hcju/isabelleps huggingface dataset:
intros_4: fixes c1 :: "'a com"   and s :: "'a option"   and s'' :: "'a option"   and c2 :: "'a com"   and s' :: "'a option" assumes "Sem c1 s s''"   and "Sem c2 s'' s'" shows "Sem (Seq c1 c2) s s'"
assms_2: shows "f \<in> space\<^sub>N (\<LL> (ennreal p) M)"
distinct_5: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com" shows "com.SKIP \<noteq> IF x41 THEN x42 ELSE x43"
apath_Nil_iff: fixes u :: "'a"   and v :: "'a" shows "S.apath u [] v = (u = v \<and> u \<in> pverts sG)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<^bold>* (a \<^bold>* c) = a \<^bold>* (b \<^bold>* c)"
IH: assumes "\<not> no_matching_Goto \<gamma> p rs"   and "rs \<noteq> []" shows "\<exists>rs1 m chain rs2. rs = rs1 @ Rule m (Goto chain) # rs2 \<and> no_matching_Goto \<gamma> p rs1 \<and> matches \<gamma> m p"
simps_2: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and d :: "nat"   and l :: "nat"   and rA :: "'a \<times> 'a mat" shows "sub3 sel_fun mf d (Suc l) rA = sub2 sel_fun mf d l (sub3 sel_fun mf d l rA)"
Cd_def: fixes d :: "'b" shows "Singleton_for_d.Cd d \<equiv> set_option \<circ> MaxR.MaxR_opt (Pd_singletons_for_ds X {Xd x} d)"
fps_XDp_def: fixes c :: "'a"   and a :: "'a fps" shows "fps_XDp c a = fps_XD a + fps_const c * a"
map_add_lr_disj: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>0' :: "'c \<Rightarrow> 'd option"   and h\<^sub>1' :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>0' ++ h\<^sub>1'"   and "h\<^sub>1 \<bottom> h\<^sub>1'" shows "dom h\<^sub>1 \<subseteq> dom h\<^sub>0'"
sup1_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion>1 (b \<squnion>1 c) = a \<squnion>1 b \<squnion>1 c"
assms: shows "\<not> SN_on (E\<^sup>* O R O E\<^sup>*) {s}"
Mutrec: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and n :: "nat"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "finite G"   and "card G = n"   and "verified G"   and "A \<in> G" shows "{} \<rhd> Call : A"
T6: shows "C InAngle A B Q \<and> A B Q CongA D E F"
Solve: fixes A :: "'b list set"   and X :: "'b list set" assumes "finite (UNIV // \<approx>A)"   and "X \<in> UNIV // \<approx>A" shows "\<exists>rhs. Solve X (Init (UNIV // \<approx>A)) = {(X, rhs)} \<and> invariant {(X, rhs)}"
zero_eq_1_divide_iff: fixes a :: "'a" shows "((0::'a) = (1::'a) / a) = (a = (0::'a))"
simps_2: fixes f :: "ref \<Rightarrow> ref"   and l :: "dag"   and a :: "ref"   and r :: "dag" shows "dag_map f (Node l a r) = Node (dag_map f l) (f a) (dag_map f r)"
subseq_append_le_same_iff: fixes xs :: "'b list"   and ys :: "'b list" shows "subseq (xs @ ys) ys = (xs = [])"
hyps: fixes A :: "'b set" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> local.subspace s" shows "local.independent (\<Union> (some_basis ` set ss))"
observe_execution_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and t :: "transition"   and s' :: "nat"   and p :: "value option list"   and r' :: "nat \<Rightarrow>f value option"   and es :: "(String.literal \<times> value list) list"   and obs :: "value option list list" assumes "step e s r (fst h) (snd h) = Some (t, s', p, r')"   and "observe_execution e s' r' es = obs" shows "observe_execution e s r (h # es) = p # obs"
select_def: fixes f :: "'d \<Rightarrow> 'c option"   and S :: "'d set" shows "select f S = {z |z. \<exists>x\<in>S. f x = Some z}"
cast_document_ptr_not_node_ptr_1: fixes document_ptr :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr" shows "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^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 document_ptr \<noteq> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr"
trancl_eq: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "wf_sub_rel subs" shows "subs\<^sup>+ = subs"
4: shows "(f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T)\<^sup>\<star> = (f \<sqinter> q)\<^sup>T\<^sup>\<star> * (f \<sqinter> - q)\<^sup>\<star>"
divide_rat_def: fixes q :: "rat"   and r :: "rat" shows "q div r = q * inverse r"
eq_implies: fixes x :: "'a"   and y :: "'a" assumes "x = y"   and "x \<in> X" shows "x \<sqsubseteq> y"
simps_1: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" shows "inferred_clause_sets R S 0 = simplify S"
widen_Class_Iface_eq: fixes G :: "prog"   and C :: "qtname"   and I :: "qtname" shows "G\<turnstile>Class C\<preceq>Iface I = G\<turnstile>C\<leadsto>I"
igConsIPresIGWlsSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igConsIPresIGWlsSTR MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWlsSTR MOD \<and> igOpIPresIGWlsSTR MOD"
assms_2: shows "(ea, eb) \<in> set (routing_ipassmt_wi tbl)"
distinct_23: shows "ToLowerCaseOp \<noteq> CharactersOp"
card_greaterThanAtMost_int: fixes l :: "int"   and u :: "int" shows "card {l<..u} = nat (u - l)"
fls_deriv_subdegree': fixes f :: "'a fls" assumes "of_int (fls_subdegree f) * fls_nth f (fls_subdegree f) \<noteq> (0::'a)" shows "fls_subdegree (fls_deriv f) = fls_subdegree f - 1"
finite_subst_domain_subst: fixes x :: "'c"   and y :: "('d, 'c) Term.term" shows "finite (subst_domain (subst x y))"
2: fixes g0 :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g0 \<in> G" shows "\<not> is_sig_red (\<preceq>\<^sub>t) (=) (G - {g0}) g0"
tickFree_rev: fixes t :: "'a event list" shows "tickFree (rev t) = tickFree t"
P_map: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) G"   and l1 :: "'b \<Rightarrow> 'j"   and l2 :: "'d \<Rightarrow> 'k"   and co1 :: "'e \<Rightarrow> 'l"   and co2 :: "'f \<Rightarrow> 'm"   and contra1 :: "'n \<Rightarrow> 'g"   and contra2 :: "'o \<Rightarrow> 'h" assumes "P x" shows "P (map_G l1 l2 co1 co2 contra1 contra2 x)"
nonzero_hnorm_inverse: fixes a :: "'a star" assumes "a \<noteq> 0" shows "hnorm (inverse a) = inverse (hnorm a)"
exec_elim_cases_13: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Throw,s\<rangle> \<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Normal sc__" and "t = Abrupt sc__"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
wf_ineq_append'': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t V (S @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
fmdrop_set_fmdom: fixes A :: "('c, 'd) fmap" shows "fmdrop_set (fmdom' A) A = {$$}"
stable_Join_ensures1: fixes F :: "'c program"   and A :: "'c set"   and G :: "'c program"   and B :: "'c set" assumes "F \<in> stable A"   and "G \<in> A ensures B" shows "F \<squnion> G \<in> A ensures B"
moebius_mu_power': fixes a :: "nat"   and n :: "nat" shows "moebius_mu (a ^ n) = (if a = 1 \<or> n = 0 then 1::'a else if n = 1 then moebius_mu a else (0::'a))"
assms_2: shows "\<forall>asset\<in>support_set p. prices Mkt asset n \<in> borel_measurable (F n)"
un_z: shows "hom_boundary (int n) (nsphere n) (equator n) un = z"
Un_Int_eq_1: fixes S :: "'a set"   and T :: "'a set" shows "(S \<union> T) \<inter> S = S"
at_within_empty: fixes "open" :: "'c set \<Rightarrow> bool"   and a :: "'c" assumes "class.topological_space open" shows "topological_space.at_within open a {} = bot"
upd_space: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "i < n" shows "upd i < n"
flat_ordI: fixes x :: "'b"   and a :: "'b"   and y :: "'b" assumes "x \<noteq> a \<Longrightarrow> x = y" shows "flat_ord a x y"
wf_until_aux_Nil: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi>' :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi>' :: "'a MFOTL.formula" shows "wf_until_aux \<sigma> n R pos \<phi>' I \<psi>' [] 0"
self_append_conv: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs = xs @ ys) = (ys = [])"
seq_eq_bottom_iff: fixes x :: "'b"   and y :: "'a" shows "(seq\<cdot>x\<cdot>y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
WT: shows "exception_\<I> (\<I> \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g \<A> \<surd>"
simps: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "ldropWhile P xs = (case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then ldropWhile P xs' else xs)"
get_ancestors_parent_child_rel: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_ancestors child \<rightarrow>\<^sub>r ancestors"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "((ptr, child) \<in> (Shadow_DOM.parent_child_rel h)\<^sup>*) = (ptr \<in> set ancestors)"
assms_6: shows "\<bar>x\<bar> \<le> r"
whileAnnoGFix_def: fixes gs :: "('f \<times> 'd set) list"   and b :: "'d set"   and I :: "'g \<Rightarrow> 'd set"   and V :: "'g \<Rightarrow> ('d \<times> 'd) set"   and c :: "'g \<Rightarrow> ('d, 'e, 'f) com" shows "whileAnnoGFix gs b I V c = while gs b (c undefined)"
closed_seqs_memE: fixes s :: "nat \<Rightarrow> 'a"   and R :: "('a, 'b) ring_scheme"   and k :: "nat" assumes "s \<in> carrier R\<^bsup>\<omega>\<^esup>" shows "s k \<in> carrier R"
freeFreshDerivative: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)"   and "x \<sharp> \<alpha>"   and "x \<sharp> P" shows "x \<sharp> P'"
aext_upred_eq: fixes e :: "('d, 'c) uexpr"   and f :: "('d, 'c) uexpr"   and a :: "'c \<Longrightarrow> 'b" shows "(e =\<^sub>u f) \<oplus>\<^sub>p a = (e \<oplus>\<^sub>p a =\<^sub>u f \<oplus>\<^sub>p a)"
subcategory_axioms: shows "subcategory (\<cdot>) (\<lambda>f. \<F>C.arr f \<and> (\<F>C.ide (\<F>C.dom f) \<and> Diag (\<F>C.DOM (\<F>C.dom f))) \<and> \<F>C.ide (\<F>C.cod f) \<and> Diag (\<F>C.DOM (\<F>C.cod f)))"
simps_1: fixes T :: "ty" shows "ground_type (T\<lfloor>\<rceil>) = ground_type T"
map_props: shows "distinct (map \<alpha> l0) \<and> sorted_wrt Ra (map \<alpha> l0)"
cosine_law_triangle: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(dist b c)\<^sup>2 = (dist a b)\<^sup>2 + (dist a c)\<^sup>2 - 2 * dist a b * dist a c * cos (angle b a c)"
integral_mult_left_zero: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'b"   and c :: "'b" shows "LINT x|M. f x * c = integral\<^sup>L M f * c"
halfspaces_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>halfspaces_rel = \<langle>\<langle>R\<rangle>sctn_rel\<rangle>list_set_rel O br below_halfspaces top"
Sup_upto_llist_def: fixes Xs :: "'b set llist"   and j :: "enat" shows "Sup_upto_llist Xs j = \<Union> (lnth Xs ` {i. enat i < llength Xs \<and> enat i \<le> j})"
visit_subset_dfs: fixes succs :: "'c \<Rightarrow> 'c list"   and is_node :: "'c \<Rightarrow> bool"   and invariant :: "'d \<Rightarrow> bool"   and ins :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empt :: "'d"   and S :: "'d"   and xs :: "'c list"   and y :: "'c" assumes "DFS succs is_node invariant ins memb empt"   and "invariant S"   and "list_all is_node xs"   and "is_node y"   and "memb y S" shows "memb y (gen_dfs succs ins memb S xs)"
filter_code_def: fixes l :: "'b list"   and f :: "'b \<Rightarrow> bool" shows "filter_code l f = filter f l"
S2Hist: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> $S2 rmhist p \<longrightarrow> unchanged (rmhist!p)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "P F"   and "P T"   and "\<And>v. P (Le v)"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (And v va)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
first_edge_first: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and w' :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w"   and "(v, w') \<in> H.edges_of_walk xs" shows "H.first_edge_of_walk xs = (v, w')"
rule_\<alpha>_type: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and N :: "symbol"   and \<alpha> :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "(N, \<alpha>) \<in> \<RR>" shows "CFG.is_sentence \<NN> \<TT> \<alpha>"
maximal_independent_subset: fixes V :: "('c, 'd) vec set" obtains B :: "('c, 'd) vec set" where "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B"
add_less_cancel_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus c a) (plus c b) = less a b"
nth_prime_partition: fixes x :: "nat" assumes "2 \<le> x" shows "\<exists>k. x \<in> {nth_prime k..<nth_prime (Suc k)}"
many_strongly_connected_iff_6_eq: fixes x :: "'a" shows "many_strongly_connected x = (x\<^sup>T ; x\<^sup>\<star> = x\<^sup>+)"
ResB: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "y \<noteq> a"   and "y \<noteq> x" shows "<\<nu>y>P \<longmapsto> a<\<nu>x> \<prec> (<\<nu>y>P')"
elims: fixes g :: "('e, 'f) generator"   and x :: "'f"   and y :: "'e list" assumes "unstream g x = y" obtains s :: "'f" where "x = s" and "y = (case generator g s of Done \<Rightarrow> [] | Skip x \<Rightarrow> unstream g x | Yield x s' \<Rightarrow> x # unstream g s')"
simps_3: fixes x :: "'b"   and t :: "'b tree" shows "exec (Insert x) [t] = Splay_Tree.insert x t"
flowpipe_nonneg_1: fixes X0 :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and hl :: "real"   and hu :: "real"   and CX :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and X1 :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set" assumes "flowpipe X0 hl hu CX X1" shows "0 \<le> hl"
norm_memb_eq_rep: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "arr f"   and "t \<in> f" shows "\<^bold>\<parallel>t\<^bold>\<parallel> = rep f"
ap_spmf_interchange: fixes u :: "('e \<Rightarrow> 'd) spmf"   and x :: "'e" shows "u \<diamondop> return_spmf x = return_spmf (\<lambda>f. f x) \<diamondop> u"
less_eq_scene_def: fixes X :: "'c scene"   and Y :: "'c scene" shows "X \<subseteq>\<^sub>S Y = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on Y \<oplus>\<^sub>S s\<^sub>3 on X = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 \<oplus>\<^sub>S s\<^sub>3 on X on Y)"
edom_mono: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "x \<sqsubseteq> y" shows "edom x \<subseteq> edom y"
UNIV_accent: shows "UNIV = {Acute, Grave, Macron}"
polyadd: fixes bs :: "'b list"   and p :: "'b poly"   and q :: "'b poly" shows "Ipoly bs (p +\<^sub>p q) = Ipoly bs p + Ipoly bs q"
Comp_assoc: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "concrete_category Obj Hom Id Comp"   and "A \<in> Obj"   and "B \<in> Obj"   and "C \<in> Obj"   and "D \<in> Obj"   and "f \<in> Hom A B"   and "g \<in> Hom B C"   and "h \<in> Hom C D" shows "Comp D C A h (Comp C B A g f) = Comp D B A (Comp D C B h g) f"
defined_args_valid': fixes x :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and y :: "'a state \<times> 'a state \<Rightarrow> Set('b)" shows "\<delta> (x \<doteq> y) = (\<upsilon> x and \<upsilon> y)"
ok_SKIP1: fixes F :: "'c program" shows "\<bottom> ok F"
\<tau>move2_InstanceOfD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and T :: "ty"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (e instanceof T) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
chamber_system_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "f \<turnstile> ChamberComplex.chamber_system X = ChamberComplex.chamber_system Y \<inter> (f \<turnstile> X)"
order_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "order r A = ((\<forall>x\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> x \<longrightarrow> x = y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> z))"
i_set_mult_def: shows "i_set_mult \<equiv> \<lambda>k. {x. i_set_multp k x}"
redundancy_implies_partial_saturation: fixes S1 :: "'at Literal set set"   and A :: "'at"   and S2 :: "'at Literal set set" assumes "partial_saturation S1 A S1"   and "S2 \<subseteq> S1"   and "all_fulfill (\<lambda>x. redundant x S2) S1" shows "partial_saturation S2 A S2"
Path_append: fixes p :: "ref"   and f :: "ref \<Rightarrow> ref"   and q :: "ref"   and as :: "ref list"   and bs :: "ref list" shows "Path p f q (as @ bs) = (\<exists>y. Path p f y as \<and> Path y f q bs)"
lens_compat_def: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f" shows "(X ##\<^sub>L Y) = (\<forall>s\<^sub>1 s\<^sub>2. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2)"
distinct_89: fixes x4 :: "val"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "Val x4 \<noteq> Call x101 x102 x103 x104"
Wrap_def: fixes x :: "'e" shows "Wrap x \<equiv> x"
cf_def: shows "cf \<equiv> \<lambda>(u, v). if (u, v) \<in> E then c (u, v) - f (u, v) else if (v, u) \<in> E then f (v, u) else (0::'a)"
comp_monotone': fixes w :: "'b list"   and r :: "'b list"   and w' :: "'b list" assumes "w \<bowtie> r" shows "w \<and>\<^sub>p w' \<bowtie> r"
m1: shows "(if 1 = 0 then mat_extension dims vars2 M0 else if 1 = 1 then mat_extension dims vars2 M1 else undefined) = exM1"
cblinfun_cspan_UNIV: fixes basisA :: "'a set"   and basisB :: "'b set"   and basis :: "('a \<Rightarrow>\<^sub>C\<^sub>L 'b) set" assumes "cspan basisA = UNIV"   and "cspan basisB = UNIV"   and "\<And>a b. \<lbrakk>a \<in> basisA; b \<in> basisB\<rbrakk> \<Longrightarrow> \<exists>F\<in>basis. \<forall>a'\<in>basisA. F *\<^sub>V a' = (if a' = a then b else (0::'b))" shows "cspan basis = UNIV"
div_mult_div_if_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a"   and d :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b a"   and "dvd.dvd times d c" shows "times (divide a b) (divide c d) = divide (times a c) (times b d)"
lincomb_insert2: fixes S :: "complex vec set"   and a :: "complex vec \<Rightarrow> complex"   and v :: "complex vec" assumes "finite S"   and "S \<subseteq> carrier_vec n"   and "a \<in> insert v S \<rightarrow> UNIV"   and "v \<notin> S"   and "v \<in> carrier_vec n" shows "lincomb a (insert v S) = a v \<cdot>\<^sub>v v + lincomb a S"
depth_snd: fixes P :: "'b trm" shows "depth P < depth (Snd P)"
A: shows "xs \<cong> {ipurge_tr_rev_aux I D U xs, rev (ipurge_tr_aux (I\<inverse>) D U (rev xs)), \<lambda>y ys. \<exists>v\<in>sinks_aux (I\<inverse>) D U (rev ys). (D y, v) \<in> I}"
path_trans_2: fixes s :: "'s"   and l :: "'l"   and sh :: "'s"   and p :: "'l list"   and s' :: "'s" assumes "step s l sh"   and "local.path sh p s'" shows "local.path s (l # p) s'"
ode_d_raw_def: fixes n :: "nat"   and x :: "'c"   and dn :: "'c"   and d :: "'c" shows "ode_d_raw n x dn d = einterpret (ode_d_expr DIM('c) n) (list_of_eucl x @ list_of_eucl dn @ list_of_eucl d)"
cases: fixes a1 :: "bexp"   and a2 :: "char list \<Rightarrow> val"   and a3 :: "bool" assumes "tbval a1 a2 a3" obtains v :: "bool" and s :: "char list \<Rightarrow> val" where "a1 = Bc v" and "a2 = sa__" and "a3 = v"   | b :: "bexp" and s :: "char list \<Rightarrow> val" and bv :: "bool" where "a1 = bexp.Not b" and "a2 = sb__" and "a3 = (\<not> bv)" and "tbval b sb__ bv"   | b1 :: "bexp" and s :: "char list \<Rightarrow> val" and bv1 :: "bool" and b2 :: "bexp" and bv2 :: "bool" where "a1 = And b1 b2" and "a2 = sc__" and "a3 = (bv1 \<and> bv2)" and "tbval b1 sc__ bv1" and "tbval b2 sc__ bv2"   | a1a :: "aexp" and s :: "char list \<Rightarrow> val" and i1 :: "int" and a2a :: "aexp" and i2 :: "int" where "a1 = Less a1a__ a2a__" and "a2 = sd__" and "a3 = (i1 < i2)" and "taval a1a__ sd__ (Iv i1)" and "taval a2a__ sd__ (Iv i2)"   | a1a :: "aexp" and s :: "char list \<Rightarrow> val" and r1 :: "real" and a2a :: "aexp" and r2 :: "real" where "a1 = Less a1a a2a" and "a2 = s" and "a3 = (r1 < r2)" and "taval a1a s (Rv r1)" and "taval a2a s (Rv r2)"
suffixes_eq_snoc: fixes ys :: "'b list"   and xs :: "'b list list"   and x :: "'b list" shows "(suffixes ys = xs \<cdot> [x]) = ((ys = \<epsilon> \<and> xs = \<epsilon> \<or> (\<exists>z zs. ys = z # zs \<and> xs = suffixes zs)) \<and> x = ys)"
relpow_Suc_D2: fixes x :: "'a"   and z :: "'a"   and n :: "nat"   and R :: "('a \<times> 'a) set" assumes "(x, z) \<in> R ^^ Suc n" shows "\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n"
right_total_UNIV_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "right_total A" shows "rel_set A (Collect (Domainp A)) UNIV"
init_pnet_opnet: fixes p :: "net_tree"   and s :: "'s net_state" assumes "wf_net_tree p"   and "s \<in> init (pnet np p)" shows "netgmap sr s \<in> netmask (net_tree_ips p) ` init (opnet onp p)"