hcju/isabelleps huggingface dataset:
SMP_D_aux1: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "t \<in> SMP (set M)"   and "has_all_wt_instances_of \<Gamma> (subterms\<^sub>s\<^sub>e\<^sub>t (set M)) (set M)"   and "is_TComp_var_instance_closed \<Gamma> M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)" shows "\<forall>x\<in>fv t. \<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var y) = \<Gamma> (Var x)"
simps_2: fixes p :: "'b"   and i :: "'b memref"   and "is" :: "'b memref list" shows "last_prog p (i # is) = (case i of Prog\<^sub>s\<^sub>b x p' xa \<Rightarrow> last_prog p' is | _ \<Rightarrow> last_prog p is)"
Switch1: fixes p :: "'b"   and c :: "'c"   and av :: "'a" assumes "Abort p c av s s' \<and> Init p c av t t'"   and "a = Switch id1 p c av" shows "thesis"
arr_iff_in_hom: fixes f :: "(bool, bool list) concrete_category.arr" shows "J.arr f = \<guillemotleft>f : J.dom f \<rightarrow>\<^sub>J J.cod f\<guillemotright>"
i: shows "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
step_Abrupt: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and x :: "'b" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Abrupt x" shows "s' = Abrupt x"
qfree_plus_inf: fixes \<phi> :: "atom fm" assumes "qfree \<phi>" shows "qfree (inf\<^sub>+ \<phi>)"
chine_composites_3: shows "Chn (\<lparr>Chn = \<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi>)\<rparr> \<star> \<rho>) = chine_hcomp \<lparr>Chn = \<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi>)\<rparr> \<rho>"
failK_def: shows "failK \<equiv> \<Lambda> s f. f"
fst_msum_aform: fixes n :: "nat"   and f :: "'b \<times> 'b pdevs"   and g :: "'b \<times> 'b pdevs" shows "fst (msum_aform n f g) = fst f + fst g"
fempty_fminus: fixes A :: "'a fset" shows "{||} |-| A = {||}"
diff_fun_submanifold: fixes S :: "'a set" assumes "open S" shows "diff_fun k (charts_submanifold S) \<phi>"
subst_subst: fixes x :: "name"   and y :: "name"   and v :: "lam"   and t :: "lam"   and u :: "lam" assumes "x \<noteq> y"   and "x \<sharp> v" shows "t[y::=v][x::=u[y::=v]] = t[x::=u][y::=v]"
DEFL_eq_stateT: shows "DEFL('a\<cdot>('b, 'c) stateT) = DEFL('c \<rightarrow> ('a \<times> 'c)\<cdot>'b)"
hom_prod: fixes f :: "'b \<Rightarrow> int"   and X :: "'b set" shows "of_int (prod f X) = (\<Prod>x\<in>X. of_int (f x))"
lift_post_var: fixes x :: "'b \<Longrightarrow> 'd" shows "\<lceil>x\<rceil>\<^sub>> = $x\<acute>"
set_of_partition_on_map: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "(`) ((`) f) ` {P. partition_on A P} = {P. partition_on (f ` A) P}"
locally_sound_mode_use_def: fixes lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "locally_sound_mode_use lc = (\<forall>c' mds' mem'. \<langle>c', mds', mem'\<rangle> \<in> loc_reach lc \<longrightarrow> (\<forall>x. (x \<in> mds' GuarNoReadOrWrite \<longrightarrow> doesnt_read_or_modify c' x) \<and> (x \<in> mds' GuarNoWrite \<longrightarrow> doesnt_modify c' x)))"
assms_2: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "0 \<le> f x"
inject_1: fixes x1 :: "'e"   and y1 :: "'e" shows "(Var x1 = Var y1) = (x1 = y1)"
principal_ideal_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "class.ring plus zero minus uminus times" shows "ring.principal_ideal plus zero uminus times S = (\<exists>a. ring.ideal_generated plus zero uminus times {a} = S)"
leg_simps_1: shows "C.arr \<mu>\<nu>.leg0"
sub_distribD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and s :: "'a" assumes "sub_distrib t"   and "sound P" shows "t P s \<ominus> 1 \<le> t (\<lambda>s'. P s' \<ominus> 1) s"
Standard_starfun2: fixes x :: "'a star"   and y :: "'b star"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "x \<in> Standard"   and "y \<in> Standard" shows "(*f2* f) x y \<in> Standard"
y'_def: shows "y' = filter (\<lambda>z. z \<in> set x) y"
before_in_def: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "x < y in xs = (index xs x < index xs y \<and> y \<in> set xs)"
vr: shows "valid_tmap (binsert fst (i, a) t)"
Suc: shows "is_quantum_predicate (wlp_while_n (M 0) (M 1) WS k P)"
map_nth: fixes xs :: "'a list" shows "map ((!) xs) [0..<List.length xs] = xs"
snd_uminus: fixes x :: "'b \<times> 'a" shows "snd (- x) = - snd x"
is_Inf_lower: fixes A :: "'b set"   and inf :: "'b"   and x :: "'b" assumes "is_Inf A inf"   and "x \<in> A" shows "inf \<sqsubseteq> x"
arbitrary_union_of_Union: fixes \<U> :: "'b set set"   and P :: "'b set \<Rightarrow> bool" assumes "\<And>S. S \<in> \<U> \<Longrightarrow> (arbitrary union_of P) S" shows "(arbitrary union_of P) (\<Union> \<U>)"
iH: fixes i :: "nat" shows "map_pmf (flip i) (bv n) = bv n"
subspace_substandard: fixes P :: "'a \<Rightarrow> bool" shows "subspace {x. \<forall>i\<in>Basis. P i \<longrightarrow> x \<bullet> i = 0}"
tuple_prj_spc: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "\<langle>\<p>\<^sub>1[f, g] \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g]\<rangle> = f \<down>\<down> g"
csrp_pj: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e set" assumes "Ring R"   and "ideal R I"   and "x \<in> carrier (R /\<^sub>r I)" shows "pj R I (csrp_fn R I x) = x"
Invariant: fixes s :: "'a" assumes "P s"   and "b s" shows "P (c s)"
lex_translate_origin: fixes a :: "real \<times> real"   and b :: "real \<times> real" assumes "NO_MATCH (0::'a) a" shows "Counterclockwise_2D_Arbitrary.lex a b = Counterclockwise_2D_Arbitrary.lex 0 (b - a)"
border_rq_nemp: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "w\<^sup><\<inverse>x  \<noteq> \<epsilon>"
simps_24: fixes x51 :: "nat"   and x52 :: "nat"   and x21 :: "nat"   and x22 :: "nat" shows "disconnect(x51, x52) \<noteq> x21:deliver(x22)"
EndPhase2_HInv3_t: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase2 s s' t"   and "HInv3_inner s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_inner s' p q d"
isfields_def: fixes P :: "'a prog"   and C :: "char list" shows "isfields P C \<equiv> filter (\<lambda>((F, D), b, T). b = Static \<and> D = C) (fields P C)"
exp_similiar_sq_mtx_diag_eq: fixes P :: "'a sq_mtx"   and A :: "'a sq_mtx"   and f :: "'a \<Rightarrow> real" assumes "mtx_invertible P"   and "A = P\<^sup>-\<^sup>1 * (\<d>\<i>\<a>\<g> i. f i) * P" shows "exp A = P\<^sup>-\<^sup>1 * exp (\<d>\<i>\<a>\<g> i. f i) * P"
subspace_on_def: fixes T :: "'a set" shows "subspace.subspace T = ((0::'a) \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. x + y \<in> T) \<and> (\<forall>c. \<forall>x\<in>T. c *\<^sub>R x \<in> T))"
lt_of_key_order_def: shows "lt_of_key_order \<equiv> map_fun key_compare id lt_of_comp"
smult_0_left: fixes p :: "'b poly" shows "smult (0::'b) p = 0"
mult_isol_var: fixes u :: "'a"   and x :: "'a"   and v :: "'a"   and y :: "'a" assumes "u \<le> x"   and "v \<le> y" shows "\<lparr>u\<rparr>v\<lparr>x \<cdot> y\<rparr>"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c pmf"   and b :: "'d pmf" shows "rel_pmf R\<inverse>\<inverse> a b = rel_pmf R b a"
idem: fixes d :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "deflation d" shows "d\<cdot>(d\<cdot>x) = d\<cdot>x"
setnMT: fixes a :: "'a list"   and b :: "'a list" assumes "set a = set b"   and "a \<noteq> []" shows "b \<noteq> []"
f1_1: shows "f\<^sub>1 \<in> \<R>\<^sup>2"
fac_acc_body1_fac_body_final_eq: shows "fac_acc_body1 = unwrapA oo fac_body_final oo wrapA"
minSpanningForest_def: fixes F :: "nat uprod set" shows "minSpanningForest F \<equiv> spanningForest F \<and> (\<forall>F'. spanningForest F' \<longrightarrow> sum w F \<le> sum w F')"
case_eq_if: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and option :: "'g option" shows "(case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
ladder_n_of_cut: fixes L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat"   and n :: "nat" assumes "L \<noteq> []"   and "index < length L - 1" shows "ladder_n (ladder_cut L n) index = ladder_n L index"
LTL_FG_almost_all_suffixes: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl" shows "w \<Turnstile> F G \<phi> = (\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile> \<phi>)"
leq_def: shows "leq = restrict_rel F {(g, g'). g' \<le> g}"
fun_belowI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. f x \<sqsubseteq> g x" shows "f \<sqsubseteq> g"
projection_idempotent: fixes l :: "'a list"   and X :: "'a set" shows "l \<upharpoonleft> X = l \<upharpoonleft> X \<upharpoonleft> X"
prod_mset_Un: fixes A :: "'a multiset"   and B :: "'a multiset" shows "prod_mset (A + B) = prod_mset A * prod_mset B"
fmdom_add: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" shows "fmdom (m ++\<^sub>f n) = fmdom m |\<union>| fmdom n"
snd_mismatch_def: fixes u\<^sub>0 :: "'b list"   and u\<^sub>1 :: "'b list" assumes "binary_code u\<^sub>0 u\<^sub>1" shows "binary_code.c\<^sub>1 u\<^sub>0 u\<^sub>1 = (u\<^sub>1 \<cdot> u\<^sub>0) ! \<^bold>|binary_code.\<alpha> u\<^sub>0 u\<^sub>1\<^bold>|"
M: fixes m :: "int" assumes "f $$ m \<noteq> (0::'a)" shows "M \<le> m"
map\<^sub>1_simps_2: fixes f :: "'a" assumes "C.ide f" shows "src\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]) = F.map\<^sub>0 (src\<^sub>C f)"
Diff_UNIV: fixes A :: "'b set" shows "A - UNIV = {}"
open_Uf: fixes t :: "'a" assumes "t \<in> S - U" shows "open (Uf t)"
assign_t_simp: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> Suc 0 \<Down> s' = (s' = s(x := aval a s))"
R2_def: shows "R2 = {(w1, w2). \<exists>\<iota> \<iota>' b b' c1 c1'. w1 = [while\<^bsub>\<iota>\<^esub> b do c1 od] \<and> w2 = [while\<^bsub>\<iota>'\<^esub> b' do c1' od] \<and> \<iota> \<notin> set (PPc c) \<and> \<iota>' \<notin> set (PPc c) \<and> ([c1], [c1']) \<in> R \<and> b \<equiv>\<^bsub>d\<^esub> b'}"
commute: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<union># b = b \<union># a"
fork_1: shows "s' = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r')]), r' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"
cart_eq_inner_axis: fixes a :: "(real, 'a) vec"   and i :: "'a" shows "a $ i = a \<bullet> axis i 1"
disj_matrices_add_x: fixes A :: "'a matrix"   and B :: "'a matrix"   and C :: "'a matrix" assumes "disj_matrices A B"   and "disj_matrices A C" shows "disj_matrices (B + C) A"
ngba_g_E_rtrancl: fixes A :: "('d, 'c) ngba" shows "(g_E (ngba_g A))\<^sup>* = {(p, q). q \<in> reachable A p}"
merge_states_aux_def: fixes s1 :: "nat"   and s2 :: "nat"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "merge_states_aux s1 s2 e = (\<lambda>(uid, (origin, dest), t). (uid, (if origin = s1 then s2 else origin, if dest = s1 then s2 else dest), t)) |`| e"
dsafe_Diff: fixes \<theta> :: "('a, 'b) trm" assumes "dfree \<theta>" shows "dsafe (Differential \<theta>)"
gt_state_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and s' :: "('a, 'b) state" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update" shows "PivotUpdateMinVars.gt_state eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update s s' \<equiv> \<exists>x\<^sub>i x\<^sub>j l\<^sub>i. let dir = if \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i then Positive else Negative in min_lvar_not_in_bounds s = Some x\<^sub>i \<and> l\<^sub>i = the (Direction.LB dir s x\<^sub>i) \<and> min_rvar_incdec_eq dir s (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i) = Inr x\<^sub>j \<and> s' = pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s"
mult_less_cancel_right: 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 abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" 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))"
extend_def: shows "extend \<equiv> \<lambda>K. cbox (\<Sum>j\<in>Basis. if j = i then (a \<bullet> i) *\<^sub>R i else (interval_lowerbound K \<bullet> j) *\<^sub>R j) (\<Sum>j\<in>Basis. if j = i then (b \<bullet> i) *\<^sub>R i else (interval_upperbound K \<bullet> j) *\<^sub>R j)"
assms_1: shows "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"
HPhase1or2ReadElse_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadElse s s' p d q"   and "HInv6 s"   and "Inv2c s" shows "HInv6 s'"
a: shows "R[a\<turnstile>c>b] = ImpR x.<c>.M e"
multi_member_split: fixes x :: "'d"   and M :: "'d multiset" assumes "x \<in># M" shows "\<exists>A. M = add_mset x A"
inf_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<sqinter> z \<le> y \<sqinter> z"
comm_def: shows "comm \<equiv> \<lambda>(guar, post). {c. (\<forall>i. Suc i < length c \<longrightarrow> c ! i -c\<rightarrow> c ! Suc i \<longrightarrow> (snd (c ! i), snd (c ! Suc i)) \<in> guar) \<and> (fst (last c) = None \<longrightarrow> snd (last c) \<in> post)}"
ys: shows "filter (\<lambda>j. j \<notin> set xs) [0..<dim_col A'] = ys"
c3: fixes ps :: "'a list"   and x :: "'a"   and xs' :: "'a list"   and y :: "'a"   and ys' :: "'a list" assumes "xs = ps @ x # xs'"   and "ys = ps @ y # ys'"   and "x \<noteq> y" shows "thesis"
smalls_big_append: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs\<^sub>1 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and cs\<^sub>2 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and s\<^sub>3 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "small_steps \<pi> cs\<^sub>1 (Some cs\<^sub>2)"   and "\<pi>: cs\<^sub>2 \<Rightarrow> s\<^sub>3" shows "\<pi>: cs\<^sub>1 \<Rightarrow> s\<^sub>3"
filter_replicate: fixes P :: "'c \<Rightarrow> bool"   and n :: "nat"   and x :: "'c" shows "filter P (replicate n x) = (if P x then replicate n x else [])"
inj_Cons: fixes A :: "('b list \<times> 'b) set" shows "inj_on (\<lambda>(xs, n). n # xs) A"
ideD_1: fixes a :: "'a" assumes "Left.ide a" shows "Left.arr a"
r_neg2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> (\<ominus> x \<oplus> y) = y"
d_Bisimulation_Up_To_USdB_def: fixes d :: "'d"   and R :: "(('exp, 'id) MWLfCom list \<times> ('exp, 'id) MWLfCom list) set" shows "d_Bisimulation_Up_To_USdB d R \<equiv> sym R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' W m2. SRabbr (V ! i) m1 W m2 \<and> m1 =\<^bsub>d\<^esub> m1' \<longrightarrow> (\<exists>W' m2'. SRabbr (V' ! i) m1' W' m2' \<and> (W, W') \<in> R \<union> \<approx>\<^bsub>d\<^esub> \<and> m2 =\<^bsub>d\<^esub> m2'))"
l_delete_shadow_root_get_disconnected_nodes\<^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_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_disconnected_nodes\<^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_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes\<^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_disconnected_nodes get_disconnected_nodes_locs"
execute_bind_case: fixes f :: "'b Heap"   and g :: "'b \<Rightarrow> 'a Heap"   and h :: "heap" shows "execute (f \<bind> g) h = (case execute f h of None \<Rightarrow> None | Some (x, h') \<Rightarrow> execute (g x) h')"
sPP: shows "subst (PP (encF N (Var xx))) \<langle>PP (encF N t)\<rangle> xx = PP (encF N \<langle>PP (encF N t)\<rangle>)"
leg_simps_3: shows "C.arr doms.\<pi>.leg1"
conv_abscissa_deriv_le: fixes f :: "'a fds" shows "conv_abscissa (fds_deriv f) \<le> conv_abscissa f"
before_in_mtf: fixes z :: "'a"   and xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "z \<in> set xs" shows "x < y in mtf z xs = (y \<noteq> z \<and> (if x = z then y \<in> set xs else x < y in xs))"
spr_sim_simps: fixes t :: "'s Trace" shows "fst (spr_sim t) = spr_abs t"
lookup_tensor0: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" shows "lookup (tensor0 ds) is = (0::'a)"
Lxy_project: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" assumes "x \<noteq> y"   and "x \<in> set xs"   and "y \<in> set xs"   and "distinct xs"   and "x < y in xs" shows "Lxy xs {x, y} = [x, y]"
assms_1: shows "s \<subseteq> \<Union> (set ts)"
Pf_mono: fixes N :: "nat"   and n :: "nat"   and t :: "'s" assumes "N \<le> n" shows "T.prob t {\<omega> \<in> space (T t). enat n < sfirst (HLD H) (t ## \<omega>)} \<le> T.prob t {\<omega> \<in> space (T t). enat N < sfirst (HLD H) (t ## \<omega>)}"
positive_psi_l: fixes k :: "nat" assumes "k < K" shows "positive (proj_psi_l k)"
round_robin_Some_NoneD: fixes n0 :: "nat"   and \<sigma> :: "'t list \<times> nat"   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"   and t :: "'t"   and \<sigma>' :: "'t list \<times> nat" assumes "\<alpha>.round_robin n0 \<sigma> s = \<lfloor>(t, None, \<sigma>')\<rfloor>" shows "\<exists>x ln n. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> 0 < ln $ n \<and> \<not> waiting (wset s t) \<and> may_acquire_all (locks s) t ln"
SCp_swap_applies_1: fixes G :: "'b formula"   and H :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "G, H, \<Gamma> \<Rightarrow> \<Delta>" shows "H, G, \<Gamma> \<Rightarrow> \<Delta>"
collapse: fixes fundef :: "'a fundef" shows "Fundef (body fundef) (arity fundef) = fundef"
ennreal_max_0': fixes x :: "real" shows "ennreal (max x 0) = ennreal x"
Field_card_of: fixes A :: "'d set" shows "Field |A| = A"
Range_converse: fixes r :: "('c \<times> 'd) set" shows "Range (r\<inverse>) = Domain r"
minus_equation_iff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (uminus b = a)"
sorted_drop: fixes xs :: "'a list"   and n :: "nat" assumes "sorted xs" shows "sorted (drop n xs)"
Compl_anti_mono: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "- B \<subseteq> - A"
dg_SemiCAT_components_3: fixes \<alpha> :: "V" shows "dg_SemiCAT \<alpha>\<lparr>Dom\<rparr> = (\<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomDom\<rparr>)"
stable_transient: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> stable A"   and "F \<in> transient A" shows "\<exists>act\<in>Acts F. A \<subseteq> - Domain act"
locally_sound_mode_use_def: fixes lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "locally_sound_mode_use lc = (\<forall>c' mds' mem'. \<langle>c', mds', mem'\<rangle> \<in> loc_reach lc \<longrightarrow> (\<forall>x. (x \<in> mds' GuarNoRead \<longrightarrow> doesnt_read c' x) \<and> (x \<in> mds' GuarNoWrite \<longrightarrow> doesnt_modify c' x)))"
h: shows "\<Gamma> \<turnstile> t \<Leftrightarrow> u : T\<^sub>1\<rightarrow>T\<^sub>2"
VARfromCorrectSYSTEM_def: shows "VARfromCorrectSYSTEM \<equiv> \<forall>v. \<forall>Z\<in>AbstrLevel level0 \<union> AbstrLevel level1. v \<in> VAR Z \<longrightarrow> (\<forall>x\<in>VARfrom v. x \<in> IN Z)"
matrix_inv_Gauss_Jordan_PA: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "invertible A" shows "matrix_inv A = fst (Gauss_Jordan_PA A)"
absorb_iff2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># b) = (a \<union># b = b)"
is_regular_spairD3_2: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "lt (monom_mult ((1::'b) / punit.lc (rep_list p)) (lcs (punit.lt (rep_list p)) (punit.lt (rep_list q)) - punit.lt (rep_list p)) p) \<noteq> lt (monom_mult ((1::'b) / punit.lc (rep_list q)) (lcs (punit.lt (rep_list p)) (punit.lt (rep_list q)) - punit.lt (rep_list q)) q)"
finfun_Bex_def: fixes A :: "'a \<Rightarrow>f bool"   and P :: "'a \<Rightarrow> bool" shows "finfun_Bex A P = Bex {x. A $ x} P"
list_of_oalist_except_min: fixes ko :: "'o"   and xs :: "'x" shows "list_of_oalist (except_min ko xs) = (list.tl (sort_oalist_aux ko (list_of_oalist xs)), ko)"
invar_var_goal_def: fixes I :: "'b \<Rightarrow> bool"   and R :: "('c \<times> 'c) set"   and v :: "'b \<Rightarrow> 'c"   and s :: "'b"   and s' :: "'b" shows "invar_var_goal I R v s s' \<equiv> I s' \<and> (v s', v s) \<in> R"
t1_space_empty: fixes X :: "'c topology" assumes "topspace X = {}" shows "t1_space X"
fps_inverse_mult: fixes f :: "'a fps"   and g :: "'a fps" shows "inverse (f * g) = inverse f * inverse g"
cobounded2: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "less_eq_prod (inf_prod a b) b"
1: shows "filter_sup F G = \<up>x \<and> F \<inter> G = \<up>y"
C: fixes \<mu> :: "'a" assumes "\<guillemotleft>\<mu> : v \<star> g \<Rightarrow> u\<guillemotright>" shows "trnr\<^sub>\<epsilon> u (trnr\<^sub>\<eta> v \<mu>) = \<mu>"
IfaceT: fixes I :: "qtname" assumes "statT = IfaceT I"   and "dynimethd G I dynC sig = x" shows "thesis"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.arr f"   and "S.dom f = a"   and "S.cod f = b" shows "S.in_hom f a b"
filter_rev_def: shows "filter_rev \<equiv> filter_rev_aux []"
coerce_eq_ID: shows "COERCE('e, 'e) = ID"
odd: fixes p :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "odd_of_distinct p i \<noteq> 0" shows "odd i"
idx_inj_on_dom: fixes xs :: "'a list" assumes "distinct xs" shows "inj_on (idx xs) (dom (idx xs))"
primitive_pythagorean_tripleE_int: fixes z :: "int"   and x :: "int"   and y :: "int" assumes "z\<^sup>2 = x\<^sup>2 + y\<^sup>2"   and "coprime x y" obtains u :: "int" and v :: "int" where "coprime u v" and "\<not> (odd u \<and> odd v)" and "x = 2 * u * v \<and> y = u\<^sup>2 - v\<^sup>2 \<or> x = u\<^sup>2 - v\<^sup>2 \<and> y = 2 * u * v"
distinct_1: fixes x2 :: "'a" shows "Marker \<noteq> Msg x2"
nth_fls_subdegree_nonzero: fixes f :: "'b fls" assumes "f \<noteq> 0" shows "f $$ fls_subdegree f \<noteq> (0::'b)"
Sup_pred_def: fixes A :: "'a pred set" shows "\<Squnion> A = Pred (\<Squnion> (eval ` A))"