hcju/isabelleps huggingface dataset:
uninst_allVars_simps_3: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and g :: "'d"   and u :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and p :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option" shows "CFG_SSA_base.allVars \<alpha>n inEdges' defs (uses(g := u g)) p g = CFG_SSA_base.allVars \<alpha>n inEdges' defs u p g"
L_lower_bound_card: fixes P :: "'a set set" assumes "bp P" shows "card L \<le> card P"
nu_ideal1: fixes x :: "'a"   and y :: "'a" assumes "\<nu> x = x"   and "y \<le> x" shows "\<nu> y = y"
continuous_at_ereal: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real" assumes "continuous F f" shows "continuous F (\<lambda>x. ereal (f x))"
column_def: fixes j :: "'f"   and A :: "(('d, 'f) vec, 'e) vec" shows "column j A = (\<chi>i. A $ i $ j)"
active_thread_ex_red: fixes t :: "'t"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "t \<in> active_threads s" shows "\<exists>ta. Ex (redT_syntax1 s t ta)"
arr_map: fixes f :: "'c" assumes "C.arr f" shows "S.arr (local.map f)"
simps_4: fixes \<alpha>1 :: "'d"   and t1 :: "('b, 'c, 'd) Tree"   and \<alpha>2 :: "'d"   and t2 :: "('b, 'c, 'd) Tree" shows "(tAct \<alpha>1 t1 =\<^sub>\<alpha> tAct \<alpha>2 t2) = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"
subgroupE_2: fixes H :: "'a set" assumes "subgroup H G" shows "H \<noteq> {}"
inf_le1: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) x"
mult_less_cancel_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a c) (times b c) = ((less_eq zero c \<longrightarrow> less a b) \<and> (less_eq c zero \<longrightarrow> less b a))"
sb: shows "sys_mem_store_buffers (mutator m') s = mw_Mutate r f opt_r' # ws"
fun_left_idem: fixes x :: "'a"   and z :: "'b" shows "f x (f x z) = f x z"
ABind_def: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and v :: "Vars.var"   and e :: "exp" shows "ArityAnalysis.ABind Aexp v e = (\<Lambda> ae. fup\<cdot>(Aexp e)\<cdot>(ae v))"
"9_2": fixes z :: "real" assumes "1 < z" shows "summable (\<lambda>i. 1 / (z + real i) powr (1 + \<epsilon>))"
assms_4: shows "A B C D' E' F' SumA G' H' I'"
strict_Some: fixes f :: "'c \<Rightarrow> 'b err"   and x :: "'c err"   and y :: "'b" shows "(strict f x = OK y) = (\<exists>z. x = OK z \<and> f z = OK y)"
minimal: fixes y :: "(bool \<times> 'a) list" assumes "cancels_to_1 l' y" shows "y \<notin> {l'. cancels_to_1\<^sup>*\<^sup>* l l'}"
inorder_split: fixes t :: "('c \<times> nat) tree" shows "Tree2.inorder (split t) = Tree2.inorder t"
setparts_2: shows "set (drop (Suc i) as') \<subseteq> set as'"
instrs_append_Ghost\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "instrs (sb @ [Ghost\<^sub>s\<^sub>b A L R W]) = instrs sb @ [Ghost A L R W]"
canonical_retraction_face_distance_map: fixes A :: "'a set set"   and C :: "'a set"   and D :: "'a set"   and F :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<in> A"   and "order.greater_eq C F" shows "face_distance F (canonical_retraction A C ` D) = face_distance F D"
tangent_space_linear_on: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "X \<in> tangent_space p" shows "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X"
quasi_inverse_transpose_2: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "f \<star>\<^sub>D g \<cong>\<^sub>D h"   and "D.equivalence_map f" shows "g \<cong>\<^sub>D f\<^sup>~\<^sup>D \<star>\<^sub>D h"
eq_on_def: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b"   and X :: "'a set" shows "(s = t on X) = (\<forall>x\<in>X. s x = t x)"
step_fun_call: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and g :: "'fun"   and gd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and \<Sigma> :: "'dyn list"   and frame\<^sub>f :: "('fun, 'dyn) frame"   and frame\<^sub>g :: "('fun, 'dyn) frame"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.ICall g"   and "Finca_get F g = Some gd"   and "arity gd \<le> length \<Sigma>"   and "frame\<^sub>f = Frame f pc \<Sigma>"   and "frame\<^sub>g = Frame g 0 (take (arity gd) \<Sigma>)" shows "Sinca.step (State F H (frame\<^sub>f # st)) (State F H (frame\<^sub>g # frame\<^sub>f # st))"
extend_border: fixes j :: "nat"   and s :: "'a list"   and i :: "nat" assumes "j \<le> length s"   and "s ! (i - 1) = s ! (j - 1)"   and "strict_border (take (i - 1) s) (take (j - 1) s)"   and "\<ff> s j \<le> i + 1" shows "\<ff> s j = i + 1"
exhaust: fixes y :: "w_node" obtains x1 :: "nat" where "y = (_ x1 _)"   | "y = (_Entry_)"   | "y = (_Exit_)"
advice_sets_subformulas: fixes X :: "'a ltln set"   and \<phi> :: "'a ltln"   and Y :: "'a ltln set" shows "(X \<subseteq> subformulas\<^sub>\<mu> \<phi> \<and> Y \<subseteq> subformulas\<^sub>\<nu> \<phi>) = (\<exists>xs ys. X = set xs \<and> Y = set ys \<and> (xs, ys) \<in> set (advice_sets \<phi>))"
span_listE: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and bs :: "('c \<Rightarrow>\<^sub>0 'd) list" assumes "p \<in> phull (set bs)" obtains qs :: "'d list" where "length qs = length bs" and "p = sum_list (map2 (\<cdot>) qs bs)"
add_closed: fixes g :: "'v"   and h :: "'v" assumes "g \<in> V"   and "h \<in> V" shows "g + h \<in> V"
\<theta>\<^sub>h'_def: shows "\<theta>\<^sub>h' = TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
simps_31: fixes r :: "'b rexp"   and r' :: "'b rexp"   and s :: "'b rexp"   and s' :: "'b rexp" shows "le_rexp (Times r r') (Times s s') = (if r = s then le_rexp r' s' else le_rexp r s)"
Eps_case_prod_eq: fixes x :: "'a"   and y :: "'b" shows "(SOME (x', y'). x = x' \<and> y = y') = (x, y)"
P7: shows "A C B LtA C A B' \<and> A B C LtA C A B'"
mkarr_inj_on_Ide: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Ide t"   and "Ide u"   and "mkarr t = mkarr u" shows "t = u"
induct: fixes P :: "'d node_action \<Rightarrow> bool"   and node_action :: "'d node_action" assumes "\<And>x1 x2. P (x1:*cast(x2))"   and "\<And>x1 x2. P (x1:deliver(x2))"   and "\<And>x1 x2 x3. P (x1\<not>x2:arrive(x3))"   and "\<And>x1 x2. P (connect(x1, x2))"   and "\<And>x1 x2. P (disconnect(x1, x2))"   and "\<And>x1 x2 x3. P (x1:newpkt(x2, x3))"   and "P \<tau>\<^sub>n" shows "P node_action"
Rep_BinomialHeap: fixes x :: "('c, 'd) BinomialHeap" shows "Rep_BinomialHeap x \<in> {q. BinomialHeapStruc.invar q}"
a_get_ancestors_di_locs_def: fixes get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_di_locs get_parent_locs get_host_locs get_disconnected_document_locs = get_parent_locs \<union> get_host_locs \<union> get_disconnected_document_locs"
computation_negative_flip: shows "rat_poly_plus (rat_poly_inv (rat_poly_times B (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B))) (rat_poly_inv (rat_poly_times A (rat_poly_times B A))) = rat_poly_inv (rat_poly_times B (rat_poly_times B B))"
Square_body_def: shows "local.Square_body \<equiv> \<acute>R :== \<acute>N * \<acute>N"
pathstart_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b" shows "pathstart (subpath u v g) = g u"
Src_Dom_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "Src (Dom t) = Src t \<and> Src (Cod t) = Src t"
elims_2: fixes x :: "'a btree"   and xa :: "'a" assumes "isin x xa" obtains ts :: "('a btree \<times> 'a) list" and t :: "'a btree" and y :: "'a" where "x = Node ts t" and "xa = y" and "case split ts y of (x, []) \<Rightarrow> isin t y | (x, (sub, sep) # rs) \<Rightarrow> if y = sep then True else isin sub y"
peek_and_def2: fixes P :: "'b assn"   and p :: "state \<Rightarrow> bool"   and Y :: "vals"   and s :: "state" shows "(P \<and>. p) Y s = (\<lambda>Z. P Y s Z \<and> p s)"
compact_Inter: fixes \<F> :: "'a set set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S"   and "\<F> \<noteq> {}" shows "compact (\<Inter> \<F>)"
\<mu>_distr: shows "distr M borel X = \<mu>"
poly_deg_monom_mult: fixes c :: "'a"   and p :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a"   and t :: "'b \<Rightarrow>\<^sub>0 'c" assumes "c \<noteq> (0::'a)"   and "p \<noteq> 0" shows "poly_deg (MPoly_Type_Class.punit.monom_mult c t p) = deg_pm t + poly_deg p"
BaPv: shows "B *\<^sub>v (adjoint P *\<^sub>v v) \<in> carrier_vec n"
is_a_proper_intersec_def: fixes P :: "Points"   and A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" shows "is_a_proper_intersec P A B C D \<equiv> A \<noteq> B \<and> C \<noteq> D \<and> line A B \<noteq> line C D \<and> col P A B \<and> col P C D"
cases: fixes a1 :: "(val \<times> val) list"   and a2 :: "(val \<times> val) list" assumes "a1 \<lesssim> a2" obtains t1 :: "(val \<times> val) list" and t2 :: "(val \<times> val) list" where "a1 = t1" and "a2 = t2" and "\<forall>v1 v2. (v1, v2) \<in> set t1 \<longrightarrow> (\<exists>v3 v4. (v3, v4) \<in> set t2 \<and> v1 \<sqsubseteq> v3 \<and> v3 \<sqsubseteq> v1 \<and> v2 \<sqsubseteq> v4 \<and> v4 \<sqsubseteq> v2)"
assms: shows "\<forall>v. v \<le>\<^sub>F\<^sub>I w \<longrightarrow> path v p"
seqSubstSimps_3: fixes M :: "'a"   and I :: "('a, 'b, 'c) input"   and \<sigma> :: "(name list \<times> 'a list) list" shows "M\<lparr>I[<\<sigma>>] = (M[<\<sigma>>])\<lparr>seqSubs' I \<sigma>"
entail_union: fixes N :: "'g set"   and N1 :: "'g set"   and N2 :: "'g set" shows "(N \<Turnstile>G N1 \<and> N \<Turnstile>G N2) = (N \<Turnstile>G N1 \<union> N2)"
simps_1: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and s :: "state" shows "approximating_bigstep_fun \<gamma> p [] s = s"
elims: fixes E :: "'b set set"   and x :: "'b"   and y :: "'b set" assumes "Graph_E E"   and "Graph_E.neighbors E x = y" obtains v :: "'b" where "x = v" and "y = {u. {u, v} \<in> E}"
emeasure_point_measure_finite: fixes A :: "'b set"   and X :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "finite A"   and "X \<subseteq> A" shows "emeasure (point_measure A f) X = sum f X"
Domainp_rel_converter_le: fixes A :: "'a \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'h \<Rightarrow> bool" shows "Domainp (rel_converter A B C R) \<le> pred_converter (Collect (Domainp A)) (Domainp B) (Domainp C) (Collect (Domainp R))"
M_def: fixes n :: "nat"   and fs :: "'a vec list"   and k :: "nat" assumes "gram_schmidt_fs_Rn n fs" shows "gram_schmidt_fs_Rn.M n fs k = mat k k (\<lambda>(i, j). gram_schmidt_fs.\<mu> n fs i j)"
simps: fixes a :: "dB" shows "NF a = ((\<exists>ts x. a = Var x \<degree>\<degree> ts \<and> listall NF ts) \<or> (\<exists>t. a = Abs t \<and> NF t))"
finsert_commute: fixes x :: "'c"   and y :: "'c"   and A :: "'c fset" shows "finsert x (finsert y A) = finsert y (finsert x A)"
less_is: shows "\<forall>i\<in>load_tmps is. i < n"
irreducible_degree_field: fixes p :: "'b poly" assumes "irreducible p" shows "0 < degree p"
elims: fixes x :: "'d \<Rightarrow> nat \<Rightarrow> 'e"   and xa :: "nat"   and xb :: "nat"   and xc :: "'d"   and xd :: "'c \<Rightarrow> bool"   and xe :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and xf :: "'c"   and y :: "'c" assumes "idx_iteratei_aux x xa xb xc xd xe xf = y" obtains get :: "'d \<Rightarrow> nat \<Rightarrow> 'e" and sz :: "nat" and i :: "nat" and l :: "'d" and c :: "'c \<Rightarrow> bool" and f :: "'e \<Rightarrow> 'c \<Rightarrow> 'c" and \<sigma> :: "'c" where "x = get" and "xa = sz" and "xb = i" and "xc = l" and "xd = c" and "xe = f" and "xf = \<sigma>" and "y = (if i = 0 \<or> \<not> c \<sigma> then \<sigma> else idx_iteratei_aux get sz (i - 1) l c f (f (get l (sz - i)) \<sigma>))"
norm_ge_square: fixes a :: "real"   and x :: "'a" shows "(a \<le> norm x) = (a \<le> 0 \<or> a\<^sup>2 \<le> x \<bullet> x)"
asymp_equiv_inverse: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "(\<lambda>x. inverse (f x)) \<sim>[F] (\<lambda>x. inverse (g x))"
intro: fixes S :: "'c set"   and G :: "('c, 'd) pre_digraph" assumes "S \<in> pre_digraph.sccs_verts G" shows "max_reachable_set_axioms G S"
one_eq_Numeral1_nring: shows "(1::'a) = Numeral1"
simps_3: fixes x41 :: "nat"   and x42 :: "e list"   and x43 :: "e list"   and y41 :: "nat"   and y42 :: "e list"   and y43 :: "e list" shows "(Label x41 x42 x43 = Label y41 y42 y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
restrict_fupd: fixes i :: "'c"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "i \<notin> I" shows "restrict (f(i := x)) I = restrict f I"
host_apply_impl_correct: fixes s :: "s"   and tf :: "tf"   and h :: "host"   and vs :: "v list"   and m' :: "s \<times> v list" assumes "host_apply_impl s tf h vs = Some m'" shows "\<exists>hs. host_apply s tf h vs hs = Some m'"
segment_disj: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "a \<noteq> b"   and "bij f" shows "segment f a b \<inter> segment f b a = {}"
spawn: fixes \<iota> :: "nat"   and V :: "('exp, 'id) MWLsCom list"   and m :: "'id \<Rightarrow> 'val" shows "SRabbr (spawn\<^bsub>\<iota>\<^esub> V) m V None m"
Cons_3: shows "\<forall>var\<in>vars_constraints (map snd (hh # t)). var < start"
degree_Mp_mult_monic: fixes f :: "int poly"   and g :: "int poly" assumes "monic f"   and "monic g" shows "degree_m (f * g) = degree f + degree g"
0: shows "w\<^sub>1 =\<^sub>F w\<^sub>2"
lift_SD_iff_nonagent: fixes p :: "'b pmf"   and q :: "'b pmf"   and i :: "'a" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "i \<in> agents' - agents" shows "SD (R' i) p q"
3022: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- y \<sqinter> - x) = top"
flip_altsublist_chain_set: fixes S :: "'b set"   and x :: "'b list"   and zs :: "'b list list"   and y :: "'b list" assumes "PreCoxeterSystem S"   and "x \<in> lists S"   and "binrelchain PreCoxeterSystem.flip_altsublist_adjacent (x # zs @ [y])" shows "set y = set x"
Elem_nat2Nat_inf: fixes n :: "nat" shows "Elem (nat2Nat n) HOLZF.Inf"
ccProd_union2: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and S'' :: "Vars.var set" shows "S G\<times> (S' \<union> S'') = S G\<times> S' \<squnion> S G\<times> S''"
domain_eq: shows "Domainp (pcr_ell2 (=)) = has_ell2_norm"
extract_rpath_def: fixes src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option" shows "extract_rpath src dst PRED \<equiv> while\<^sub>T\<^bsup>\<lambda>(v, p). v \<in> dom PRED \<and> isPath v p dst \<and> distinct (pathVertices v p) \<and> (\<forall>v'\<in>set (pathVertices v p). min_dist src v \<le> min_dist src v') \<and> min_dist src v + length p = min_dist src dst\<^esup> (\<lambda>(v, p). v \<noteq> src) (\<lambda>(v, p). assert (v \<in> dom PRED) \<bind> (\<lambda>_. let u = the (PRED v); p = (u, v) # p; v = u in return (v, p))) (dst, []) \<bind> (\<lambda>(uu_, p). return p)"
is_natural_1: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (G f) (\<tau> (partial_magma.dom A f)) = \<tau> f"
VSBrowns_def: shows "VSBrowns = opt_group \<circ> QE_dnf opt_group (step_augment gen_qe brownsHeuristic) \<circ> VSLuckiestBlocks \<circ> opt_group"
add_uminus_minus_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + - B = A - B"
iMOD_div_self: fixes m :: "nat"   and r :: "nat" assumes "0 < m" shows "[ r, mod m ] \<oslash> m = [r div m\<dots>]"
SC_is_legal: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "sequentially_consistent P (E, ws)" shows "legal_execution P \<E> (E, ws)"
prime_nat_def: shows "Primes.prime_nat = normalization_semidom_class.prime"
restrict_def: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d dual" shows "map_dual g (map_dual f v) = map_dual (g \<circ> f) v"
trigonometric_set_mul_absolutely_integrable: fixes f :: "real \<Rightarrow> real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>x. trigonometric_set n x * f x) absolutely_integrable_on {- pi..pi}"
cases: fixes a1 :: "'var \<Rightarrow> 'val"   and a2 :: "nat"   and a3 :: "'var \<Rightarrow> 'val"   and a4 :: "nat" assumes "((a1, a2), a3, a4) \<in> cop" obtains \<sigma> :: "'var \<Rightarrow> 'val" and n :: "nat" and \<sigma>' :: "'var \<Rightarrow> 'val" and n' :: "nat" where "a1 = \<sigma>''__" and "a2 = n__" and "a3 = \<sigma>'''__" and "a4 = n'" and "((\<sigma>''__, n__), \<sigma>'''__, n') \<in> cp" and "path \<sigma>''__ n__ \<in> dom local.att"   | \<sigma> :: "'var \<Rightarrow> 'val" and k :: "nat" and \<sigma>' :: "'var \<Rightarrow> 'val" and k' :: "nat" and n :: "nat" where "a1 = \<sigma>" and "a2 = n" and "a3 = \<sigma>'" and "a4 = k'" and "((\<sigma>, k), \<sigma>', k') \<in> cp" and "n cd\<^bsup>path \<sigma>\<^esup>\<rightarrow> k" and "path \<sigma> (Suc k) \<noteq> path \<sigma>' (Suc k')" and "path \<sigma> n \<in> dom local.att"
nat_distrib_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> (b + c) = a \<cdot> b + a \<cdot> c"
assms_5: shows "nprv (insert \<phi>4 F) \<psi>"
eval_r': shows "eval r [encode r] = eval f [the (eval h [encode r])]"
simps_1: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and uu :: "'d"   and uv :: "'d fs_set"   and P :: "'a"   and \<phi> :: "'b" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.L_satisfies satisfies (AC (uu, uv, P)) \<phi> = satisfies P \<phi>"
higher_deriv_ln_Gamma_real: fixes x :: "real"   and j :: "nat" assumes "0 < x" shows "(deriv ^^ j) ln_Gamma x = (if j = 0 then ln_Gamma x else Polygamma (j - 1) x)"
pick_inj_on_goodAbs: shows "inj_on pick (Collect goodAbs)"
jumpF_neg_has_sgnx: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" assumes "jumpF f F < 0" shows "(f has_sgnx - 1) F"
assms_2: shows "i + (1::'a) \<noteq> j"
assms: shows "x \<parallel> y \<cdot> d z = x \<cdot> d z \<parallel> (y \<cdot> d z)"
one_Quantity_ext_def: shows "1 = \<lparr>mag = 1::'a, dim = 1, \<dots> = 1::'c\<rparr>"
continuous_on_product_coordinates: fixes i :: "'a" shows "continuous_on UNIV (\<lambda>x. x i)"
mapl_T_alt_def: fixes l1 :: "'n \<Rightarrow> 't" shows "mapl_T l1 = map_T l1 id id id id"
filterlim_real_sequentially: shows "filterlim real at_top sequentially"
foldl_Nil: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'e" shows "foldl f a [] = a"
ivl_disj_un_singleton_2: fixes u :: "'b" shows "{..<u} \<union> {u} = {..u}"
foldl_assoc: fixes x :: "'b \<Rightarrow> bool"   and y :: "'b \<Rightarrow> bool"   and zs :: "('b \<Rightarrow> bool) list" shows "foldl (\<and>*) (x \<and>* y) zs = (x \<and>* foldl (\<and>*) y zs)"
combine_common_factor: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and e :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.semiring plus times" shows "plus (times a e) (plus (times b e) c) = plus (times (plus a b) e) c"
src_inv: fixes \<mu> :: "'a" assumes "local.iso \<mu>" shows "src (local.inv \<mu>) = src \<mu>"
is_unit_const_poly_iff: fixes c :: "'a" shows "([:c:] dvd 1) = (c dvd (1::'a))"
game_hash_def: fixes w :: "'domain"   and w' :: "'domain" shows "game_hash w w' = seed_gen \<bind> (\<lambda>seed. return_spmf (hash seed w = hash seed w' \<and> w \<noteq> w'))"
Entry_path: fixes n :: "'b" assumes "valid_node n" shows "\<exists>as. Entry -as\<rightarrow>\<^sub>\<surd>* n"
part_K_I_I: assumes "I \<circ> K \<circ> I = I" shows "K \<circ> I = I"
iT_Div_0_if: fixes I :: "nat set" shows "I \<oslash> 0 = (if I = {} then {} else [\<dots>0])"
cases: fixes a1 :: "key"   and a2 :: "agent"   and runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option" assumes "(a1, a2) \<in> azC runz" obtains Rs :: "fid_t" and A :: "agent" and B :: "agent" and al :: "atom list" and C :: "agent" where "a1 = sesK (Rs__ $ sk)" and "a2 = C__" and "runz Rs__ = Some (Serv, [A__, B__], al__)" and "C__ \<in> {A__, B__, Sv}"   | Rs :: "fid_t" and A :: "agent" and B :: "agent" and al :: "atom list" and C :: "agent" where "a1 = sesK (Rs $ sk)" and "a2 = Ca__" and "runz Rs = Some (Serv, [A, B], al)" and "A \<in> bad \<or> B \<in> bad"   | K :: "key" and C :: "agent" where "a1 = K" and "a2 = C" and "(K, C) \<in> keySetup"
mat_of_cols_carrier_1: fixes n :: "nat"   and vs :: "'a vec list" shows "mat_of_cols n vs \<in> carrier_mat n (length vs)"
induct: fixes P :: "'a word \<Rightarrow> 'a simple_rule list \<Rightarrow> 'a wordinterval \<Rightarrow> bool"   and a0 :: "'a word"   and a1 :: "'a simple_rule list"   and a2 :: "'a wordinterval" assumes "\<And>uu_ uv_. P uu_ [] uv_"   and "\<And>d m rs acc_dropped. \<lbrakk>simple_match_ip (dst m) d \<Longrightarrow> P d rs acc_dropped; \<not> simple_match_ip (dst m) d \<Longrightarrow> P d rs acc_dropped\<rbrakk> \<Longrightarrow> P d (SimpleRule m Accept # rs) acc_dropped"   and "\<And>d m rs acc_dropped. \<lbrakk>simple_match_ip (dst m) d \<Longrightarrow> P d rs (wordinterval_union (ipcidr_tuple_to_wordinterval (src m)) acc_dropped); \<not> simple_match_ip (dst m) d \<Longrightarrow> P d rs acc_dropped\<rbrakk> \<Longrightarrow> P d (SimpleRule m Drop # rs) acc_dropped" shows "P a0 a1 a2"
expectation_indicator: fixes M :: "'b measure"   and A :: "'b set" assumes "prob_space M"   and "A \<in> sets M" shows "integral\<^sup>L M (rind A) = Sigma_Algebra.measure M A"