hcju/isabelleps huggingface dataset:
closure_of_subtopology_open: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" assumes "openin X U \<or> S \<subseteq> U" shows "subtopology X U closure_of S = U \<inter> X closure_of S"
init_s: fixes t :: "real" assumes "t \<le> 0" shows "delayed_safe_distance.other.s t = other.s \<delta>"
hyps_1: fixes q :: "ref" assumes "Dag q (rep \<propto> low) (rep \<propto> high) lrt"   and "\<forall>pt\<in>set_of lrt. rep pt = pt" shows "Dag q (rep \<propto> lowa) (rep \<propto> higha) lrt"
rtrancl_path_simps_1: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" shows "rtrancl_path R x [] y = (y = x)"
isCont_Arccos_lemma: fixes z :: "complex" assumes "Im (z + \<i> * csqrt (1 - z\<^sup>2)) = 0"   and "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "False"
g_def: shows "g \<equiv> Cn 4 r_ifz [z, recf.Id 4 1, Cn 4 S [recf.Id 4 0]]"
main: fixes j :: "nat" shows "- (fs ! i - v) $ j + fs ! i $ j = - (fs ! i - gso i) $ j + fs ! i $ j"
subset: shows "outstanding_refs is_volatile_Read\<^sub>s\<^sub>b (r # sb) \<subseteq> A"
scale_one: fixes x :: "'t \<Rightarrow>\<^sub>0 'c" shows "1 \<odot> x = x"
simps_3: fixes n :: "nat"   and s :: "dB"   and k :: "nat" shows "liftn n (Abs s) k = Abs (liftn n s (k + 1))"
perm_1: fixes pi :: "(name \<times> name) list" shows "pi \<bullet> \<zero> = \<zero>"
less_imp_inverse_less: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less zero a" shows "less (inverse b) (inverse a)"
observable_move_call: fixes f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   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 a' :: "'edge"   and ms :: "'node list"   and S :: "'node SDG_node set"   and ms' :: "'node list" assumes "pred (f a) s"   and "transfer (f a) s = s'"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "a' \<in> get_return_edges a"   and "\<forall>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "hd ms \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "length ms = length s"   and "length s' = Suc (length s)"   and "hd ms = sourcenode a"   and "ms' = targetnode a # targetnode a' # tl ms" shows "S,f \<turnstile> (ms,s) -a\<rightarrow> (ms',s')"
l8_5: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Per Bet Cong A B B"
pmf_of_list_aux: fixes xs :: "('b \<times> real) list" assumes "\<And>x. x \<in> set (map snd xs) \<Longrightarrow> 0 \<le> x"   and "sum_list (map snd xs) = 1" shows "\<integral>\<^sup>+ x. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV = 1"
fmlookup_map: fixes f :: "'e \<Rightarrow> 'c"   and m :: "('d, 'e) fmap"   and x :: "'d" shows "fmlookup (fmmap f m) x = map_option f (fmlookup m x)"
simps_2: fixes l :: "'b btree"   and a :: "'b"   and r :: "'b btree" shows "bal_up\<^sub>i (Up\<^sub>i l a r) = (height l = height r \<and> bal l \<and> bal r)"
Iexp: shows "I = insert (Min I) (I - {Min I})"
Int_atLeastAtMost: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word"   and d :: "'b word" shows "signed.atLeastAtMost a b \<inter> signed.atLeastAtMost c d = signed.atLeastAtMost (signed.max a c) (signed.min b d)"
assms: shows "LeftDerivationLadder \<alpha> D L \<gamma>"
map: fixes C' :: "char list"   and fs :: "fdecl list" assumes "map_of (map ((\<lambda>((F, D), a). (F, D, a)) \<circ> (\<lambda>(F, y). ((F, C'), y))) fs) F = \<lfloor>(D, b, T)\<rfloor>" shows "D = C' \<and> (F, b, T) \<in> set fs"
simps_2: fixes S :: "(char list \<Rightarrow> int) set"   and x :: "char list"   and e :: "aexp"   and P :: "(char list \<Rightarrow> int) set" shows "step S (x ::= e {P}) = x ::= e {{s'. \<exists>s\<in>S. s' = s(x := aval e s)}}"
2: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "P a b"   and "P b c"   and "a \<le> b"   and "b \<le> c" shows "P a c"
coprime_cong_mult_nat: fixes a :: "nat"   and b :: "nat"   and m :: "nat"   and n :: "nat" assumes "[a = b] (mod m)"   and "[a = b] (mod n)"   and "coprime m n" shows "[a = b] (mod m * n)"
exhaust_sel: fixes state :: "('a, 'b, 'c) state" obtains "state = State (state.fun_env state) (state.heap state) (callstack state)"
restrict_size_le: fixes k :: "'a ifex"   and var :: "'a"   and val :: "bool" shows "size (restrict_top k var val) \<le> size k"
goodEnv_imp_qGoodEnv_pickE: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" assumes "goodEnv rho" shows "qGoodEnv (pickE rho)"
basic_identity\<^sub>E_infix_def: fixes x :: "\<kappa>"   and y :: "\<kappa>" shows "x \<^bold>=\<^sub>E y \<equiv> \<lparr>basic_identity\<^sub>E,x,y\<rparr>"
nth_tl: fixes n :: "nat"   and xs :: "'a list" assumes "n < |tl xs|" shows "tl xs ! n = xs ! Suc n"
exhaust: fixes y :: "pr_operation" obtains "y = PUSH"   | "y = RELABEL"
Id_on_eqI: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "a = b"   and "a \<in> A" shows "(a, b) \<in> Id_on A"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "idom_divide_hom hom"   and "inj_idom_hom hom" shows "inj_idom_divide_hom hom"
tND2: fixes p :: "('b set set, 'c) Combinators list"   and x :: "('b set set, 'c) Combinators"   and y :: "('b set set, 'c) Combinators"   and a :: "'b set set"   and b :: "'b set set"   and c :: "'b set set"   and d :: "'b set set" assumes "allNetsDistinct p"   and "x \<in> set p"   and "y \<in> set p"   and "a = srcNet x"   and "b = destNet x"   and "c = srcNet y"   and "d = destNet y"   and "b \<noteq> d"   and "singleCombinators [x]"   and "x \<noteq> DenyAll"   and "singleCombinators [y]"   and "y \<noteq> DenyAll" shows "twoNetsDistinct a b c d"
vec_nth: fixes x :: "('c, 'b) Finite_Cartesian_Product.vec" shows "($h) x \<in> UNIV"
compute_shift_map_keys: fixes t :: "'a"   and f :: "'d \<Rightarrow> 'c"   and xs :: "(('a \<times> 'b) \<times> 'd) list" shows "pprod.shift_map_keys t f (fmap_of_list xs) = fmap_of_list (map (\<lambda>(k, v). (pprod.splus t k, f v)) xs)"
xi_3: shows "additive_strength (inv f) xi \<le> 0"
card_of_Un_Plus_ordIso: fixes A :: "'a set"   and B :: "'a set" assumes "A \<inter> B = {}" shows "|A \<union> B| =o |A <+> B|"
LLq_num: fixes \<phi> :: "'fmla"   and q :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {zz}"   and "q \<in> num"   and "\<forall>p\<in>num. prv (subst \<phi> p zz)" shows "prv (all zz (imp (LLq (Var zz) q) \<phi>))"
simps_1: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" shows "RM M2 M1 \<Omega> V m 0 = {}"
oSuc_plus_omega: fixes x :: "ordinal" shows "oSuc x + \<omega> = x + \<omega>"
reaches_steps: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "E\<^sup>*\<^sup>* x y" shows "\<exists>xs. hd xs = x \<and> last xs = y \<and> Graph_Defs.steps E xs"
inf_truncate_append: fixes z :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "inf_truncate z i @ [z (Suc i)] = inf_truncate z (Suc i)"
wf_semiJoin_3: fixes J :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and QQ :: "(nat set \<times> 'a option list set) set"   and st :: "nat set"   and t :: "'a option list" assumes "1 \<le> card J"   and "wf_query n J Q Qn"   and "non_empty_query Q"   and "covering J Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"   and "QQ = (\<lambda>tab. semiJoin tab (st, t)) ` Q" shows "covering J QQ"
pop_refine: fixes s :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list"   and p :: "'a set list"   and D :: "'a set"   and pE :: "('a \<times> 'a) set" assumes "(s, p, D, pE) \<in> GS_rel"   and "p \<noteq> []"   and "pE \<inter> last p \<times> UNIV = {}" shows "pop_impl s \<le> \<Down> GS_rel (RETURN (pop (p, D, pE)))"
alpha_qVar_iff: fixes X :: "('b, 'c, 'd, 'e, 'f) qTerm"   and xs' :: "'d"   and x' :: "'e" shows "(X #= qVar xs' x') = (X = qVar xs' x')"
0: fixes b :: "'a local.term" assumes "Ide a \<and> Diag a"   and "Ide b \<and> Diag b" shows "Can (a \<^bold>\<Down> b) \<and> a \<^bold>\<Down> b \<in> Hom (a \<^bold>\<otimes> b) \<^bold>\<lfloor>a \<^bold>\<otimes> b\<^bold>\<rfloor>"
cleanChans_channel_inv: fixes dchans :: "integer list" assumes "set (gState.channels g) \<subseteq> Collect channel_inv" shows "set (cleanChans dchans (gState.channels g)) \<subseteq> Collect channel_inv"
hbexE: fixes A :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "HBex A P" obtains x :: "hf" where "x \<^bold>\<in> A" and "P x"
assms_3: shows "uex x \<circ> H = H \<circ> uex x"
lm021: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> injectionsUniverse"
Cons_lenlex_iff: fixes m :: "'b"   and ms :: "'b list"   and n :: "'b"   and ns :: "'b list"   and r :: "('b \<times> 'b) set" shows "((m # ms, n # ns) \<in> lenlex r) = (length ms < length ns \<or> length ms = length ns \<and> (m, n) \<in> r \<or> m = n \<and> (ms, ns) \<in> lenlex r)"
nat_of_plus_sat: fixes x :: "'a sat"   and y :: "'a sat" shows "nat_of (x + y) = min (nat_of x + nat_of y) LENGTH('a)"
h_3: shows "pathfinish h \<in> U - {pathfinish g}"
append_rows_nth: fixes A :: "'a mat"   and a :: "nat"   and n :: "nat"   and B :: "'a mat"   and b :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat a n"   and "B \<in> carrier_mat b n"   and "i < a + b"   and "j < n" shows "(A @\<^sub>r B) $$ (i, j) = (if i < dim_row A then A $$ (i, j) else B $$ (i - a, j))"
Cast: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l, sh)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l, sh)\<rangle>"
fps_const_power: fixes c :: "'a"   and n :: "nat" shows "fps_const c ^ n = fps_const (c ^ n)"
satisfies_bounded_disjuncts: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b \<phi>) = (\<exists>\<psi>\<in>disjuncts \<phi>. \<AA> \<Turnstile>\<^sub>b \<psi>)"
algebraic_int_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> algebraic_int (f x)" shows "algebraic_int (sum f A)"
nth_app_eq1: fixes i :: "nat"   and sla :: "'a list"   and sl :: "'a list" assumes "i < length sla" shows "(sla @ sl) ! i = sla ! i"
bounded_box: fixes a :: "'b"   and b :: "'b" shows "bounded (box a b)"
wellformed_mk_symmetric: fixes G :: "('c, 'd) pre_digraph" assumes "wf_digraph G" shows "pair_wf_digraph (mk_symmetric G)"
intro: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>a b c. f a b = f a c \<Longrightarrow> b = c"   and "\<And>b a c. f b a = f c a \<Longrightarrow> b = c" shows "cancel_semigroup_axioms f"
rhs: shows "sample_uniform q = spmf_of_set {..<q}"
root_int_main_4: fixes n :: "int"   and p :: "nat"   and y :: "int"   and b :: "bool" assumes "0 \<le> n"   and "root_int_main p n = (y, b)"   and "0 < p" shows "n < (y + 1) ^ p"
cong_sym_poly: fixes a :: "'b poly"   and b :: "'b poly"   and m :: "'b poly" assumes "[a = b] (mod m)" shows "[b = a] (mod m)"
maxt34: shows "wmax (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2) \<equiv>\<^sub>E max (rl1 * ru2) (ru1 * ru2)"
wf_measure: fixes f :: "'c \<Rightarrow> nat" shows "wf (Wellfounded.measure f)"
ret: shows "returns_to {x \<in> S. s x = 0} x"
Bex_def: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "Bex A P = (\<exists>x. x \<in> A \<and> P x)"
intro: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog" assumes "heap_is_wellformed = l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes get_disconnected_nodes get_shadow_root get_tag_name" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes get_shadow_root get_tag_name heap_is_wellformed"
A: shows "Sup {limsup (\<lambda>n. ereal (dist x (u n) - dist y (u n))) |u. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi} \<le> ereal (dist x y)"
finiteUniqAndConsistentTrailsWithGivenVariableSuperset: fixes V :: "nat set" assumes "finite V" shows "finite {M. vars (elements M) \<subseteq> V \<and> uniq (elements M) \<and> consistent (elements M)}"
map_iterator_genord_rule_P: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and R :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'f \<Rightarrow> bool"   and I :: "'e set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "map_iterator_genord it m R"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> it - {k} \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v'); \<forall>k' v'. k' \<notin> it \<and> m k' = Some v' \<longrightarrow> R (k', v') (k, v)\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<notin> it \<and> m k = Some v \<and> k' \<in> it \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v')\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
S4bS5S6LeadstoS6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and sigma :: "behavior" assumes "sigma \<Turnstile> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<leadsto> S5 rmhist p"   and "sigma \<Turnstile> S5 rmhist p \<leadsto> S6 rmhist p" shows "sigma \<Turnstile> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<or> S5 rmhist p \<or> S6 rmhist p \<leadsto> S6 rmhist p"
subgroups_Inter_pair: fixes I :: "nat set"   and J :: "nat set" assumes "subgroup I G"   and "subgroup J G" shows "subgroup (I \<inter> J) G"
is_glb_singleton: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.po below" shows "po.is_glb below {x} x"
option_rel_Some2: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d" shows "rel_option A x (Some y) = (\<exists>x'. x = Some x' \<and> A x' y)"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < k"   and "p A i j" shows "q A i j"
intro: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path G P v0 p \<sigma>"   and "vmc_path_no_deadend_axioms G v0" shows "vmc_path_no_deadend G P v0 p \<sigma>"
lookup_update: fixes k1 :: "'c"   and v :: "'b"   and xs :: "('c, 'b) alist"   and k2 :: "'c" shows "DAList.lookup (DAList.update k1 v xs) k2 = (if k1 = k2 then Some v else DAList.lookup xs k2)"
map\<^sub>1_def: fixes f :: "'a" shows "\<tau>'\<tau>.map\<^sub>1 f = \<a>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 (trg\<^sub>C f), \<tau>\<^sub>0 (trg\<^sub>C f), F f] \<cdot>\<^sub>D (map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>C f), G f, \<tau>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D (map\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, map\<^sub>0 (src\<^sub>C f), \<tau>\<^sub>0 (src\<^sub>C f)]"
distinct_29: fixes x51 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x52 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x53 :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and x54 :: "acom"   and x61 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and x62 :: "bexp"   and x63 :: "acom" shows "{x51/x52/x53} CONSEQ x54 \<noteq> {x61} WHILE x62 DO x63"
append_self_conv: fixes xs :: "'d list"   and ys :: "'d list" shows "(xs @ ys = xs) = (ys = [])"
map\<^sub>0_def: fixes a :: "('a E.term, 'a) arr" shows "DN.map\<^sub>0 a \<equiv> src\<^sub>B (DN a)"
normal_nnf: fixes \<phi> :: "'a fm" assumes "normal \<phi>" shows "normal (nnf \<phi>)"
prems_2: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun f Y)"
ring_seq_smult_comp_assoc: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "a \<in> carrier Zp" shows "a \<odot>\<^bsub>function_ring (carrier Zp) Zp\<^esub> f \<circ> s = a \<odot>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> (f \<circ> s)"
cont_field: shows "\<forall>b\<in>basis. \<forall>(k, \<gamma>2)\<in>subdiv. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
downset_prop2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "less_eq y x" shows "y \<in> ord.downset less_eq x"
insertAssertion_def: shows "insertAssertion \<equiv> \<lambda>x \<Psi>'. frame_rec (\<lambda>\<Psi>. \<langle>\<epsilon>, \<Psi>' \<otimes> \<Psi>\<rangle>) (\<lambda>x F. FRes x) x"
cut_greater_Diff: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A - B) \<down>> t = A \<down>> t - B \<down>> t"
mult_sheaf_spec_def: fixes U :: "nat set set"   and s :: "nat set \<Rightarrow> (nat \<times> nat) set"   and s' :: "nat set \<Rightarrow> (nat \<times> nat) set" shows "cring0.mult_sheaf_spec U s s' \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.mult_rel ({0}\<setminus>\<pp>) {0} ring0.subtraction ring0.subtraction 0 (s \<pp>) (s' \<pp>)"
least_power_props_2: fixes a :: "'a" shows "0 < least_power p a"
natural_isomorphism_axioms: shows "natural_isomorphism A A A.map A.map A.map"
simps: fixes v :: "'d"   and p :: "'c" shows "valid_out_port (v, p) = (v |\<in>| vertices \<and> p |\<in>| outPorts (nodeOf v))"
M_op: shows "op_sem oper \<in> (M \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) \<rightarrow>\<^sub>M stock_measure t'"
diff_right_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a c) b = minus (minus a b) c"
digraph_map_rev_swap: fixes a :: "'b"   and b :: "'b" assumes "arc_to_ends G a = arc_to_ends G b"   and "{a, b} \<subseteq> arcs G" shows "digraph_map G (G'.rev_swap a b)"
VVpstar: fixes v :: "'a"   and w :: "'a" assumes "v \<in> S"   and "\<not> deadend v"   and "v \<in> VV p**"   and "v \<rightarrow> w" shows "w \<in> S \<union> T"
indep_setD: fixes A :: "'a set set"   and B :: "'a set set"   and a :: "'a set"   and b :: "'a set" assumes "indep_set A B"   and "a \<in> A"   and "b \<in> B" shows "prob (a \<inter> b) = prob a * prob b"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "('a E.term, 'a) arr" assumes "B.ide a"   and "B.ide a'"   and "\<guillemotleft>g : UP a' \<Rightarrow> UP a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<Rightarrow>\<^sub>B a\<guillemotright> \<and> UP f = g"
fresh_minus_perm: fixes a :: "atom"   and p :: "perm" shows "a \<sharp> - p = a \<sharp> p"
bij_from_nat: shows "bij_betw mod_type_class.from_nat {0..<CARD('b)} UNIV"
tt_mult_scalar: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "punit.tt (local.punit.mult_scalar p q) = local.punit.splus (punit.tt p) (punit.tt q)"
steps_imp_not_SN_on: fixes t :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'b"   and R :: "('c \<times> 'c) set"   and x :: "'b" assumes "\<And>x. (t x, t (f x)) \<in> R" shows "\<not> SN_on R {t x}"
closedin_trans: fixes T :: "'c set"   and S :: "'c set"   and U :: "'c set" assumes "closedin (top_of_set T) S"   and "closedin (top_of_set U) T" shows "closedin (top_of_set U) S"
is_red_minus: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F (p - q)" shows "is_red F p \<or> is_red F q"
lemma_4_12_i_iii: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (1::'a)" shows "\<exists>n. a ^ n \<le> b"
Gamma_hurwitz_zeta_aux_integral: fixes z :: "complex"   and a :: "real"   and n :: "nat" assumes "0 < Re z"   and "0 < a" shows "Gamma z / (of_nat n + complex_of_real a) powr z = CLINT s:{0<..}|lebesgue. complex_of_real s powr (z - 1) / complex_of_real (exp ((real n + a) * s))"
assms_1: shows "(st, a, st') \<in> pnet_sos S T"
cexpr_sem_map_vars: fixes \<sigma> :: "nat \<Rightarrow> val"   and f :: "nat \<Rightarrow> nat"   and e :: "cexpr" shows "cexpr_sem \<sigma> (map_vars f e) = cexpr_sem (\<sigma> \<circ> f) e"
unitAssertWeaken: fixes \<Psi> :: "'b" shows "\<one> \<hookrightarrow> \<Psi>"
closed_sum_right_subset: fixes A :: "'b set"   and B :: "'b set" assumes "(0::'b) \<in> A" shows "B \<subseteq> A +\<^sub>M B"
ext_inject: fixes vars :: "(String.literal, variable) assoc_list"   and channels :: "channel list"   and timeout :: "bool"   and procs :: "pState list"   and more :: "'b"   and vars' :: "(String.literal, variable) assoc_list"   and channels' :: "channel list"   and timeout' :: "bool"   and procs' :: "pState list"   and more' :: "'b" shows "(\<lparr>gState.vars = vars, channels = channels, timeout = timeout, procs = procs, \<dots> = more\<rparr> = \<lparr>gState.vars = vars', channels = channels', timeout = timeout', procs = procs', \<dots> = more'\<rparr>) = (vars = vars' \<and> channels = channels' \<and> timeout = timeout' \<and> procs = procs' \<and> more = more')"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "x \<sqsubseteq> xa" obtains xb :: "'a" and y :: "'a" where "x = Some xb" and "xa = Some y__" and "xb \<sqsubseteq> y__"   | y :: "'a option" where "x = None" and "xa = y"
cl_insert_absorb: fixes X :: "'a set"   and x :: "'a" assumes "X \<subseteq> \<E>"   and "x \<in> \<E>.cl X" shows "\<E>.cl (insert x X) = \<E>.cl X"