hcju/isabelleps huggingface dataset:
rotate: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "ChamberComplexApartmentSystemTriangle X \<A> B' A B D E C z"
simps: fixes callee :: "'i \<Rightarrow> 'm \<Rightarrow> ('l \<times> 'i, 'j, 'k) gpv"   and gpv :: "('h, 'm, 'l) gpv"   and s :: "'i" shows "inline1 callee gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1 callee (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))"
vec_lambda_eta: fixes g :: "('a, 'b) vec" shows "(\<chi>i. g $ i) = g"
prems_10: fixes r :: "'a" assumes "start = Some r" shows "r \<in> set (interp_ins (ops @ [a]))"
minarc_meet_bot_iff: fixes x :: "'a" shows "(minarc x \<sqinter> x = bot) = (x = bot)"
inv_idpc: shows "\<forall>x \<sigma>. P (\<sigma>\<lparr>break_status := x\<rparr>) = P \<sigma>"
join_assoc: fixes M :: "'c measure measure measure"   and N :: "'c measure" assumes "sets M = sets (subprob_algebra (subprob_algebra N))" shows "join (distr M (subprob_algebra N) join) = join (join M)"
resumption_ord_simps_1: fixes out :: "'g"   and c :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption"   and r :: "('f, 'g, 'h) Resumption.resumption" shows "resumption_ord (Resumption.resumption.Pause out c) r = (\<exists>c'. r = Resumption.resumption.Pause out c' \<and> rel_fun (=) resumption_ord c c')"
subterms_inclusion_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and E1 :: "'b trm set"   and E2 :: "'b trm set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.subterms_inclusion E1 E2 = (\<forall>x1\<in>E1. \<exists>x2\<in>E2. occurs_in x1 x2)"
Seq_def: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "a ;; b \<equiv> \<lambda>ab. a ab \<circ> b ab"
assms_1: shows "ev (\<lambda>xs. P (shd xs)) xs"
fun_app_lifted_def: fixes f\<^sub>T :: "['d| 'f \<Rightarrow> ['d| 'e]]"   and x\<^sub>T :: "['d| 'f]" shows "f\<^sub>T . x\<^sub>T \<equiv> f\<^sub>T \<bind> (\<bind>) x\<^sub>T"
subsets_are_rows_R: fixes subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list" shows "\<forall>i<length subsets. row (alt_matrix_A_R signs subsets) i = vec (length signs) (\<lambda>j. z_R (subsets ! i) (signs ! j))"
one_fresh_preserv_automatic: fixes t :: "lam"   and s :: "lam"   and a :: "name" assumes " t \<longrightarrow>\<^sub>1 s"   and "a \<sharp> t" shows "a \<sharp> s"
wf_nFAND: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and LESS :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> bool"   and assigns :: "nat \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'd"   and nvars :: "'c \<Rightarrow> 'b"   and Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c"   and CONS :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and SNOC :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and Length :: "'c \<Rightarrow> nat"   and extend :: "'a \<Rightarrow> bool \<Rightarrow> 'e \<Rightarrow> 'e"   and size :: "'e \<Rightarrow> 'b"   and zero :: "'b \<Rightarrow> 'e"   and alphabet :: "'b \<Rightarrow> 'e list"   and eval :: "'d \<Rightarrow> nat \<Rightarrow> bool"   and downshift :: "'d \<Rightarrow> 'd"   and upshift :: "'d \<Rightarrow> 'd"   and add :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and cut :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and len :: "'d \<Rightarrow> nat"   and restrict :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and Restrict :: "'a \<Rightarrow> nat \<Rightarrow> ('f, 'a) aformula"   and lformula0 :: "'f \<Rightarrow> bool"   and FV0 :: "'a \<Rightarrow> 'f \<Rightarrow> nat set"   and find0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> bool"   and wf0 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and decr0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> 'f"   and satisfies0 :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and nullable0 :: "'f \<Rightarrow> bool"   and lderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and rderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and idx :: "'b"   and \<phi>s :: "('f, 'a) aformula list" assumes "Formula SUC LESS assigns nvars Extend CONS SNOC Length extend size zero alphabet eval downshift upshift add cut len restrict Restrict lformula0 FV0 find0 wf0 decr0 satisfies0 nullable0 lderiv0 rderiv0" shows "Formula_Operations.wf SUC wf0 idx (nFAND \<phi>s) = (\<forall>\<phi>\<in>set \<phi>s. Formula_Operations.wf SUC wf0 idx \<phi>)"
l_delete_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<equiv> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
group_ringD: shows "group_ring = {x. supp (aezfun x) \<subseteq> G}"
assms_2: shows "\<pi> 0 \<noteq> local.return"
simps_1: fixes f :: "'e \<Rightarrow> ('c, 'd) state"   and x :: "'e" shows "traverse f (Tree_Relabelling.tree.Leaf x) = pure Tree_Relabelling.tree.Leaf \<diamondop> f x"
card_insert_disjoint': fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) - Suc 0 = card A"
arith1: fixes n :: "nat"   and t :: "nat"   and q :: "nat" assumes "Suc n * t = Suc n * q" shows "t = q"
rel_interior_subset_closure: fixes S :: "'b set" shows "rel_interior S \<subseteq> closure S"
mtx_invertibleD_1: fixes A :: "'a sq_mtx" assumes "mtx_invertible A" shows "A\<^sup>-\<^sup>1 * A = 1"
star_inductl_one: fixes x :: "'a"   and y :: "'a" assumes "(1::'a) + x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<le> y"
reindex_cong: fixes l :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd"   and h :: "'b \<Rightarrow> 'd" assumes "bij l"   and "g \<circ> l = h" shows "Sum_any g = Sum_any h"
top_greatest: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and top :: "'f"   and a :: "'f" assumes "class.order_top less_eq less top" shows "less_eq a top"
dg_prod_is_arrE: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "pdigraph_base \<alpha> I \<AA>"   and "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" obtains "vsv f" and "\<D>\<^sub>\<circ> f = I" and "vsv a" and "\<D>\<^sub>\<circ> a = I" and "vsv b" and "\<D>\<^sub>\<circ> b = I" and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>"
ideD_2: fixes a :: "('a, 'a set) Maps.arr" assumes "Maps.ide a" shows "Maps.dom a = a"
ide_cod: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.ide (Left_a.cod f)"
set_compre_subset: fixes B :: "'a set"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "B \<subseteq> A" shows "{x \<in> B. P x} \<subseteq> {x \<in> A. P x}"
While_transC_invert: fixes tst :: "'test"   and c1 :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(while tst do c1, s) \<rightarrow>c (c', s')" obtains "tval tst s" and "c' = c1 ;; while tst do c1" and "s' = s"
jumpF_eq: shows "jumpF_pathstart (part_circlepath z r st tt) z0 = jumpF (\<lambda>i. f i / g i) (at_right st)"
less_multiset_ext\<^sub>D\<^sub>M_iff_mult: fixes N :: "'b multiset"   and A :: "'b set"   and M :: "'b multiset"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "set_mset N \<subseteq> A"   and "set_mset M \<subseteq> A"   and "\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. R x y \<longrightarrow> R y z \<longrightarrow> R x z" shows "less_multiset_ext\<^sub>D\<^sub>M R M N = ((M, N) \<in> mult {(x, y). x \<in> A \<and> y \<in> A \<and> R x y})"
up_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" shows "up M \<equiv> \<lambda>i j. if 0 < i then if j = 0 then \<infinity> else min (dbm_add (M i 0) (M 0 j)) (M i j) else M i j"
it_to_it_def: fixes it :: "('e list \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'e list \<Rightarrow> 'e list) \<Rightarrow> 'e list \<Rightarrow> 'e list" shows "it_to_it it \<equiv> foldli (it (\<lambda>_. True) (\<lambda>x l. l @ [x]) [])"
frontier_of_Int_subset: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "X frontier_of (S \<inter> T) \<subseteq> X frontier_of S \<union> X frontier_of T"
path_from_given_point_2: fixes x :: "'a"   and p :: "'a" assumes "path x"   and "point p" shows "start_points (x\<^sup>T\<^sup>\<star> ; p \<cdot> x) \<le> p"
set_mset_replicate_mset_subset: fixes n :: "nat"   and x :: "'k" shows "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
smooth_onD: fixes m :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "m-smooth_on S f"   and "enat n \<le> m" shows "higher_differentiable_on S f n"
valid_append_atomics_reusables: fixes w :: "nat \<times> nat"   and as :: "'b list"   and t :: "'b tree" assumes "0 < fst w"   and "valid as t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "snd w \<le> length as" shows "\<forall>t\<in>set (atomics as (max (fst w) (Suc (r t))) (snd w) @ reusables t w). valid as t"
bounded_antilinear_eq_on: fixes A :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c"   and G :: "'b set"   and t :: "'b" assumes "bounded_antilinear A"   and "bounded_antilinear B"   and "\<And>x. x \<in> G \<Longrightarrow> A x = B x"   and "t \<in> closure (cspan G)" shows "A t = B t"
affine_hull_insert_span_gen: fixes a :: "'c"   and s :: "'c set" shows "affine hull insert a s = (+) a ` span ((+) (- a) ` s)"
image_comp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and r :: "'d set" shows "f ` g ` r = (f \<circ> g) ` r"
q11_upto_def: shows "q11 = (\<lambda>k. k\<^sup>2 mod 11) ` {..<11}"
dim_vec_vCons: fixes a :: "'a"   and v :: "'a vec" shows "dim_vec (vCons a v) = Suc (dim_vec v)"
up_filter: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.order less_eq less" shows "ord.filter less_eq {y. less_eq x y}"
product_in_factors: fixes x :: "i"   and y :: "i" assumes "O x y" shows "P (x \<otimes> y) x \<and> P (x \<otimes> y) y"
insert_somewhere: fixes ref :: "'a option"   and r :: "'a"   and list :: "'a list"   and oid :: "'a" assumes "ref = None \<or> ref = Some r \<and> r \<in> set list" shows "\<exists>xs ys. list = xs @ ys \<and> insert_spec list (oid, ref) = xs @ oid # ys"
real_polynomial_function_eq: fixes p :: "'a \<Rightarrow> real" shows "real_polynomial_function p = polynomial_function p"
Postdomination_axioms: shows "Postdomination sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main (_Exit_)"
simps_2: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and X :: "'b ltln set" shows "(\<psi>\<^sub>1 and\<^sub>n \<psi>\<^sub>2)[X]\<^sub>\<nu> = \<psi>\<^sub>1[X]\<^sub>\<nu> and\<^sub>n \<psi>\<^sub>2[X]\<^sub>\<nu>"
extended_distance_symmetric: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c Bonk_Schramm_extension_unfolded"   and y :: "'c Bonk_Schramm_extension_unfolded" assumes "class.metric_space dist uniformity open" shows "metric_space.extended_distance dist x y = metric_space.extended_distance dist y x"
hyps: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "length u = n"   and "length v = n"   and "length w = n" shows "foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip (map (\<lambda>xy. fst xy + snd xy) (zip u v)) w) (0::'a) = foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u w) (0::'a) + foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip v w) (0::'a)"
closed_open: fixes S :: "'c set" shows "closed S = open (- S)"
x0_hb_bound7': fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and x :: "real"   and i :: "nat" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "x\<^sub>1 \<le> x"   and "i < k" shows "1 < bs ! i * x + (hs ! i) x"
qGood_iff_good_asTerm: fixes qX :: "('c, 'd, 'varSort, 'var, 'e) qTerm" shows "good (asTerm qX) = qGood qX"
simps: fixes i :: "'a"   and j :: "nat" shows "is_valid_event_id (i, j) = (j < length (events i))"
line_suitable_junctors_2: fixes \<A> :: "'b set"   and F :: "'b formula"   and G :: "'b formula" shows "line_suitable \<A> (F \<^bold>\<and> G) = (line_suitable \<A> F \<and> line_suitable \<A> G)"
bal_list_tm_def: fixes n :: "nat"   and xs :: "'b list" shows "bal_list_tm n xs = bal_tm n xs \<bind> (\<lambda>(t, uu_). return t)"
of_rat_mult: fixes a :: "rat"   and b :: "rat" shows "of_rat (a \<cdot> b) = of_rat a \<cdot> of_rat b"
prod_mOp_mem: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and X :: "'d \<Rightarrow> 'e" assumes "\<forall>j\<in>I. aGroup (A j)"   and "X \<in> carr_prodag I A" shows "prod_mOp I A X \<in> carr_prodag I A"
rep_eq: fixes x :: "'k word" shows "the_nat x = nat (take_bit LENGTH('k) (Word.rep x))"
poly2_def: fixes p :: "'a poly poly"   and x :: "'a"   and y :: "'a" shows "poly2 p x y = poly (poly p [:y:]) x"
ro: shows "read_only_unowned (snd (snd (c n))) (fst (c n))"
listsum_Nil: fixes ds :: "nat list" shows "listsum ds [] = tensor0 ds"
A_Exists_1: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<A>(\<^bold>\<exists>!x. \<phi> x) \<^bold>\<equiv> (\<^bold>\<exists>!x. \<^bold>\<A>\<phi> x) in v]"
comm_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "a \<cdot> b = b \<cdot> a"
set_times_mono4: fixes a :: "'b"   and C :: "'b set"   and D :: "'b set" assumes "a \<in> C" shows "a *o D \<subseteq> D * C"
all_finite_subset_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and P :: "'c set \<Rightarrow> bool" shows "(\<forall>B. finite B \<and> B \<subseteq> f ` A \<longrightarrow> P B) = (\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B))"
flow_flow_of_current: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "('a \<times> 'a) \<times> 'a \<times> 'a \<Rightarrow> ennreal" assumes "web_flow (web_of_network \<Delta>) f" shows "flow \<Delta> (flow_of_current \<Delta> f)"
distinct_1: fixes x1 :: "'a" shows "Atom x1 \<noteq> \<bottom>"
Sup_bool_def: fixes A :: "bool set" shows "\<Squnion> A = (True \<in> A)"
L: fixes a :: "'a"   and b :: "'b" shows "norm (a ** b) \<le> norm a * norm b * L"
simps_48: fixes x61 :: "('a, 'b, 'c) expr"   and x62 :: "('a, 'b, 'c) expr"   and x4 :: "('a, 'b, 'c) expr" shows "Assign x61 x62 \<noteq> Ref x4"
psubst_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "X #[rho] \<equiv> asTerm pick X #[[pickE rho]]"
acyclic_subcls1: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "wf_prog wf_md P" shows "acyclic (subcls1 P)"
map_converter_id: fixes conv :: "('h, 'i, 'j, 'k) converter" shows "map_converter id id id id conv = conv"
extract_real_def: fixes val :: "val" shows "extract_real val = (case val of RealVal x4 \<Rightarrow> x4 | _ \<Rightarrow> 0)"
member_def: shows "DList_Set.member \<equiv> map_fun list_of_dlist id (equal_base.list_member ceq')"
mset_list_of_mset: fixes m :: "'b multiset" shows "mset (list_of_mset m) = m"
l3_inv2D: fixes x :: "l3_state" assumes "x \<in> l3_inv2" shows "Keys_bad (ik x) (l3_state.bad x)"
valid_path_def: fixes f :: "real \<Rightarrow> 'a" shows "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1}"
isos_compose: fixes f :: "'c"   and f' :: "'c" assumes "C.iso f"   and "C.iso f'"   and "C.seq f' f" shows "C.iso (f' \<cdot>\<^sub>C f)"
shiftr_over_and_dist: fixes a :: "'a word"   and b :: "'a word"   and c :: "nat" shows "a && b >> c = (a >> c) && (b >> c)"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "r2.redT a1 a2 a3" obtains t :: "'t" and x :: "'x2" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" and x' :: "'x2" and m' :: "'m2" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "a1 = s__" and "a2 = (ta__, ta)" and "a3 = s'__" and "r2.r_syntax ta__ x__ (shr s__) ta x' m'" and "thr s__ ta__ = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "r2.actions_ok s__ ta__ ta" and "redT_upd s__ ta__ ta x' m' s'__"   | s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x2" and n :: "'l" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<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)"
pCons_induct2: fixes P :: "'a poly \<Rightarrow> 'b poly \<Rightarrow> bool"   and p :: "'a poly"   and q :: "'b poly" assumes "P 0 0"   and "\<And>a p. a \<noteq> (0::'a) \<or> p \<noteq> 0 \<Longrightarrow> P (pCons a p) 0"   and "\<And>b q. b \<noteq> (0::'b) \<or> q \<noteq> 0 \<Longrightarrow> P 0 (pCons b q)"   and "\<And>a p b q. \<lbrakk>a \<noteq> (0::'a) \<or> p \<noteq> 0; b \<noteq> (0::'b) \<or> q \<noteq> 0; P p q\<rbrakk> \<Longrightarrow> P (pCons a p) (pCons b q)" shows "P p q"
converse_Id_on: fixes A :: "'b set" shows "(Id_on A)\<inverse> = Id_on A"
at_prm_eq_trans: fixes pi1 :: "('a \<times> 'a) list"   and pi2 :: "('a \<times> 'a) list"   and pi3 :: "('a \<times> 'a) list" assumes " pi1 \<triangleq> pi2 "   and " pi2 \<triangleq> pi3 " shows " pi1 \<triangleq> pi3 "
Compositionality_If: fixes d :: "'d"   and b :: "'exp"   and b' :: "'exp"   and c1 :: "('exp, 'id) MWLfCom"   and c2 :: "('exp, 'id) MWLfCom"   and c1' :: "('exp, 'id) MWLfCom"   and c2' :: "('exp, 'id) MWLfCom" assumes "b \<equiv>\<^bsub>d\<^esub> b' \<or> [c1] \<approx>\<^bsub>d\<^esub> [c2] \<or> [c1'] \<approx>\<^bsub>d\<^esub> [c2']"   and "[c1] \<approx>\<^bsub>d\<^esub> [c1']"   and "[c2] \<approx>\<^bsub>d\<^esub> [c2']" shows "[if b then c1 else c2 fi] \<approx>\<^bsub>d\<^esub> [if b' then c1' else c2' fi]"
VDMAssign: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var"   and e :: "Expr" shows "G \<rhd> Assign x e : \<lambda>s t. t = (update (fst s) x (evalE e (fst s)), snd s)"
thing1: shows "0 \<le> 9223372034707292161 + - (2 ^ 31)"
bot_entails_all: fixes B :: "'a literal multiset"   and N1 :: "'a literal multiset set" assumes "B \<in> {{#}}" shows "{B} \<TTurnstile>\<G>e N1"
distinct_1: fixes x1 :: "fdecl"   and x2 :: "mdecl" shows "fdecl x1 \<noteq> mdecl x2"
assms_1: fixes U :: "'a set" assumes "U \<in> S" shows "openin T1 (f -` U \<inter> topspace T1)"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "B.cod (\<tau> f) = G (A.cod f)"
multi_union_self_other_eq: fixes A :: "'c multiset"   and X :: "'c multiset"   and Y :: "'c multiset" assumes "A + X = A + Y" shows "X = Y"
delta_sol: shows "PE.delta = PE_lf (dual PE.delta, PE.delta)"
cnj_inverse: fixes x :: "octo" shows "Octonions.cnj (inverse x) = inverse (Octonions.cnj x)"
set_break_down_right: fixes M :: "'a multiset"   and N :: "'a set multiset" shows "{#x \<in># M \<times># N. fst x \<in> snd x#} = (\<Sum>n\<in>#N. {#m \<in># M. m \<in> n#} \<times># {#n#})"
less_def: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubset> y) = (x \<sqsubseteq> y \<and> x \<noteq> y)"
sigma_sets_subset': fixes a :: "'c set set"   and M :: "'c measure"   and \<Omega>' :: "'c set" assumes "a \<subseteq> sets M"   and "\<Omega>' \<in> sets M" shows "sigma_sets \<Omega>' a \<subseteq> sets M"
g_arith_nonneg: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "a \<le> 824 / 10\<^sup>2"   and "0 \<le> b"   and "b \<le> 751 / 10\<^sup>2" shows "0 \<le> g_arith a b"
valid_put_unis: fixes p :: "('b, 'c) form"   and e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and m :: "nat" assumes "\<forall>e f g. eval e f g p" shows "eval e f g (put_unis m p)"
learn_cp_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" shows "learn_cp \<psi> U s \<equiv> learn_lim \<psi> U s \<and> (\<forall>f\<in>U. \<forall>n. \<psi> (the (s (f \<triangleright> n))) \<in> U)"
valid_union_conc: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and c :: "'f list multiset"   and c' :: "'f list multiset" shows "valid fg (c + c') = (valid fg c \<and> valid fg c' \<and> mon_c fg c \<inter> mon_c fg c' = {})"
simps_2: fixes A :: "nat set"   and k :: "nat" assumes "esize A \<le> enat k" shows "nth_least_ext A k = Suc (Max A + (k - card A))"
preserves_comp_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f' :: "'d"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A (A f' f)" shows "\<tau> (A f' f) = B (\<tau> f') (F f)"
scaleR1: fixes e :: "real"   and xr :: "'a"   and P :: "'a" assumes "0 < e"   and "ccw (0::'a) xr P" shows "ccw (0::'a) (e *\<^sub>R xr) P"