hcju/isabelleps huggingface dataset:
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = inf a b" shows "less_eq a b"
AndL: fixes F :: "'a formula"   and G :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "F, G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<and> G, \<Gamma> \<Rightarrow> \<Delta>"
pt_var_inst: shows "pt TYPE('a) TYPE(NominalSequents.var)"
actual_validity_def: shows "actual_validity \<equiv> \<lambda>\<phi>. [\<phi> in dw]"
object_ptr_kinds_preserved: fixes SW :: "((_) heap, 'z, 'aa) prog set"   and setter :: "((_) heap, 'z, 'ab) prog"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w object_ptr. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
induce_eq_iff_induced: fixes H :: "('a, 'b) pre_digraph" assumes "induced_subgraph H G" shows "G \<restriction> verts H = H"
sel_1: fixes x1 :: "'a simple_match"   and x2 :: "simple_action" shows "match_sel (SimpleRule x1 x2) = x1"
nat_pow_pow: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier R" shows "[m] \<cdot> ([n] \<cdot> x) = [(n * m)] \<cdot> x"
monom_mult_one_left: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" shows "punit.monom_mult (1::'b) (0::'a) p = p"
\<phi>_F_char: fixes g :: "'d"   and y' :: "'d"   and y :: "'d" assumes "\<guillemotleft>g : y' \<rightarrow>\<^sub>D y\<guillemotright>" shows "\<phi> y' (F g) = \<eta>o y \<cdot>\<^sub>D g"
eq_refl: fixes y :: "'a"   and x :: "'a" assumes "y \<in> U"   and "x = y" shows "x \<le>\<^sub>o\<^sub>w y"
cases_scheme: fixes r :: "'a view_scheme" obtains ext :: "real_int" and lan :: "nat_int" and own :: "cars" and more :: "'a" where "r = \<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr>"
fbox_def: fixes x :: "'a"   and y :: "'a" shows "antidomain_semiringl.fbox antirange_op (\<lambda>x y. y \<cdot> x) x y = ar (ar y \<cdot> x)"
degree_times_X: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "p \<noteq> \<zero>\<^bsub>P\<^esub>" shows "deg R (X_poly R \<otimes>\<^bsub>P\<^esub> p) = deg R p + 1"
hd_dropWhile: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" assumes "dropWhile P xs \<noteq> []" shows "\<not> P (hd (dropWhile P xs))"
emeasure_pair_measure_Times: fixes A :: "'b set"   and N :: "'b measure"   and B :: "'c set" assumes "A \<in> sets N"   and "B \<in> sets lborel" shows "emeasure (N \<Otimes>\<^sub>M lborel) (A \<times> B) = emeasure N A * emeasure lborel B"
limp_rel_def: fixes T :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" shows "limp_rel T S = {(y, x) |y x. \<forall>z. (x, z) \<in> S \<longrightarrow> (y, z) \<in> T}"
mat_empty_row_length: fixes nr :: "nat"   and nc :: "nat"   and M :: "'b list list" assumes "mat nr nc M"   and "M = []" shows "mat 0 nc M"
Trivial_fup_Aexp_below_fv: fixes e :: "exp"   and a :: "Arity\<^sub>\<bottom>" shows "Aexp_bot_syn a e \<sqsubseteq> (\<lambda>x. up\<cdot>0) f|` fv e"
f_shrink_last_length: fixes xs :: "'b list"   and k :: "nat" shows "length (xs \<div>\<^sub>l k) = length xs div k"
Reals_of_int: fixes z :: "int" shows "of_int z \<in> \<real>"
Inf_pres_comp: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c" assumes "Inf_pres f"   and "Inf_pres g" shows "Inf_pres (f \<circ> g)"
is_arrD_3: fixes \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
weak_map_of_SomeI: fixes k :: "'a"   and x :: "'b"   and l :: "('a \<times> 'b) list" assumes "(k, x) \<in> set l" shows "\<exists>x. map_of l k = \<lfloor>x\<rfloor>"
one_poly_def: shows "one_poly \<equiv> [(1, 1::'d)]"
fpr_sub_path': fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and path' :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list" shows "find_path_rec_f P ctx cl path' = Some path \<longrightarrow> (\<forall>fqn ctxcld path'' path_fqn. cl = cl_fqn fqn \<and> find_cld_f P ctx fqn = Some ctxcld \<and> find_path_rec_f P (fst ctxcld) (superclass_name_f (snd ctxcld)) path'' = Some path_fqn \<longrightarrow> (\<exists>path'''. path_fqn = path'' @ path''' \<and> path = path' @ ctxcld # path'''))"
dep3_event_permute_1: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and R :: "'b set"   and S :: "'b set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.dep3_event \<P> Q R S x" shows "MinkowskiPrimitive.dep3_event \<P> Q S R x"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_unlabel_steps_iff_1: fixes t :: "('e, 'f) Term.term"   and A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "(send\<langle>t\<rangle> \<in> set (unlabel A)) = (receive\<langle>t\<rangle> \<in> set (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)))"
reals_complete: fixes S :: "real set" assumes "\<exists>X. X \<in> S"   and "\<exists>Y. isUb UNIV S Y" shows "\<exists>t. isLub UNIV S t"
mem_interior_convex_shrink: fixes S :: "'a set"   and c :: "'a"   and x :: "'a"   and e :: "real" assumes "convex S"   and "c \<in> interior S"   and "x \<in> S"   and "0 < e"   and "e \<le> 1" shows "x - e *\<^sub>R (x - c) \<in> interior S"
assm: shows "v=v1--v2 \<and> v2=v3--v4"
DeMoivre: fixes n :: "nat" shows "cis pi ^ n = cis (real n * pi)"
PB: shows "four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2 * four_block_mat B1 UR LL B2 \<in> carrier_mat (n + m) (n + m)"
group_closure_mult_left: fixes s :: "int"   and S :: "int set"   and t :: "int" assumes "s \<in> group_closure S" shows "s * t \<in> group_closure S"
sum_list_nonneg_eq_0_iff: fixes xs :: "'a multiset list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> {#} \<subseteq># x" shows "(monoid_add.sum_list (+) {#} xs = {#}) = (\<forall>x\<in>set xs. x = {#})"
Zassenhaus_1: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and H1 :: "'c set"   and K :: "'c set"   and K1 :: "'c set" assumes "Group.group G"   and "subgroup H G"   and "H1 \<lhd> G\<lparr>carrier := H\<rparr>"   and "subgroup K G"   and "K1 \<lhd> G\<lparr>carrier := K\<rparr>" shows "G\<lparr>carrier := H1 <#>\<^bsub>G\<^esub> H \<inter> K\<rparr> Mod (H1 <#>\<^bsub>G\<^esub> H \<inter> K1) \<cong> G\<lparr>carrier := H \<inter> K\<rparr> Mod (H1 \<inter> K <#>\<^bsub>G\<^esub> H \<inter> K1)"
pasz: shows "path_image \<gamma> \<subseteq> S - {z}"
IH: fixes V :: "'a set" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t (fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)) (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
fmdom'_fmrestrict_set_fmadd: fixes X :: "'d set"   and f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" shows "fmdom' (fmrestrict_set X (f ++\<^sub>f g)) = X \<inter> (fmdom' f \<union> fmdom' g)"
targetnodes_is_tl_sourcenodes_App_n': fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'"   and "as \<noteq> []" shows "targetnodes as = tl (sourcenodes as) @ [n']"
rawpsubst_dsj: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubst (dsj \<phi>1 \<phi>2) txs = dsj (rawpsubst \<phi>1 txs) (rawpsubst \<phi>2 txs)"
fmapET_def: shows "fmapET = (\<Lambda> f m. bindET\<cdot>m\<cdot>(\<Lambda> x. unitET\<cdot>(f\<cdot>x)))"
f_accessibleI_1: fixes q0 :: "'a"   and Q0 :: "'a set"   and \<delta> :: "('a, 'b) ta_rule set" assumes "q0 \<in> Q0" shows "q0 \<in> f_accessible \<delta> Q0"
distinct_17: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Skip \<noteq> x41 ;; x42"
quasinorm_sum_1: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm"   and f :: "'c" shows "eNorm (N1 +\<^sub>N N2) f = \<Sqinter> {eNorm N1 f1 + eNorm N2 f2 |f1 f2. f = f1 + f2}"
similar_mat_wit_trans: fixes A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat"   and C :: "'b mat"   and P' :: "'b mat"   and Q' :: "'b mat" assumes "similar_mat_wit A B P Q"   and "similar_mat_wit B C P' Q'" shows "similar_mat_wit A C (P * P') (Q' * Q)"
producesD: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset"   and A :: "'a" assumes "production N C = {A}" shows "C \<in> N \<and> C \<noteq> {#} \<and> Pos A = Max_mset C \<and> \<not> interp N C \<Turnstile> C \<and> S C = {#}"
measurable_convergent: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "\<And>i. f i \<in> borel_measurable M" shows "Measurable.pred M (\<lambda>x. convergent (\<lambda>i. f i x))"
less_imp_le_pred: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "m \<le> n - Suc 0"
that: assumes "set (awalk_verts u p) \<subseteq> verts G"   and "set p \<subseteq> arcs G"   and "cas u p v"   and "awhd u p = u"   and "awlast u p = v"   and "u \<in> verts G"   and "v \<in> verts G" shows "thesis"
tcopy_correct: fixes n :: "nat" shows "{\<lambda>tp. tp = ([], Oc \<up> Suc n)} tcopy {\<lambda>tp. tp = ([Bk], <(n, n)>)}"
prems: shows "wf'\<^sub>s\<^sub>s\<^sub>t V (x # S)"
gr0_imp_self_le_mult1: fixes k :: "nat"   and m :: "nat" assumes "0 < k" shows "m \<le> m * k"
finite_has_minimal: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.order less_eq less"   and "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. less_eq b m \<longrightarrow> m = b"
mapM_Some: fixes f :: "'f \<Rightarrow> 'e option"   and xs :: "'f list"   and ys :: "'e list" assumes "mapM f xs = Some ys" shows "ys = map (\<lambda>x. the (f x)) xs \<and> (\<forall>x\<in>set xs. f x \<noteq> None)"
valid_eval: fixes env :: "v sem_env"   and s :: "'a state"   and e :: "exp0" shows "evaluate True env s e (eval' env e s)"
injectivity_3: fixes X :: "'a set"   and Y :: "'a set"   and k :: "nat"   and n :: "nat"   and k' :: "nat"   and xs :: "'a list"   and xs' :: "'a list"   and ys :: "'a list"   and ys' :: "'a list"   and zs :: "'a list"   and zs' :: "'a list" assumes "X \<inter> Y = {}"   and "k \<in> {0..n} \<and> k' \<in> {0..n}"   and "(length xs = k \<and> distinct xs \<and> set xs \<subseteq> X) \<and> length xs' = k' \<and> distinct xs' \<and> set xs' \<subseteq> X"   and "(length ys = n - k \<and> distinct ys \<and> set ys \<subseteq> Y) \<and> length ys' = n - k' \<and> distinct ys' \<and> set ys' \<subseteq> Y"   and "interleavings xs ys zs \<and> interleavings xs' ys' zs'"   and "zs = zs'" shows "ys = ys'"
mono: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b" shows "sup c d \<le> sup a b"
interaction_bounded_by_contD_ignore: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat"   and out :: "'d"   and c :: "('e, 'd, 'f) rpv"   and input :: "'f" assumes "interaction_bounded_by consider gpv n"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "interaction_bounded_by consider (c input) n"
inf_sup_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> x \<squnion> y = x"
5: fixes l :: "nat"   and s :: "item list"   and m :: "nat" assumes "isFresh l s"   and "l \<le> m" shows "isFresh m s"
has_real_derivative: fixes f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and F :: "real filter" assumes "(f has_derivative f') F" obtains c :: "real" where "(f has_real_derivative c) F"
measurable_range_rep: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "range f = (\<Union>r\<in>proj_rep_set n. {f r})"
delete_def: fixes x :: "'b"   and t :: "'b tree234" shows "delete x t = tree\<^sub>d (del x t)"
l_neg: fixes x :: "complex vec" assumes "x \<in> carrier_vec d" shows "\<ominus>\<^bsub>V\<^esub> x + x = 0\<^sub>v d"
red_single_higher: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "punit.red_single p q f t" shows "punit.higher q (local.punit.splus t (punit.lt f)) = punit.higher p (local.punit.splus t (punit.lt f))"
sums_unique2: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "f sums a"   and "f sums b" shows "a = b"
Forall_def: fixes x :: "'f"   and P :: "('d, 'e, 'f) formula" shows "Forall x P = ! (Exists x (! P))"
grouped_by_fst_def: fixes xs :: "('b \<times> 'c) list" shows "grouped_by_fst xs = map (\<lambda>u. filter (\<lambda>x. fst x = u) xs) (remdups (map fst xs))"
not_less: fixes x :: "'e word"   and y :: "'e word" shows "(\<not> x <s y) = (y \<le>s x)"
sqmatrix_mult_distrib_left: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix"   and C :: "('c, 'd) sqmatrix" shows "A \<cdot> (B + C) = A \<cdot> B + A \<cdot> C"
IH: fixes k :: "nat"   and I :: "'a \<Rightarrow> bool option" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" shows "are_all_operators_applicable (trace_parallel_plan_strips I \<pi> ! k) (\<pi> ! k) \<and> are_all_operator_effects_consistent (\<pi> ! k)"
hseq_char': fixes \<nu> :: "'d"   and \<mu> :: "'d" shows "D.hseq \<nu> \<mu> = (\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null)"
emeasure_Collect_distr: fixes X :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "X \<in> M \<rightarrow>\<^sub>M N"   and "Measurable.pred N P" shows "emeasure (distr M N X) {x \<in> space N. P x} = emeasure M {x \<in> space M. P (X x)}"
Im_Ln_tendsto_at_bot: fixes a :: "real" shows "((\<lambda>x. Im (Ln (Complex a x))) \<longlongrightarrow> - pi / 2) at_bot"
map_of_is_SomeI: fixes xys :: "('d \<times> 'e) list"   and x :: "'d"   and y :: "'e" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = Some y"
simps_2: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "x # xs \<cong> {y # ys, [], P} = (P x xs \<and> x = y \<and> xs \<cong> {ys, [], P})"
pairwise_def: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)" shows "List_Factoring.pairwise A = (\<forall>init. distinct init \<longrightarrow> (\<forall>qs\<in>{xs. set xs \<subseteq> set init}. \<forall>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. T\<^sub>p_on_rand' A (fst A (Lxy init {x, y}) \<bind> (\<lambda>is. return_pmf (Lxy init {x, y}, is))) (Lxy qs {x, y}) = ALGxy A qs init x y))"
exhaust_sel: fixes char :: "char" obtains "char = Char (digit0 char) (digit1 char) (digit2 char) (digit3 char) (digit4 char) (digit5 char) (digit6 char) (digit7 char)"
Goal: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"   and "x \<sharp> P" shows "P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'"
singular_face_def: fixes p :: "nat"   and k :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" shows "singular_face p k f \<equiv> restrict (f \<circ> simplical_face k) (standard_simplex (p - Suc 0))"
da_elim_cases_56: fixes Env :: "env"   and B :: "lname set"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In2 (e1.[e2])\<guillemotright> A" obtains E1 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" and "Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A"
seq_rident: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "seq x one = x"
A82_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. C \<noteq> sA93 \<longrightarrow> sA82 \<notin> Sources level1 C"
distributed_convolution: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real"   and f :: "real \<Rightarrow> ennreal"   and g :: "real \<Rightarrow> ennreal" assumes "indep_var borel X borel Y"   and "distributed M lborel X f"   and "distributed M lborel Y g" shows "distributed M lborel (\<lambda>x. X x + Y x) (\<lambda>x. \<integral>\<^sup>+ y. f (x - y) * g y \<partial>lborel)"
dfs_cond_def: shows "dfs_cond \<equiv> {(\<Sigma>, W). W \<noteq> []}"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "z \<sqsubseteq> x"
implAuth_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> asym.valid \<union> range LtK \<union> Tags"   and "asym.implAuth A B M \<in> synth (analz H)" shows "asym.implAuth A B M \<in> H \<or> M \<in> synth (analz H) \<and> (A, B) \<in> broken H"
ivl_disj_un_one_4: fixes l :: "'c"   and u :: "'c" assumes "l \<le> u" shows "{..<l} \<union> {l..u} = {..u}"
cs1m: shows "cs\<^bsup>\<pi>\<^sub>1\<^esup> m = cs\<^bsup>\<pi>\<^esup> m"
v_props_1: shows "\<V> \<subseteq> two_chain_vertical_boundary two_chain"
some_src_eqI: fixes a :: "'a"   and \<mu> :: "'a" assumes "a \<in> sources \<mu>"   and "some_src a = a" shows "some_src \<mu> = a"
decr_lemma: fixes d :: "int"   and x :: "int"   and z :: "int" assumes "0 < d" shows "x - (\<bar>x - z\<bar> + 1) * d < z"
upshift_def: fixes P :: "nat fset" shows "upshift P = Suc |`| P"
a_rcosets_part_G: shows "\<Union> (a_rcosets H) = carrier G"
less_eq_QDelta_def: fixes qd1 :: "QDelta"   and qd2 :: "QDelta" shows "(qd1 \<le> qd2) = (qdfst qd1 < qdfst qd2 \<or> qdfst qd1 = qdfst qd2 \<and> qdsnd qd1 \<le> qdsnd qd2)"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'a" shows "opn.initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> opn.someinit i | Some s \<Rightarrow> s, snd \<sigma>)"
subst_compose_fv: fixes \<theta> :: "'e \<Rightarrow> ('d, 'c) Term.term"   and x :: "'e"   and \<sigma> :: "'c \<Rightarrow> ('d, 'f) Term.term" assumes "fv (\<theta> x) = {}" shows "fv ((\<theta> \<circ>\<^sub>s \<sigma>) x) = {}"
refines_def: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "prog \<sqsubseteq> prog' \<equiv> \<forall>P. sound P \<longrightarrow> wp prog P \<tturnstile> wp prog' P"
trans_def: fixes r :: "('a \<times> 'a) set" shows "trans r = (\<forall>x y z. x \<^bsub>r\<^esub>\<preceq> y \<longrightarrow> y \<^bsub>r\<^esub>\<preceq> z \<longrightarrow> x \<^bsub>r\<^esub>\<preceq> z)"
mbisim: shows "s1 \<approx>m (ls2, (ts2', m2), ws2, is2)"
Min_in: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set" assumes "finite A"   and "A \<noteq> {}" shows "ordered_powerprod_lin.Min A \<in> A"
cvalid: shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
Gp_rcs: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set"   and x :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> K "   and "H \<subseteq> K"   and "x \<in> K" shows "H \<bullet>\<^bsub>\<natural>\<^bsub>G\<^esub>K\<^esub> x = H \<bullet>\<^bsub>G\<^esub> x"
update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "ip \<noteq> dip" shows "update rt dip (dsn, dsk, flag, hops, nhip) ip = rt ip"
kruskal_spanning_invariant_def: fixes minarc :: "'b \<Rightarrow> 'b"   and sum :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "class.m_kleene_algebra minarc sum plus inf less_eq less sup bot top uminus one times conv star" shows "m_kleene_algebra.kruskal_spanning_invariant inf less_eq uminus one times conv star f g h \<equiv> conv g = g \<and> h = conv h \<and> inf g (uminus (uminus h)) = h \<and> m_kleene_algebra.spanning_forest less_eq uminus one times conv star f (inf (uminus h) g)"
sdp_Bind: fixes p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s. sub_distrib_pconj (p (f s))" shows "sub_distrib_pconj (Bind f p)"
unfold_congs_2: fixes r :: "('c, 'd, 'e) automaton_scheme"   and r' :: "('c, 'd, 'e) automaton_scheme"   and v' :: "('c \<times> 'd \<times> 'c) set"   and f :: "('c \<times> 'd \<times> 'c) set \<Rightarrow> ('c \<times> 'd \<times> 'c) set"   and f' :: "('c \<times> 'd \<times> 'c) set \<Rightarrow> ('c \<times> 'd \<times> 'c) set" assumes "r = r'"   and "automaton.trans r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "trans_update f r = trans_update f' r'"
"3_2": shows "v\<^sub>1 \<preceq>\<^sub>F v\<^sub>2"
intsFrom_split: fixes n :: "int"   and m :: "int" assumes "n \<le> m" shows "[MkI\<cdot>n..] = [MkI\<cdot>n..MkI\<cdot>(m - 1)] ++ [MkI\<cdot>m..]"
leq_refl: fixes x :: "'a spmf" shows "ord_spmf (=) x x"
F_anal_valid: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeII_chain i j F two_chain s" shows "\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> j) i"
dom_prob_proj: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'e) fmap) set"   and vs :: "'c set" shows "prob_dom (prob_proj PROB vs) \<subseteq> vs"
slice_kind_def: fixes S :: "'node SDG_node set"   and a :: "'edge" shows "slice_kind S a \<equiv> slice_kind_aux (sourcenode a) (targetnode a) (HRB_slice S) (kind a)"
Nmul: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "INum (x *\<^sub>N y) = INum x * INum y"
subseq_append_iff: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "subseq xs (ys @ zs) = (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> subseq xs1 ys \<and> subseq xs2 zs)"
all_def: fixes f :: "'a \<Rightarrow> ??.Quickcheck_Narrowing.property" shows "??.Quickcheck_Narrowing.all f = (case narrowing 100 of ??.Quickcheck_Narrowing.narrowing_cons.Narrowing_cons ty cs \<Rightarrow> Universal ty (\<lambda>t. f (??.Quickcheck_Narrowing.conv cs t)) (partial_term_of TYPE('a)))"
power_distrib: fixes h :: "'a"   and e :: "nat" assumes "h \<in> carrier G" shows "\<^bold>g [^] e \<otimes> h [^] e = (\<^bold>g \<otimes> h) [^] e"
subst_fv_imgI: fixes \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and v :: "'d" assumes "\<sigma> v \<noteq> Var v" shows "fv (\<sigma> v) \<subseteq> range_vars \<sigma>"
degree_isnpolyh_Suc: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "Polynomial_Expression.degree p = 0"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> [a, b]\<^sub>\<circ>" shows "[a, b]\<^sub>\<circ>\<lparr>a\<rparr> = b"
fix_eq4: fixes f :: "'c"   and F :: "'c \<rightarrow> 'c" assumes "f = fix\<cdot>F" shows "f = F\<cdot>f"
semInt_prSwap: fixes Delta :: "('c, 'd, 'e, 'f, 'g) signature"   and SEM :: "('c, 'd, 'e, 'f, 'g, 'h) semDom" assumes "FixSyn TYPE('b) Delta"   and "FixSyn.wlsSEM Delta SEM" shows "FixSyn.prSwap Delta (FixSyn.semInt Delta SEM) SEM"
prefix_def: fixes xs :: "'a list"   and ys :: "'a list" shows "prefix xs ys = (\<exists>zs. ys = xs @ zs)"
init_item_def: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "init_item r k = Item r 0 k k"