hcju/isabelleps huggingface dataset:
rel_FGcontra_def: fixes L1 :: "'au \<Rightarrow> 'bh \<Rightarrow> bool"   and Co1 :: "'av \<Rightarrow> 'bi \<Rightarrow> bool"   and Co2 :: "'bd \<Rightarrow> 'bq \<Rightarrow> bool"   and Co3 :: "'aw \<Rightarrow> 'bj \<Rightarrow> bool"   and Co4 :: "'ax \<Rightarrow> 'bk \<Rightarrow> bool"   and Co5 :: "'ay \<Rightarrow> 'bl \<Rightarrow> bool"   and Contra1 :: "'az \<Rightarrow> 'bm \<Rightarrow> bool"   and Contra2 :: "'ba \<Rightarrow> 'bn \<Rightarrow> bool"   and Contra3 :: "'bb \<Rightarrow> 'bo \<Rightarrow> bool"   and Contra4 :: "'bc \<Rightarrow> 'bp \<Rightarrow> bool"   and Contra5 :: "'bf \<Rightarrow> 'br \<Rightarrow> bool" shows "rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4 Contra5 = rel_F L1 Co1 Co3 Co1 Co4 Co5 (rel_G Contra1 Contra2 Contra3 Contra4 Co1 Co2) Contra1 Contra5"
info_disc_filtr: shows "disc_filtr M F \<and> sets (F \<bottom>) = {{}, space M}"
sel_1: fixes x21 :: "tree"   and x22 :: "tree" shows "ltree (Branching x21 x22) = x21"
scalingI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P c x. \<lbrakk>sound P; 0 \<le> c\<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x" shows "scaling t"
snoc_conv_cons: fixes ys :: "'a list"   and y :: "'a" shows "\<exists>x xs. ys @ [y] = x # xs"
assms_4: shows "InvariantClCurrentLevel (getCl state) (getM state)"
equal_ratfps_def: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "equal_class.equal x y = (x = y)"
keys_def_alt: fixes t :: "('b, 'c) RBT.rbt" shows "RBT.keys t = map fst (RBT.entries t)"
simps_6: fixes t :: "pdf_type"   and x :: "pdf_type" shows "op_type (Cast t) x = (case (x, t) of (UNIT, b) \<Rightarrow> None | (BOOL, INTEG) \<Rightarrow> Some INTEG | (BOOL, REAL) \<Rightarrow> Some REAL | (BOOL, PRODUCT pdf_type1 x) \<Rightarrow> Map.empty x | (BOOL, _) \<Rightarrow> None | (INTEG, REAL) \<Rightarrow> Some REAL | (INTEG, PRODUCT pdf_type1 x) \<Rightarrow> Map.empty x | (INTEG, _) \<Rightarrow> None | (REAL, INTEG) \<Rightarrow> Some INTEG | (REAL, PRODUCT pdf_type1 x) \<Rightarrow> Map.empty x | (REAL, _) \<Rightarrow> None | (PRODUCT pdf_type1 x, b) \<Rightarrow> Map.empty x)"
assms_6: shows "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)"
set_less_eq_aux'_infinite_complement_partitionD: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubseteq>'' B" shows "A \<in> infinite_complement_partition \<and> B \<in> infinite_complement_partition"
elims: fixes x :: "'a option list"   and xa :: "'a option list"   and y :: "'a option list" assumes "merge x xa = y" obtains t1 :: "'a option list" and t2 :: "'a option list" where "x = t1" and "xa = t2" and "y = map merge_option (zip t1 t2)"
assms_2: shows "eucl_rel_poly x' y (q', r')"
arrI: fixes f1 :: "'a"   and f2 :: "'a \<times> 'a" assumes "arr f1"   and "VxV.arr f2" shows "VxVxV.arr (f1, f2)"
c1_1: fixes t :: "'b"   and x :: "'a" assumes "t \<in> cbox a b"   and "x \<in> S" shows "((\<lambda>x. f x t) has_derivative fx x t) (at x within S)"
ifte_ifte_distrib: fixes x :: "'a"   and y :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "ifte x (ifte y a b) (ifte y c d) = ifte y (ifte x a c) (ifte x b d)"
intro: assumes "u = (THE x. \<forall>y. P y x)" shows "CEMC_axioms"
g_bij: shows "bij_betw g {..<length ss1} {..<length ss2}"
dilating_fun_image_left: fixes f :: "nat \<Rightarrow> nat"   and r :: "'b run"   and m :: "nat"   and n :: "nat"   and c :: "clock" assumes "dilating_fun f r" shows "{k. f m \<le> k \<and> k < f n \<and> hamlet (Rep_run r k c)} = f ` {k. m \<le> k \<and> k < n \<and> hamlet (Rep_run r (f k) c)}"
visits_finsert: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list"   and aa :: "nat"   and ba :: "nat"   and b :: "transition" assumes "visits s e s' r t" shows "visits s (finsert ((aa, ba), b) e) s' r t"
assigned_nodes_def: shows "assigned_nodes = (\<lambda>slot. Heap_Error_Monad.bind (get_tag_name slot) (\<lambda>tag. Heap_Error_Monad.bind (if tag \<noteq> ''slot'' then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_root_node (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r slot)) (\<lambda>root. if is_shadow_root_ptr_kind root then Heap_Error_Monad.bind (get_host (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root))) (\<lambda>host. Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)) (filter_M (\<lambda>slotable. Heap_Error_Monad.bind (find_slot False slotable) (\<lambda>found_slot. return (found_slot = Some slot))))) else return []))))"
cases: fixes a1 :: "state"   and a2 :: "com"   and a3 :: "state" assumes "a1 -a2\<rightarrow> a3" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = s__" and "a2 = Do f__" and "a3 = t__" and "t__ \<in> f__ s__"   | s0 :: "state" and c1 :: "com" and s1 :: "state" and c2 :: "com" and s2 :: "state" where "a1 = s0" and "a2 = (c1__; c2__)" and "a3 = s2" and "s0 -c1__\<rightarrow> s1" and "s1 -c2__\<rightarrow> s2"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and t :: "state" and c2 :: "com" where "a1 = sa__" and "a2 = IF b__ THEN c1a__ ELSE c2a__" and "a3 = ta__" and "b__ sa__" and "sa__ -c1a__\<rightarrow> ta__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and t :: "state" and c1 :: "com" where "a1 = sb__" and "a2 = IF ba__ THEN c1 ELSE c2" and "a3 = tb__" and "\<not> ba__ sb__" and "sb__ -c2\<rightarrow> tb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = sc__" and "a2 = WHILE bb__ DO c__" and "a3 = sc__" and "\<not> bb__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and t :: "state" and u :: "state" where "a1 = sd__" and "a2 = WHILE b DO ca__" and "a3 = u" and "b sd__" and "sd__ -ca__\<rightarrow> tc__" and "tc__ -WHILE b DO ca__\<rightarrow> u"   | s :: "state" and p :: "pname" and t :: "state" where "a1 = se__" and "a2 = CALL p" and "a3 = td__" and "se__ -body p\<rightarrow> td__"   | f :: "state \<Rightarrow> state" and s :: "state" and c :: "com" and t :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = s" and "a2 = LOCAL f; c; g" and "a3 = g s t" and "f s -c\<rightarrow> t"
assms_2: shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
induced_function_integral_aux_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "conservative_mpt M T"   and "A \<in> sets M"   and "integrable M f"   and "\<And>x. 0 \<le> f x" shows "integrable M (qmpt.induced_function T A f)"
select_convs_1: fixes CatDom :: "('g, 'h, 'i) Category_scheme"   and CatCod :: "('j, 'k, 'l) Category_scheme"   and MapM :: "'h \<Rightarrow> 'k"   and more :: "'m" shows "CatDom \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = CatDom"
k: shows "\<not> b' ((c' ^^ k) (f s))"
cBex_triv_one_point2: fixes A :: "'d cset"   and a :: "'d" shows "cBex A ((=) a) = cin a A"
pa_bq_d: shows "p * A $ a $ j + A $ b $ j * q = d"
pos: fixes x :: "'s" assumes "x \<in> C" shows "0 < emeasure (measure_pmf N) {x}"
the_inv_into_G_binop_G: fixes g :: "'a"   and x :: "'a" assumes "g \<in> G"   and "x \<in> G" shows "the_inv_into G (binop g) x = binop (\<ii> g) x"
fold_rev_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('c, 'd) rbt" shows "fold_rev f t = foldr (\<lambda>(k, v). f k v) (RBT_Impl.entries t)"
thenOrdering_GT_iff: fixes x :: "Ordering"   and y :: "Ordering" shows "(thenOrdering\<cdot>x\<cdot>y = GT) = (x = GT \<or> x = EQ \<and> y = GT)"
has_typ1_unique: fixes \<tau>s :: "typ list"   and t :: "Core.term"   and \<tau>1 :: "typ"   and \<tau>2 :: "typ" assumes "\<tau>s \<turnstile>\<^sub>\<tau> t : \<tau>1"   and "\<tau>s \<turnstile>\<^sub>\<tau> t : \<tau>2" shows "\<tau>1 = \<tau>2"
mult_diff_mult: fixes x :: "'b"   and y :: "'b"   and a :: "'b"   and b :: "'b" shows "x * y - a * b = x * (y - b) + (x - a) * b"
add_inf_2: fixes a :: "'a DBMEntry" shows "\<infinity> + a = \<infinity>"
cPar2B: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a" assumes "!P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a<\<nu>x> \<prec> P')" shows "F C (P \<parallel> !P) (a<\<nu>x> \<prec> (P \<parallel> P'))"
replace4: fixes f :: "'a"   and ls :: "'a list"   and oldF :: "'a"   and fs :: "'a list" assumes "f \<in> set ls"   and "oldF \<noteq> f" shows "f \<in> set (replace oldF fs ls)"
span_scale: fixes x :: "'b"   and S :: "'b set"   and c :: "'a" assumes "x \<in> local.span S" shows "c *s x \<in> local.span S"
strand_sem_split'_2: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c' \<theta>" shows "\<lbrakk>M \<union> ik\<^sub>s\<^sub>t S; S'\<rbrakk>\<^sub>c' \<theta>"
Bag_Alist_Cons: fixes x :: "'a"   and xs :: "('a \<times> nat) list"   and n :: "nat" assumes "x \<notin> fst ` set xs"   and "distinct (map fst xs)" shows "Bag (Alist ((x, n) # xs)) = mset (replicate n x) + Bag (Alist xs)"
fun_rel_comp_dist: fixes R1 :: "('a \<times> 'e) set"   and R2 :: "('b \<times> 'f) set"   and R3 :: "('e \<times> 'c) set"   and R4 :: "('f \<times> 'd) set" shows "(R1 \<rightarrow> R2) O (R3 \<rightarrow> R4) \<subseteq> R1 O R3 \<rightarrow> R2 O R4"
closed_invs_11: fixes e :: "db" shows "closed (DBAbsV e) = (\<forall>i. freedb e i \<longrightarrow> i = 0)"
injectivity_3: fixes P :: "nat set set"   and j :: "nat"   and P' :: "nat set set"   and j' :: "nat"   and B' :: "nat set"   and k :: "nat"   and B'' :: "nat set"   and k' :: "nat"   and Q :: "nat set set"   and Q' :: "nat set set"   and f :: "nat \<Rightarrow> nat set"   and g :: "nat \<Rightarrow> nat set"   and P'' :: "nat set set"   and P''' :: "nat set set" assumes "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "B' = B''"
not_Some_eq2: fixes v :: "('a \<times> 'b) option" shows "(\<forall>x y. v \<noteq> \<lfloor>(x, y)\<rfloor>) = (v = None)"
I': shows "I' \<in> P \<and> I' \<subseteq> II"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L S.map \<ll>"
AndRel: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and \<psi>' :: "Formula.formula"   and conf :: "Formula.trm \<times> bool \<times> mconstraint \<times> Formula.trm" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "\<psi>' = formula_of_constraint conf"   and "let (t1, uu_, uu_, t2) = conf in fv_trm t1 \<union> fv_trm t2 \<subseteq> fv \<phi>'" shows "wf_mformula \<sigma> j P V n R (MAndRel \<phi> conf) (formula.And \<phi>' \<psi>')"
to_fun_minus: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "f \<in> carrier P"   and "x \<in> carrier R" shows "to_fun (\<ominus>\<^bsub>P\<^esub> f) x = \<ominus> to_fun f x"
assms_2: shows "MethodInvk (New C0 es) m ds \<in> subexprs(e)"
convex_ball: fixes x :: "'a"   and e :: "real" shows "convex (ball x e)"
vs_zero_lin_dep: fixes S :: "'a vec set" assumes "S \<subseteq> mat_kernel B"   and "KB.lin_indpt S" shows "0\<^sub>v n \<notin> S"
f2: shows "(U\<^sub>f * U\<^sub>f) $$ (i, j) = of_nat (f (i div 2)) * U\<^sub>f $$ (i - 1, j) + of_nat (1 - f (i div 2)) * U\<^sub>f $$ (i, j)"
dagger_plus_one: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> = x\<^sup>\<star>"
matrix_match_condn_2: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "matrix_match A1 A2 B1 B2 \<and> i < row_length A1 * row_length B1 \<and> j < length A2 * length B2" shows "((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))"
ifupdateirrel: shows "simple_fw fw (p\<lparr>p_oiface := output_iface (routing_table_semantics rt (p_dst p))\<rparr>) = simple_fw fw p"
a': shows "a \<otimes> h [^] e' = \<^bold>g [^] z'"
add_edge_def: fixes v :: "'b"   and v' :: "'b"   and G :: "'b list_graph" shows "FiniteListGraph.add_edge v v' G = FiniteListGraph.add_node v (FiniteListGraph.add_node v' G)\<lparr>edgesL := if (v, v') \<in> set (edgesL G) then edgesL G else (v, v') # edgesL G\<rparr>"
update_simps_6: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "ip \<in> kD rt"   and "\<pi>\<^sub>2 r \<le> sqn rt ip"   and "\<pi>\<^sub>3 r = kno"   and "sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val" shows "update rt ip r = rt(ip \<mapsto> addpre (the (rt ip)) (\<pi>\<^sub>7 r))"
map_simp_ide: fixes a :: "'c" assumes "C.ide a" shows "G\<epsilon>o\<eta>G.map a = (G \<circ> \<epsilon>) a \<cdot>\<^sub>D (\<eta> \<circ> G) a"
eval_fps_const: fixes c :: "'a"   and z :: "'a" shows "eval_fps (fps_const c) z = c"
compactin_sing: fixes X :: "'c topology"   and a :: "'c" shows "compactin X {a} = (a \<in> topspace X)"
nat_less_iff: fixes w :: "int"   and m :: "nat" assumes "0 \<le> w" shows "(nat w < m) = (w < int m)"
eq: fixes A :: "'a \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure P (Pi\<^sub>E I A) = emeasure Q (Pi\<^sub>E I A)"
set_mono_prefix: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys" shows "set xs \<subseteq> set ys"
diagseq_mono: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and n :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "subseqs.diagseq P n < subseqs.diagseq P (Suc n)"
wtPB_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "Ik.wtPB \<Phi> \<equiv> Ball \<Phi> Ik.wtC"
Star: shows "rel v r = \<Union> (word_rel v ` lang r)"
simps: fixes ps :: "nat list"   and f :: "form" shows "fv_schema (ps, f) = fv f - set ps"
Tpows'_def: shows "Tpows' = map (VectorSpaceEnd.endpow V T) [0..<Suc (degree p)]"
prems: shows "tMap f (tp \<leadsto> s) = tMap f' t'"
open_UNION_cbox: fixes M :: "'b set" assumes "open M" shows "M = (\<Union>f\<in>{f \<in> Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. cbox (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) \<subseteq> M}. cbox (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i))"
self_domain_iff_ide: fixes a :: "'a" shows "(a \<in> H.domains a) = H.ide a"
path_determines_loc: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and l3 :: "'a" assumes "graph weights"   and "graph.path weights l1 l2 xs"   and "graph.path weights l1 l3 xs" shows "l2 = l3"
fused_wiring\<^sub>w_def: shows "fused_wiring\<^sub>w \<equiv> parallel_wiring\<^sub>w |\<^sub>w parallel_wiring\<^sub>w \<circ>\<^sub>w parallel_wiring\<^sub>w"
continuous_map_cases: fixes X :: "'b topology"   and P :: "'b \<Rightarrow> bool"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_map (subtopology X (X closure_of {x. P x})) Y f"   and "continuous_map (subtopology X (X closure_of {x. \<not> P x})) Y g"   and "\<And>x. x \<in> X frontier_of {x. P x} \<Longrightarrow> f x = g x" shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"
x_2: shows "x \<le> (p - 1) div 2"
has_derivative_def: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f has_derivative f') F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (netlimit F) - f' (y - netlimit F)) /\<^sub>R norm (y - netlimit F)) \<longlongrightarrow> (0::'c)) F)"
diff_funI: fixes f :: "'a \<Rightarrow> 'c" assumes "\<And>x. x \<in> carrier \<Longrightarrow> \<exists>c1\<in>atlas. x \<in> domain c1 \<and> k-smooth_on (codomain c1) (f \<circ> inv_chart c1)" shows "diff_fun k charts f"
aj: shows "Inv X (swap z Y) = (if Suc z < length X then if Y ! z < Y ! Suc z in X then Inv X Y \<union> {(Y ! z, Y ! Suc z)} else Inv X Y - {(Y ! Suc z, Y ! z)} else Inv X Y)"
f3: shows "low_equal (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1)))))) (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2))))))"
msgrel_imp_eqv_freeright_aux: fixes U :: "freemsg" shows "freeright U \<sim> freeright U"
sgn_cblinfun_def: fixes x :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "sgn x = complex_of_real (inverse (norm x)) *\<^sub>C x"
range_left: shows "(ix (child b dir d) d - 1) * 2 ^ (lv p d - lv (child b dir d) d) < ix p d"
l_set_shadow_root_get_shadow_root\<^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 set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_shadow_root_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
irreducible_const_poly_iff: fixes c :: "'a" shows "irreducible [:c:] = irreducible c"
closure_iff_nhds_not_empty: fixes x :: "'c"   and X :: "'c set" shows "(x \<in> closure X) = (\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
circ_extra_circ: fixes y :: "'a"   and x :: "'a" shows "(y * x\<^sup>\<star>)\<^sup>\<star> = (y\<^sup>+ * x\<^sup>\<star>)\<^sup>\<star>"
fifthh: assumes "0 < b" shows "\<exists>x. \<forall>y<x. b * y + c < 0"
h_def: shows "h = subgroup_indicator G {\<one>} x"
CFG_axioms: shows "CFG sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main"
add_mult_distrib_mat_vec: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and v :: "'c vec" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "(A + B) *\<^sub>v v = A *\<^sub>v v + B *\<^sub>v v"
simps: fixes S :: "'a set"   and I :: "'a set" shows "init (Kripke S I) = I"
is_predAtm_def: fixes atom :: "'a atom" shows "is_predAtm atom = (\<exists>x11 x12. atom = predAtm x11 x12)"
range_from_nat_into_subset: fixes A :: "'c set" assumes "A \<noteq> {}" shows "range (from_nat_into A) \<subseteq> A"
older_seniors_senior_simp: fixes x :: "nat"   and n :: "nat" shows "older_seniors (senior x n) n = older_seniors x n"
elims: fixes x :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and xa :: "'Var\<^sub>A"   and xb :: "'Val"   and y :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "abs.lc_set_var x xa xb = y" obtains c :: "'a \<times> (Mode \<Rightarrow> 'b set)" and mem :: "'Var\<^sub>A \<Rightarrow> 'Val" and xc :: "'Var\<^sub>A" and v :: "'Val" where "x = (c, mem)" and "xa = xc" and "xb = v" and "y = (c, mem(xc := v))"
hf_of_char_def: fixes x :: "char" shows "hf_of x \<equiv> hf_of (of_char x)"
NN_def2: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "yy \<notin> FvarsT t1" shows "NN t1 t2 = subst (subst N t1 xx) t2 yy"
space_measure_pmf: fixes p :: "'a pmf" shows "space (measure_pmf p) = UNIV"
evalHeap_reorder_head_append: fixes x :: "'d"   and \<Gamma> :: "('d \<times> 'e) list"   and e :: "'e"   and \<Delta> :: "('d \<times> 'e) list"   and eval :: "'e \<Rightarrow> 'f" assumes "x \<notin> domA \<Gamma>" shows "evalHeap ((x, e) # \<Gamma> @ \<Delta>) eval = evalHeap (\<Gamma> @ (x, e) # \<Delta>) eval"
winning_strategy_updates: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a"   and w :: "'a" assumes "strategy p \<sigma>"   and "winning_strategy p \<sigma> v0"   and "v \<notin> winning_region p"   and "v \<rightarrow> w" shows "winning_strategy p (\<sigma>(v := w)) v0"
hm_1: shows "homeomorphic_map X (subtopology X' (f ` topspace X)) f"
has_subprob_density_def: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "has_subprob_density M N f \<equiv> has_density M N f \<and> subprob_space M"
contract: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "(F, F, \<Gamma> \<Rightarrow> \<Delta> \<longrightarrow> F, \<Gamma> \<Rightarrow> \<Delta>) \<and> (\<Gamma> \<Rightarrow> F, F, \<Delta> \<longrightarrow> \<Gamma> \<Rightarrow> F, \<Delta>)"
lens_update_def: fixes X :: "'e \<Longrightarrow> 'd"   and f :: "'e \<Rightarrow> 'e"   and \<sigma> :: "'d" shows "update\<^bsub>X\<^esub> f \<sigma> = put\<^bsub>X\<^esub> \<sigma> (f (get\<^bsub>X\<^esub> \<sigma>))"
INF_apply: fixes f :: "'m \<Rightarrow> 'l \<Rightarrow> 'k"   and A :: "'m set"   and x :: "'l" shows "Inf (f ` A) x = (INF y\<in>A. f y x)"
simps: fixes a :: "'b"   and p :: "'b poly" shows "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons (0::'b) (pderiv p))"
subgame_path_conforms_with_strategy: fixes V' :: "'a set"   and p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "V' \<subseteq> V\<^bsub>G'\<^esub>"   and "G'.path_conforms_with_strategy p P \<sigma>"   and "lset P \<subseteq> V'" shows "ParityGame.path_conforms_with_strategy (G'.subgame V') p P \<sigma>"
below_monic_def: fixes f :: "'c \<Rightarrow> 'd" shows "below_monic f \<equiv> \<forall>x y. f x \<sqsubseteq> f y \<longrightarrow> x \<sqsubseteq> y"
interleaving_Nil_iff2: fixes xs :: "'a list"   and zs :: "'a list" shows "interleavings xs [] zs = (xs = zs)"
map_snd: fixes list :: "('b \<times> 'a) list" shows "map (\<lambda>(x, y). y) list = map snd list"