes "sqrt_approximation \<epsilon> n" shows "(0::'b) < n"
kruskal_spanning: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and g :: "'b" assumes "class.stone_kleene_relation_algebra_arc star one times sup less_eq less bot inf top conv uminus"   and "finite {x. x = uminus (uminus x)}" shows "[conv g = g]   f := bot; h := g; WHILE h \<noteq> bot INV {stone_kleene_relation_algebra_arc.kruskal_spanning_invariant star one times less_eq inf conv uminus f g h}  VAR {\<lambda>(e, f, h). card {x. x = uminus (uminus x) \<and> less_eq x (uminus (uminus h))}}  DO e := stone_kleene_relation_algebra_arc.choose_arc one times less_eq bot top conv uminus h; IF less_eq e (uminus (times (star (conv f)) (star f))) THEN f := sup (sup (inf f (uminus (times (times top e) (star (conv f))))) (conv (inf f (times (times top e) (star (conv f)))))) e  ELSE SKIP FI; h := inf (inf h (uminus e)) (uminus (conv e)) OD   [stone_kleene_relation_algebra_arc.spanning_forest star one times less_eq conv uminus f g]"
lfp_fusion4: fixes g :: "'c \<rightarrow> 'd"   and f :: "'c \<rightarrow> 'c"   and h :: "'d \<rightarrow> 'd" assumes "g oo f = h oo g"   and "g\<cdot>\<bottom> = \<bottom>" shows "g\<cdot>(fix\<cdot>f) = fix\<cdot>h"
Domain_mono: fixes r :: "('c \<times> 'd) set"   and s :: "('c \<times> 'd) set" assumes "r \<subseteq> s" shows "Domain r \<subseteq> Domain s"
add_seq_inf_par: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "less_eq (plus (seq a b) (seq b a)) (times a b)"
dynCall_def: fixes init :: "'d \<Rightarrow> 'd"   and p :: "'d \<Rightarrow> 'e"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "dynCall init p return c = DynCom (\<lambda>s. call init (p s) return c)"
insert3_def2: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and x :: "'b"   and t :: "'b tree"   and n :: "nat" assumes "RBTi2 TYPE('b) bal_i e" shows "RBTi2.insert3 bal_i x (t, n) = (let t' = RBTi2.ins3 bal_i n 0 x t in case t' of Same2 \<Rightarrow> (t, n) | Bal2 t' \<Rightarrow> (t', n + 1))"
Linear_not_Branch: fixes b :: "'g"   and r :: "('f, 'g, 'h, 'i) resumption"   and c :: "'h"   and rs :: "'i \<Rightarrow> ('f, 'g, 'h, 'i) resumption" shows "Linear b r \<noteq> Branch c rs"
sub_inf_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<le> - y" shows "- z \<sqinter> - x \<le> - z \<sqinter> - y"
inj_on_implies_partitions_of: fixes F :: "('c \<Rightarrow> 'd) set"   and A :: "'c set"   and B :: "'d set" assumes "F \<in> (A \<rightarrow>\<^sub>E B) // range_permutation A B"   and "univ (\<lambda>f. inj_on f A) F" shows "\<forall>X\<in>partitions_of A B F. card X = 1"
rep_eq: fixes x :: "'a" shows "blinfun_apply (blinfun_scaleR_left x) = (\<lambda>y. y *\<^sub>R x)"
combine_common_factor: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and e :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring plus times" shows "plus (times a e) (plus (times b e) c) = plus (times (plus a b) e) c"
word_set_set_same: fixes w :: "'a word"   and n :: "nat"   and x :: "bool"   and y :: "bool" shows "set_bit_class.set_bit (set_bit_class.set_bit w n x) n y = set_bit_class.set_bit w n y"
walk_take: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and n :: "nat" assumes "Graph G"   and "Graph.walk G xs" shows "Graph.walk G (take n xs)"
lfilled_collapse2: fixes n :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and es' :: "e list"   and LI :: "e list" assumes "Lfilled n lholed (es @ es') LI" shows "\<exists>lholed' vs'. Lfilled n lholed' es LI"
flip_at_base_simps_2: fixes a :: "'b"   and b :: "'b" assumes "sort_of (atom a) = sort_of (atom b)" shows "(a \<leftrightarrow> b) \<bullet> b = a"
rat_divide_code: fixes p :: "rat"   and q :: "rat" shows "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q in Rat.normalize (a * d, c * b))"
linordered_field_no_ub: shows "\<forall>x. \<exists>y. x < y"
r_congruent_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "rcong\<^bsub>G\<^esub> H = {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> y \<in> H}"
observable_io_targets_append: fixes M :: "('d, 'e, 'f) FSM"   and q1 :: "'f"   and vs :: "('d \<times> 'e) list"   and q2 :: "'f"   and xs :: "('d \<times> 'e) list"   and q3 :: "'f" assumes "observable M"   and "io_targets M q1 vs = {q2}"   and "io_targets M q2 xs = {q3}" shows "io_targets M q1 (vs @ xs) = {q3}"
always_eventually: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "\<forall>x. P x" shows "eventually P F"
assms_2: shows "r \<cdot> y \<le> y \<cdot> q"
seqs_rev: fixes n :: "nat" shows "rev ` seqs n = seqs n"
subst_mgt_comp: fixes \<theta> :: "'g \<Rightarrow> ('e, 'd) Term.term"   and \<delta> :: "'d \<Rightarrow> ('e, 'f) Term.term" shows "\<theta> \<preceq>\<^sub>\<circ> \<theta> \<circ>\<^sub>s \<delta>"
of_int_abs: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "ring_1.of_int one plus zero minus \<bar>x\<bar> = abs (ring_1.of_int one plus zero minus x)"
cong_exp_trans_1: fixes a :: "'b"   and b :: "nat"   and c :: "'b"   and n :: "'b"   and d :: "'b" assumes "[a ^ b = c] (mod n)"   and "[a = d] (mod n)" shows "[d ^ b = c] (mod n)"
KGDD_by_deleting_point: fixes x :: "'a" assumes "x \<in> \<V>"   and "\<B> rep x < \<b>"   and "1 < \<B> rep x" shows "K_GDD (del_point x) (str_del_point_blocks x) {\<k>} {b - {x} |b. b \<in># \<B> \<and> x \<in> b}"
GF_advice_prop_unfold_congruent: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and X :: "'b ltln set" assumes "\<phi> \<sim>\<^sub>Q \<psi>" shows "Unf \<phi>[X]\<^sub>\<nu> \<sim>\<^sub>Q Unf \<psi>[X]\<^sub>\<nu>"
prv_ldsj_imp: fixes \<chi> :: "'fmla"   and \<phi>s :: "'fmla list" assumes "\<chi> \<in> fmla"   and "set \<phi>s \<subseteq> fmla"   and "\<And>\<phi>. \<phi> \<in> set \<phi>s \<Longrightarrow> prv (imp \<phi> \<chi>)" shows "prv (imp (ldsj \<phi>s) \<chi>)"
finite_ACI_norm_toplevel_summands: fixes B :: "'c rexp set"   and f :: "'c rexp \<Rightarrow> 'd" assumes "finite B" shows "finite {f \<guillemotleft>s\<guillemotright> |s. toplevel_summands s \<subseteq> B}"
c_snd_is_pr: shows "c_snd \<in> PrimRec1"
fc_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "fc x \<le> fc y"
that: fixes m :: "nat" assumes "t \<^sup>@ m \<le>p u"   and "u <p t \<^sup>@ m \<cdot> t" shows "thesis"
comm_non_prim: fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>"   and "v \<noteq> \<epsilon>"   and "u \<cdot> v = v \<cdot> u" shows "\<not> primitive (u \<cdot> v)"
rel_sel: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g list"   and b :: "'h list" shows "list_all2 R a b = ((a = []) = (b = []) \<and> (a \<noteq> [] \<longrightarrow> b \<noteq> [] \<longrightarrow> R (hd a) (hd b) \<and> list_all2 R (tl a) (tl b)))"
SUP_image: fixes g :: "'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "Sup (g ` f ` A) = Sup ((g \<circ> f) ` A)"
wm_to_state_eq: fixes P :: "ast_problem"   and M :: "object atom formula set"   and p :: "predicate"   and as :: "object list" assumes "wf_ast_problem P" shows "wm_to_state M (p, as) = (Atom (predAtm p as) \<in> M)"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (src\<^sub>B f) = B.arr f"
eq_fin_lproject_visible: fixes u :: "'action list"   and v :: "'action list" assumes "u =\<^sub>F v" shows "lproject visible (llist_of u) = lproject visible (llist_of v)"
quot_fm_coding: fixes A :: "fm" shows "coding_tm \<guillemotleft>A\<guillemotright>"
finite_lin_indpt2: fixes A :: "'a vec set" assumes "finite A"   and "A \<subseteq> carrier_vec n"   and "\<And>a. \<lbrakk>True; lincomb a A = 0\<^sub>v n\<rbrakk> \<Longrightarrow> \<forall>v\<in>A. a v = (0::'a)" shows "lin_indpt A"
row_matrix_matrix_mult: fixes P :: "(('c, 'e) vec, 'f) vec"   and i :: "'f"   and A :: "(('c, 'd) vec, 'e) vec" shows "P $ i v* A = (P ** A) $ i"
nonzero_divide_eq_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(divide b c = a) = (b = times a c)"
map_of_del_list: fixes ps :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted1 ps" shows "map_of (del_list x ps) = (map_of ps)(x := None)"
simps_1: fixes v2 :: "('c \<times> 'a) list"   and m :: "('b \<times> ('b, 'c, 'a) namespace) list"   and n :: "'c" shows "nsLookup (Bind v2 m) (Short n) = map_of v2 n"
assms_2: fixes r :: "('a \<times> 'a) set" assumes "single_valued r" shows "single_valued (f r)"
antisym_conv3: fixes y :: "'c word"   and x :: "'c word" assumes "\<not> y <s x" shows "(\<not> x <s y) = (x = y)"
dirichlet_prod_const_left: fixes n :: "nat"   and c :: "'b"   and g :: "nat \<Rightarrow> 'b" assumes "0 < n" shows "dirichlet_prod (\<lambda>n. if n = Suc 0 then c else (0::'b)) g n = c * g n"
3: shows "z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))"
4: shows "Some r = d\<^sub>1 (\<^bold>\<lambda>z. make\<o> (\<lambda>w s. d\<^sub>\<kappa> (z\<^sup>P) = Some (\<omega>\<nu> o\<^sub>1)))"
cases: fixes x :: "tm \<times> tm" obtains x :: "name" and r :: "tm" and d :: "tm" and y :: "name" and z :: "name" where "atom x \<sharp> (r, d, y, z)" and "atom y \<sharp> (r, d, z)" and "atom z \<sharp> (r, d)" and "xa__ = (r, d)"
linear_matrix_vector_mul_eq: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" shows "Vector_Spaces.linear (*s) (*s) f = linear f"
exp_approx'': fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 \<le> x"   and "x \<le> 1" shows "\<bar>exp x - (\<Sum>k\<le>n. x ^ k / fact k)\<bar> \<le> 1 / fact n"
integrable_circle': fixes n :: "nat" shows "f (cnj s) contour_integrable_on part_circlepath 0 (R n) 0 pi"
pfa_insert_rule: fixes i :: "nat"   and n :: "nat"   and c :: "nat"   and l :: "'b list"   and a :: "'b array"   and x :: "'b" assumes "i \<le> n"   and "n < c" shows "<is_pfa c l (a, n)> pfa_insert (a, n) i x <\<lambda>(a', n'). is_pfa c (take i l @ x # drop i l) (a', n') * \<up> (n' = n + 1 \<and> a = a')>"
errMOD_igSwapIGAbsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igAbsIPresIGWls MOD"   and "igWlsDisj MOD"   and "igSwapIPresIGWls MOD"   and "igSwapIGAbs MOD" shows "igSwapIGAbsSTR (errMOD MOD)"
K_def: shows "K = component_of_term ` Keys (args_to_set (gs, x))"
nz: fixes z :: "complex" assumes "z \<in> ball 0 (2 * pi) - {0}" shows "1 - exp z \<noteq> 0"
PSimQ: shows "P \<leadsto>\<^sup>^<Rel> Q"
int_xor_zero: fixes x :: "int" shows "0 XOR x = x"
PREFER_D: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "PREFER P x" shows "P x"
ldl: shows "LeftDerivationLadder \<alpha> D L \<gamma>"
borel_measurable_ereal_cases: fixes f :: "'c \<Rightarrow> ereal"   and M :: "'c measure"   and H :: "ereal \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M" shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
litsim_def: fixes s :: "'a"   and t :: "'a" shows "(s \<doteq> t) = (s \<le>\<cdot> t \<and> t \<le>\<cdot> s)"
winning_strategy_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "winning_strategy p \<sigma> v0 \<equiv> \<forall>P. vmc_path G P v0 p \<sigma> \<longrightarrow> winning_path p P"
proper_sub_subst: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and \<rho> :: "'b \<Rightarrow> ('a, 'b) tm" assumes "proper_sub s t" shows "proper_sub (subst \<rho> s) (subst \<rho> t)"
set_plus_mono2_b: fixes C :: "'a set"   and D :: "'a set"   and E :: "'a set"   and F :: "'a set"   and x :: "'a" assumes "order.greater_eq D C"   and "order.greater_eq F E"   and "x \<in> C + E" shows "x \<in> D + F"
q_has_vector_derivative: fixes t :: "real"   and u :: "real set" shows "(q has_vector_derivative q' t) (at t within u)"
eq_Maybe_def: shows "eq = maybe\<cdot>(maybe\<cdot>TT\<cdot>(\<Lambda> y. FF))\<cdot>(\<Lambda> x. maybe\<cdot>FF\<cdot>(\<Lambda> y. eq\<cdot>x\<cdot>y))"
path_mono: fixes g :: "'b ugraph"   and g' :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "edges g \<subseteq> edges g'"   and "path g u p v" shows "path g' u p v"
nth_Cons_split: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "['b]"   and n :: "Integer" shows "P ((x : xs) !! n) = ((eq\<cdot>n\<cdot>0 = FF \<longrightarrow> P ((x : xs) !! n)) \<and> (eq\<cdot>n\<cdot>0 = TT \<longrightarrow> P ((x : xs) !! n)) \<and> (n = \<bottom> \<longrightarrow> P ((x : xs) !! n)))"
not_in_sum: fixes var :: "nat"   and f :: "nat \<Rightarrow> real mpoly" assumes "\<forall>i. var \<notin> vars (f i)" shows "\<forall>n. var \<notin> vars (sum f {0..<n})"
initial_upper_bound_def: fixes r :: "rat" shows "local.initial_upper_bound r = rat_of_int (root_rat_ceiling n r)"