hcju/isabelleps huggingface dataset:
new_shadow_root_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "ptr \<noteq> new_shadow_root_ptr" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h h'"
e_take_delay_init: fixes f :: "nat \<Rightarrow> nat option"   and i :: "nat"   and n :: "nat" assumes "f \<in> \<R>"   and "0 < the (delay i (f \<triangleright> n))" shows "e_take (the (delay i (f \<triangleright> n))) (f \<triangleright> n) = f \<triangleright> (the (delay i (f \<triangleright> n)) - 1)"
simps_2: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and rA :: "'a \<times> 'a mat" shows "sub1 mf q (Suc k) l rA = mute mf q (l + Suc k) l (sub1 mf q k l rA)"
assms_1: shows "u \<in> \<langle>G\<rangle>"
permutations_of_set_empty_iff: fixes A :: "'a set" shows "(permutations_of_set A = {}) = infinite A"
FGModuleEnd_axioms_def: fixes V :: "'a set"   and T :: "'a \<Rightarrow> 'a" shows "FGModuleEnd_axioms V T \<equiv> T ` V \<subseteq> V"
Q4'_a_def: shows "Q4'_a \<equiv> [: \<lambda>(stk, mrk). {(stk', mrk'). stk \<noteq> [] \<and> (\<forall>y. (hd stk, y) \<in> next \<longrightarrow> y \<in> mrk) \<and> stk' = tl stk \<and> mrk' = mrk} :]"
bisim1CASThrow2: fixes P :: "'b prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e3 :: "'addr expr1"   and v :: "'addr val" assumes "P,e2,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>ad\<rfloor>)" shows "P,e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3),h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 e1) + pc, \<lfloor>ad\<rfloor>)"
one_hom_1_axioms: shows "one_hom_1 hom"
simps: fixes sl_impl :: "(int poly, int \<times> int poly list, 'a) subseqs_foldr_impl"   and m2 :: "int"   and state :: "'a"   and u :: "int poly"   and luu :: "int poly"   and lu :: "int"   and d :: "nat"   and r :: "nat"   and vs :: "int poly list"   and res :: "int poly list"   and cands :: "(int \<times> int poly list) list" shows "p.reconstruction sl_impl m2 state u luu lu d r vs res cands = (case cands of [] \<Rightarrow> let d' = Suc d in if r < d' + d' then u # res else case next_subseqs_foldr sl_impl state of (cands, state') \<Rightarrow> p.reconstruction sl_impl m2 state' u luu lu d' r vs res cands | (lv', ws) # cands' \<Rightarrow> let lv = p.inv_M2 m2 lv' in if lv dvd coeff luu 0 then let vb = p.inv_Mp2 m2 (p.Mp (Polynomial.smult lu (p.prod_list_m ws))) in if vb dvd luu then let pp_vb = primitive_part vb; u' = u div pp_vb; r' = r - length ws; res' = pp_vb # res in if r' < d + d then u' # res' else let lu' = lead_coeff u'; vs' = fold remove1 ws vs; (cands'', state') = subseqs_foldr sl_impl (lu', []) vs' d in p.reconstruction sl_impl m2 state' u' (Polynomial.smult lu' u') lu' d r' vs' res' cands'' else p.reconstruction sl_impl m2 state u luu lu d r vs res cands' else p.reconstruction sl_impl m2 state u luu lu d r vs res cands')"
dagger_denest2: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> ; (y ; x\<^sup>\<star>)\<^sup>\<star>"
stut_fun2: fixes X :: "(nat \<Rightarrow> 'b) \<Rightarrow> 'c"   and Y :: "(nat \<Rightarrow> 'b) \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "STUTINV X"   and "STUTINV Y" shows "STUTINV f<X, Y>"
image_def: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
trans_R: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "R x y"   and "R y z" shows "R x z"
rep_eq: fixes x :: "'c"   and xa :: "'c fset" shows "(x |\<in>| xa) = (x \<in> fset xa)"
ordIso_ordLess_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('d \<times> 'd) set" assumes "r =o r'"   and "r' <o r''" shows "r <o r''"
mem_rcs: fixes H :: "'a set"   and x :: "'a"   and a :: "'a" assumes "G \<guillemotright> H "   and "x \<in> H \<bullet> a" shows "\<exists>h\<in>H. h \<cdot> a = x"
cPar2F: fixes \<alpha> :: "freeRes"   and P' :: "pi"   and C :: "'a" assumes "!P \<longmapsto> \<alpha> \<prec> P'"   and "\<And>C. F C (!P) (\<alpha> \<prec> P')" shows "F C (P \<parallel> !P) (\<alpha> \<prec> P \<parallel> P')"
"fun": fixes rs :: "(name \<times> sterm) list"   and t :: "sterm"   and t' :: "sterm"   and u :: "sterm" assumes "rs \<turnstile>\<^sub>s t \<longrightarrow> t'" shows "rs \<turnstile>\<^sub>s t $\<^sub>s u \<longrightarrow> t' $\<^sub>s u"
simps_47: fixes x91 :: "stmt"   and x92 :: "qtname"   and x93 :: "vname"   and x94 :: "stmt"   and x2 :: "expr" shows "Try x91 Catch(x92 x93) x94 \<noteq> Expr x2"
Ctxt_valid_def: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" shows " \<Turnstile> G  = (\<forall>A. A \<in> G \<longrightarrow>  \<Turnstile> Call : A )"
nth_list_augment: fixes xs :: "'a list"   and k :: "nat"   and v :: "'a" shows "list_augment xs k v ! k = v"
default_rel_def: fixes d :: "'b"   and X :: "('a \<times> 'b) set" shows "\<langle>X\<rangle>default_rel d = insert (None, d) ((\<lambda>(x, y). (Some x, y)) ` X)"
inf_assoc: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<inter># y \<inter># z = x \<inter># (y \<inter># z)"
assms_2: shows "3 + k + d \<le> 2 * d + (4 + k)"
minus_in_intervalI: fixes x :: "'a"   and X :: "'a interval"   and y :: "'a"   and Y :: "'a interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x - y \<in>\<^sub>i X - Y"
inv_equality: fixes G :: "('b, 'c) monoid_scheme"   and y :: "'b"   and x :: "'b" assumes "Group.group G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> x = y"
nfa_trans_is_node: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and q :: "bool list"   and a :: "bool list" assumes "wf_nfa A n"   and "nfa_is_node A q"   and "is_alph n a" shows "nfa_is_node A (nfa_trans A q a)"
\<pi>_less_self: assumes "0 < x" shows "\<pi> x < x"
scount_eq: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "scount P \<omega> = (if alw (ev P) \<omega> then \<infinity> else enat (card {i. P (sdrop i \<omega>)}))"
hyps_2: fixes t :: "btrm"   and q :: "pat"   and u :: "btrm" assumes "(\<lambda>[p\<^sub>2]. t) = (\<lambda>[q]. u)"   and "pat_type p\<^sub>2 = pat_type q"   and "distinct (pat_vars p\<^sub>2)"   and "distinct (pat_vars q)" shows "\<exists>pi. p\<^sub>2 = pi \<bullet> q \<and> t = pi \<bullet> u \<and> set pi \<subseteq> (supp p\<^sub>2 \<union> supp q) \<times> (supp p\<^sub>2 \<union> supp q)"
induct: fixes P :: "'a option \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a option"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>ao ys. P ao [] ys"   and "\<And>ao xs. P ao xs []"   and "\<And>ao x xs y ys. \<lbrakk>x < y \<Longrightarrow> P (Some x) xs (y # ys); \<lbrakk>\<not> x < y; y < x\<rbrakk> \<Longrightarrow> P (Some y) (x # xs) ys\<rbrakk> \<Longrightarrow> P ao (x # xs) (y # ys)" shows "P a0 a1 a2"
zero_one_eint_neq_1: shows "0 \<noteq> 1"
wp_arraycpy_eq: fixes x :: "char list"   and a :: "char list"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "local.wp (x[] ::= a) Q s = Q (s(x := s a))"
algorithmic_monotonicity_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and \<Gamma>' :: "(name \<times> ty) list" assumes "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T"   and " \<Gamma> \<subseteq> \<Gamma>' "   and "valid \<Gamma>'" shows "\<Gamma>' \<turnstile> s \<Leftrightarrow> t : T"
simps_2: fixes k :: "nat"   and ls :: "('a btree \<times> 'a) list"   and mts :: "('a btree \<times> 'a) list"   and mt :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list"   and tts :: "('a btree \<times> 'a) list"   and tt :: "'a btree" shows "rebalance_middle_tree k ls (Node mts mt) sep rs (Node tts tt) = (if k \<le> length mts \<and> k \<le> length tts then Node (ls @ (Node mts mt, sep) # rs) (Node tts tt) else case rs of [] \<Rightarrow> case node\<^sub>i k (mts @ (mt, sep) # tts) tt of T\<^sub>i x \<Rightarrow> Node ls x | Up\<^sub>i l a x \<Rightarrow> Node (ls @ [(l, a)]) x | (Node rts rt, rsep) # rs \<Rightarrow> case node\<^sub>i k (mts @ (mt, sep) # rts) rt of T\<^sub>i u \<Rightarrow> Node (ls @ (u, rsep) # rs) (Node tts tt) | Up\<^sub>i l a r \<Rightarrow> Node (ls @ (l, a) # (r, rsep) # rs) (Node tts tt))"
simps: fixes x :: "'b" shows "ok_val (OK x) = x"
T: shows "map (\<lambda>(i, j). bdd_binop (\<lambda>k l. the (fst (prod_dfs A B (0, 0)) ! k ! l)) (fst A ! i) (fst B ! j)) (snd (prod_dfs A B (0, 0))) = fst (binop_dfa f A B)"
units_eq_var: fixes e1 :: "'c \<times> 'd"   and e2 :: "'c \<times> 'd" assumes "e1 \<in> pm_prod_E"   and "e2 \<in> pm_prod_E"   and "e1 \<noteq> e2" shows "\<not> ps_prod_D e1 e2"
edomIff: fixes a :: "'a"   and m :: "'a \<Rightarrow> 'b" shows "(a \<in> edom m) = (m a \<noteq> \<bottom>)"
order_refl: fixes x :: "'b zmultiset" shows "x \<subseteq>#\<^sub>z x"
m_closed: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
InternalSets_starset_n: fixes X :: "'a set" shows "*s* X \<in> InternalSets"
simps_1: shows "hbt \<langle>\<rangle> = True"
holomorphic_power_series: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and w :: "complex" assumes "f holomorphic_on ball z r"   and "w \<in> ball z r" shows "(\<lambda>n. (deriv ^^ n) f z / fact n * (w - z) ^ n) sums f w"
assms_2: shows "(f', g') \<in> fundfoldpairs"
dom_locals_sxalloc_mono: fixes G :: "prog"   and s0 :: "state"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1" shows "dom (locals (snd s0)) \<subseteq> dom (locals (snd s1))"
the_ext_prop_1: fixes y' :: "'c"   and g :: "'d" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G x y' g" shows "\<guillemotleft>the_ext y' g : y \<rightarrow>\<^sub>C y'\<guillemotright>"
thetaSCh_Sbis: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and d1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and d2 :: "('test, 'atom, 'choice) cmd" assumes "compatCh ch"   and "c1 \<approx>s d1"   and "c2 \<approx>s d2" shows "thetaSCh ch c1 c2 d1 d2 \<subseteq> Sbis"
c: fixes y :: "'a" assumes "y \<in> closed_segment a b" shows "f y \<le> f c"
igWlsAbsDisj_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igWlsAbsDisj MOD \<equiv> \<forall>xs s xs' s' A. isInBar (xs, s) \<and> isInBar (xs', s') \<and> igWlsAbs MOD (xs, s) A \<and> igWlsAbs MOD (xs', s') A \<longrightarrow> xs = xs' \<and> s = s'"
size_2: fixes x :: "'b" shows "size (Phantom('a) x) = Suc 0"
eq0: shows "adjoint P * (A * adjoint A) * P = B * adjoint B"
if_pred_holds: fixes P :: "unit Predicate.pred" shows "Predicate.if_pred (Predicate.holds P) = P"
pderiv_mult: fixes p :: "'c poly"   and q :: "'c poly" shows "pderiv (p * q) = p * pderiv q + q * pderiv p"
Min_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.Min less_eq \<equiv> semilattice_set.F (ord.min less_eq)"
wuntil_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.wuntil \<phi> \<psi> \<equiv> Shallow.dis (Shallow.until \<phi> \<psi>) (Shallow.alw \<phi>)"
leg1_simps_2: shows "arr f\<^sub>0gh\<^sub>1.p\<^sub>1"
pnet_node_init': fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and i :: "nat"   and R :: "nat set" assumes "s \<in> init (pnet np \<langle>i; R\<rangle>)" obtains ns :: "'c" where "s = NodeS i ns R" and "ns \<in> init (np i)"
formulaEntailsClauseRemoveEntailedLiteralOpposites: fixes F :: "Literal list list"   and clause :: "Literal list"   and valuation :: "Literal list" assumes "formulaEntailsClause F clause"   and "formulaEntailsValuation F valuation" shows "formulaEntailsClause F (list_diff clause (oppositeLiteralList valuation))"
del_vert_props_axioms_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c" shows "del_vert_props_axioms G u \<equiv> u \<in> verts G \<and> out_arcs G u = {}"
assms_3: shows "\<forall>y\<in>Q. y \<noteq> 0 \<longrightarrow> y \<in> dgrad_p_set d m \<and> dickson_less_v d m (lt y) u"
Accessible_dfa_def: shows "Accessible_dfa = \<lparr>states = accessible, init = init M, final = final M \<inter> accessible, nxt = nxt M\<rparr>"
must_syncE: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1" assumes "t \<turnstile> \<langle>x, m\<rangle> \<wrong>1" obtains ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action" and x' :: "'x1" and m' :: "'m1" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "r1.r_syntax t x m ta x' m'" and "r1.actions_ok s t ta" and "m = shr s"
DirProd_def: fixes G :: "('d, 'f) monoid_scheme"   and H :: "('e, 'g) monoid_scheme" shows "G \<times>\<times> H = \<lparr>carrier = carrier G \<times> carrier H, mult = \<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h'), one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
cycleE: fixes xs :: "'a list" assumes "cycle xs" shows "path xs \<and> xs \<noteq> [] \<and> 2 < length xs \<and> last xs \<rightarrow> hd xs"
degree_monom_le: fixes a :: "'a"   and n :: "nat" shows "degree (monom a n) \<le> n"
Inf_less_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and S :: "'a set"   and a :: "'a" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less (Inf S) a = (\<exists>x\<in>S. less x a)"
struct_maps_def: fixes T :: "('a, 'c) partial_object_scheme"   and S :: "('b, 'd) partial_object_scheme" shows "struct_maps T S = {f \<in> carrier T \<rightarrow> carrier S. f = restrict f (carrier T)}"
intersection_of_mono: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and Q' :: "'b set \<Rightarrow> bool" assumes "(P intersection_of Q) S"   and "\<And>x. Q x \<Longrightarrow> Q' x" shows "(P intersection_of Q') S"
knows_composition1_s: fixes m :: "Secrets"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "sKS m \<notin> specKeysSecrets P"   and "sKS m \<notin> specKeysSecrets Q"   and "knows P [sE m]"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ" shows "knows PQ [sE m]"
cases: fixes x :: "'c \<times> ('d \<times> 'c) list" obtains s :: "'c" where "x = (s__, [])"   | s :: "'c" and p :: "'d \<times> 'c" and ps :: "('d \<times> 'c) list" where "x = (s, Cons p ps)"
zero_iff: fixes x :: "'a" assumes "x \<in> E" shows "(\<parallel>x\<parallel> = 0) = (x = (0::'a))"
wtPB_def: fixes \<Phi> :: "('a, 'b) lit list set" shows "M.wtPB \<Phi> \<equiv> Ball \<Phi> M.wtC"
iwlsFSb_iterAll_unique_ipresCons: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "iwlsFSb MOD"   and "ipresCons h hA MOD" shows "(wls s X \<longrightarrow> h X = iter MOD X) \<and> (wlsAbs (us, s') A \<longrightarrow> hA A = iterAbs MOD A)"
bn_eqvt: fixes p :: "perm"   and \<alpha> :: "'a" shows "p \<bullet> bn \<alpha> = bn (p \<bullet> \<alpha>)"
replacefacesAt_notin: fixes i :: "nat"   and "is" :: "nat list"   and olfF :: "face"   and newFs :: "face list"   and Fss :: "face list list" assumes "i \<notin> set is" shows "replacefacesAt is olfF newFs Fss ! i = Fss ! i"
SN_on_Image_rtrancl_iff: fixes R :: "('b \<times> 'b) set"   and X :: "'b set" shows "SN_on R (R\<^sup>* `` X) = SN_on R X"
main_2: fixes t :: "'a bro" assumes "t \<in> B h" shows "fib' h \<le> size t"
pq_def: shows "pq \<equiv> \<lambda>k. quotient_of (\<Sum>i = 0..k. rat_of_int (b i) / rat_of_int (a i))"
hf_of_int_def: fixes i :: "int" shows "hf_of i \<equiv> if 0 \<le> i then \<langle>0, hf_of (nat i)\<rangle> else \<langle>1, hf_of (nat (- i))\<rangle>"
S1: shows "c_assoc_have_key ls (c_pair (c_snd (c_snd (c_fst key))) (c_snd key)) = 0"
qmsg_no_change_on_send_or_receive: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and s :: "('a, 'b, 'c, 'd) seqp \<times> 'b list \<times> ('b list, 'b, unit, unit label) seqp"   and a :: "'b seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and s' :: "('a, 'b, 'c, 'd) seqp \<times> 'b list \<times> ('b list, 'b, unit, unit label) seqp"   and i :: "nat"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> oparp_sos i (oseqp_sos \<Gamma> i) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"   and "a \<noteq> \<tau>" shows "\<sigma>' i = \<sigma> i"
polynomial_hom: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and p :: "'c list" assumes "h \<in> ring_hom R S"   and "field R"   and "field S"   and "p \<in> carrier (poly_ring R)" shows "map h p \<in> carrier (poly_ring S)"
ord_spmf_None: fixes ord :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h spmf" shows "ord_spmf ord (return_pmf None) x"
MATCH1_matchless: fixes x :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and y :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)" assumes "MATCH1 x y" shows "(y, x) \<in> matchless"
jkbpC_def: shows "jkbpC \<equiv> \<Union> (range jkbpCn)"
verit_minus_simplify_2: fixes a :: "'a" shows "a - (0::'a) = a"
discrCf_raw_coind: fixes phi :: "('test, 'atom, 'choice) cmd \<times> 'state \<Rightarrow> bool"   and cf :: "('test, 'atom, 'choice) cmd \<times> 'state" assumes "phi cf"   and "\<And>cf i. \<lbrakk>i < brn (fst cf); phi cf\<rbrakk> \<Longrightarrow> snd cf \<approx> snd (cont_eff cf i) \<and> phi (cont_eff cf i)" shows "discrCf cf"
absorb2: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "sup a b = b"
imp_graph_code: fixes cnf :: "lit uprod set" shows "imp_graph cnf = (if finite cnf then {(l, l'). l' \<in> succ_imp_graph cnf l} else Code.abort STR ''Infinite or invalid 2CNF formula'' (\<lambda>_. imp_graph cnf))"
const_interval_tendsto: fixes a :: "'a" shows "(\<lambda>i. to_interval a) \<longlonglongrightarrow>\<^sub>i a"
assms_3: shows "(f \<longlongrightarrow> d) F"
setcompr_eq_image: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool" shows "{f x |x. P x} = \<P> f {x. P x}"
a_def: shows "a = sum ((*\<^sub>R) (b + 1)) Basis"
multiplicity_le_imp_dvd: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<And>p. normalization_semidom.prime zero times one normalize p \<Longrightarrow> factorial_semiring.multiplicity times one p x \<le> factorial_semiring.multiplicity times one p y" shows "dvd.dvd times x y"
s_ns_mul_ext_trans: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "trans s"   and "trans ns"   and "compatible_l ns s"   and "compatible_r ns s"   and "refl ns"   and "(A, B) \<in> s_mul_ext ns s"   and "(B, C) \<in> ns_mul_ext ns s" shows "(A, C) \<in> s_mul_ext ns s"
S1: shows "\<forall>x y. ((x, y) \<in> s) = (\<exists>u. p_s x y u = 0)"
neg_in_model: fixes M :: "nat \<Rightarrow> bool"   and n :: "nat"   and n_vars :: "nat" assumes "\<not> M n"   and "0 < n"   and "n < n_vars" shows "- int n \<in> set (model_to_dimacs_model M [1..<n_vars])"
proper_on_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> ereal" shows "proper_on S f = ((\<forall>x\<in>S. f x \<noteq> - \<infinity>) \<and> (\<exists>x\<in>S. f x \<noteq> \<infinity>))"
find_min_exist: fixes xs :: "('c, 'd) bintree option list"   and t :: "('c, 'd) bintree" assumes "find_min xs = Some t" shows "Some t \<in> set xs"
simps_1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> b"   and x :: "'a"   and z :: "'a" shows "is_path' r x [] z = (r x z = T)"
assms_8: shows "4 \<le> rk {A, B, C, A', B', C', P, Q}"
Svalid_def: fixes s :: "form list" shows "Svalid s = (\<forall>MI e. is_env MI e \<longrightarrow> SEval MI e s)"
vpa_snoc_Call: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and cs :: "'edge list"   and as :: "'edge list" assumes "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_path_aux cs as" shows "valid_path_aux cs (as @ [a])"
hyps_1: fixes T :: "secLevel" assumes "\<Gamma> \<turnstile> e1 : T"   and "x \<in> fv e1"   and "\<Gamma> x = Some High" shows "\<Gamma> \<turnstile> e1 : High"
cong_simp: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B =simp=> g x = h x" shows "prod g A = prod h B"
Lang_eq1: fixes x :: "'c"   and G :: "('d \<times> ('d, 'c) RHS) list"   and S :: "'d" shows "([x] \<in> Lang G S) = ((S, Leaf x) \<in> set G)"
True_steps_conc: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "((True # p, q) \<in> steps (RegExp2NA.conc L R) w) = ((\<exists>r. (p, r) \<in> steps L w \<and> q = True # r) \<or> (\<exists>u a v. w = u @ a # v \<and> (\<exists>r. (p, r) \<in> steps L u \<and> fin L r \<and> (\<exists>s. (start R, s) \<in> step R a \<and> (\<exists>t. (s, t) \<in> steps R v \<and> q = False # t)))))"
assoc'_eq_inv_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.\<a>' f g h = B.inv \<a>\<^sub>B[f, g, h]"
assms: shows "steps0 (1, tp) A n = (0, tp')"