hcju/isabelleps huggingface dataset:
of_nat_less_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m < n)"
rank_mat_mul_left: fixes A :: "'b mat"   and n :: "nat"   and d :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat n d"   and "B \<in> carrier_mat d nc" shows "vec_space.rank n (A * B) \<le> vec_space.rank d B"
closure_def: fixes as :: "'b list" shows "pEquivalence_Checking.closure as = while_option test (step as)"
isomorphic_implies_ide_1: fixes f :: "'a"   and g :: "'a" assumes "S.isomorphic f g" shows "S.ide f"
PiE_cong: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i = B i" shows "Pi\<^sub>E I A = Pi\<^sub>E I B"
unclocked_eq_2: shows "Big_Step_Unclocked.evaluate = BigStep.evaluate False"
assms: shows "(subsumee sub \<noteq> subsumer sub \<and> fst (subsumee sub) = fst (subsumer sub) \<and> subsumee sub \<in> red_vertices prb \<and> subsumee sub \<notin> subsumers (subs prb) \<and> subsumee sub \<notin> subsumees (subs prb) \<and> subsumer sub \<in> red_vertices prb \<and> subsumer sub \<notin> subsumees (subs prb) \<and> out_edges (red prb) (subsumee sub) = {} \<and> subs prb' = subs prb \<union> {sub}) \<and> \<not> marked prb (subsumer sub) \<and> \<not> marked prb (subsumee sub) \<and> confs prb (subsumee sub) \<sqsubseteq> confs prb (subsumer sub) \<and> prb' = \<lparr>red = red prb, black = black prb, subs = insert sub (subs prb), init_conf = init_conf prb, confs = confs prb, marked = marked prb, strengthenings = strengthenings prb, \<dots> = pre_RedBlack.more prb\<rparr>"
cst_integral: fixes f :: "bool stream \<Rightarrow> real"   and c :: "real" assumes "f \<in> borel_measurable (F 0)"   and "f (sconst True) = c" shows "has_bochner_integral M f c"
ms_closure_cong: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "ms_closure P xs f basis"   and "\<forall>\<^sub>F x in at_top. f x = g x" shows "ms_closure P xs g basis"
simps_1: fixes S :: "'b formula set" shows "extender 0 S = S"
mismatchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>P Q a b. \<lbrakk>(P, Q) \<in> Rel; a \<noteq> b\<rbrakk> \<Longrightarrow> ([a\<noteq>b]P, Q) \<in> Rel"   and "Rel \<subseteq> Rel'" shows "[a\<noteq>b]P \<leadsto>\<^sup>^<Rel'> [a\<noteq>b]Q"
distinct: shows "valid_unMultigraph.is_trail (del_unEdge v0 w v1 G) v1 ps v2"
ent_imp_entt: fixes P :: "assn"   and Q :: "assn" assumes "P \<Longrightarrow>\<^sub>A Q" shows "P \<Longrightarrow>\<^sub>t Q"
length_list_of_vec: fixes v :: "'b vec" shows "length (list_of_vec v) = dim_vec v"
enn2real_le_iff: fixes x :: "ennreal"   and c :: "real" assumes "x < top"   and "0 < c" shows "(enn2real x \<le> c) = (x \<le> ennreal c)"
Abs_fresh_iff_1: fixes a :: "atom"   and bs :: "atom set"   and x :: "'a" shows "a \<sharp> [bs]set. x = (a \<in> bs \<or> a \<notin> bs \<and> a \<sharp> x)"
simps_7: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and v :: "'c match_expr"   and vb :: "char list"   and rs :: "'c rule list" shows "no_matching_Goto \<gamma> p (Rule v (Call vb) # rs) = no_matching_Goto \<gamma> p rs"
game_single_def: fixes \<A> :: "'c_a \<Rightarrow> ('guess, 'call, 'ret) gpv" shows "game_single \<A> = init \<bind> (\<lambda>(c_o, c_a, s). exec_gpv (oracle c_o) (\<A> c_a) s \<bind> (\<lambda>(guess, s'). eval c_o c_a s' guess))"
degrees_def: fixes X :: "'a pdevs list" shows "degrees X = Max (insert 0 (degree ` set X))"
expands_to_real_compose: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and bs :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "(f expands_to c) bs" shows "((\<lambda>x. g (f x)) expands_to g c) bs"
commute: fixes a :: "'a"   and b :: "'a" shows "sup a b = sup b a"
possible_rules_f_def: shows "possible_rules_f = (\<lambda>Q f. (\<lambda>(q, qs). q \<rightarrow> f qs) ` (Q \<times> (lists Q \<inter> {qs. A f = Some (length qs)})))"
assms: shows "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)"
assms: shows "insert (HDomain_Incl s (SUCC k)) (insert (OrdP k) (insert (HFun_Sigma s) (insert (HPair k y IN s) H))) \<turnstile> B"
the_\<theta>_def: shows "the_\<theta> \<equiv> THE \<theta>. \<guillemotleft>\<theta> : s\<^sub>0 \<star> w \<Rightarrow> r\<^sub>0\<guillemotright>"
cone_mkCone: fixes p0 :: "'c"   and p1 :: "'c" assumes "D.is_rendered_commutative_by p0 p1" shows "D.cone (C.dom p0) (D.mkCone p0 p1)"
subcls_new_old: shows "G\<turnstile>declclass new\<prec>\<^sub>C declclass old"
trancl_rtrancl_trancl: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and c :: "'d" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>+"
Dynkin_subset: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and N :: "'c set set" assumes "Dynkin_system \<Omega> M"   and "N \<subseteq> M" shows "Dynkin \<Omega> N \<subseteq> M"
B2: fixes x :: "'a" assumes "x \<in> B" shows "f x \<le> f (T x)"
Complement_cproper_interval_set_Complement: fixes A :: "'b set"   and B :: "'b set" assumes "ID ccompare \<noteq> None" shows "cproper_interval (Some (- A)) (Some (- B)) = cproper_interval (Some B) (Some A)"
psubsetE: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subset> B" obtains "A \<subseteq> B" and "\<not> B \<subseteq> A"
frac_eq_obtains: fixes r :: "'a"   and s :: "'a"   and x :: "'a \<times> 'a" assumes "(r, s) \<in> R \<times> S"   and "x = (SOME x. x \<in> r / s)" obtains s1 :: "'a" where "s1 \<in> S" and "s1 \<cdot> s \<cdot> fst x = s1 \<cdot> snd x \<cdot> r" and "x \<in> R \<times> S"
2: shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel \<A>))"
nxt_Accessible_dfa: shows "nxt Accessible_dfa = nxt M"
tedious: assumes "1 \<le> c"   and "2 * d * c < 1"   and "0 < d" shows "False"
ring_axioms_def: fixes R :: "('c, 'd) ring_scheme" shows "ring_axioms R \<equiv> (\<forall>x y z. x \<in> carrier R \<longrightarrow> y \<in> carrier R \<longrightarrow> z \<in> carrier R \<longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z) \<and> (\<forall>x y z. x \<in> carrier R \<longrightarrow> y \<in> carrier R \<longrightarrow> z \<in> carrier R \<longrightarrow> z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y)"
Suc_a: fixes b :: "nat" shows "(a + b) * valid_countings a b = (a - b) * (a + b choose a)"
Units: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "carrier G \<subseteq> Units G"
erf_continuous_on: fixes A :: "'a set" shows "continuous_on A erf"
lemma_A: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and K :: "nat"   and Tk :: "nat"   and X :: "msg"   and Ta :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos"   and "Key K \<notin> analz (knows Spy evs)"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Crypt K \<lbrace>Agent A, Number Ta\<rbrace> \<in> parts (knows Spy evs)" shows "Says A B \<lbrace>X, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"
assms_2: shows "FAIL \<in> succ AB w (p1, p2)"
call_zero_iff: fixes x :: "'a" shows "(\<forall>u. \<langle>x, u\<rangle> = 0) = (x = (0::'a))"
ex1_implies_ex: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "\<exists>x. P x"
lfilter_cong: fixes xs :: "'c llist"   and ys :: "'c llist"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> lset ys \<Longrightarrow> P x = Q x" shows "lfilter P xs = lfilter Q ys"
MaxI2: fixes k :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "k \<in> A"   and "\<And>x. x \<in> A \<Longrightarrow> P x"   and "finite A" shows "P (Max A)"
lower_asymptotic_density_subset: fixes A :: "nat set"   and B :: "nat set" assumes "A \<subseteq> B" shows "lower_asymptotic_density A \<le> lower_asymptotic_density B"
antiparallel_edges_axioms: shows "antiparallel_edges \<Delta>"
inject: fixes x2 :: "'h"   and y2 :: "'h" shows "(Some x2 = Some y2) = (x2 = y2)"
assms_2: shows "length ys = card {a \<in> - A. a < length xs + length ys}"
that: fixes f :: "'t \<Rightarrow>\<^sub>0 'b"   and v :: "'t" assumes "f \<in> F"   and "v \<in> keys p"   and "f \<noteq> 0"   and "lt f adds\<^sub>t v" shows "thesis"
LeftDerivationFix_cut_appendix: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<beta> :: "symbol list"   and \<delta>' :: "symbol list"   and n :: "nat" assumes "LeftDerivationFix (\<alpha> @ \<delta>) i D j (\<beta> @ \<delta>')"   and "LeftDerivation \<alpha> (take n D) \<beta>"   and "n \<le> length D"   and "derivation_ge (drop n D) (length \<beta>)"   and "i < length \<alpha>"   and "j < length \<beta>" shows "LeftDerivationFix \<alpha> i (take n D) j \<beta>"
fdia_def: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "relfdia x y = rel_ad (rel_ad (x ; y))"
2: shows "isTrue (PP \<langle>\<phi>\<rangle>)"
D0_def: shows "D0 \<equiv> set bs \<times> (set gs \<union> set bs) -\<^sub>p set ps0"
induct: fixes P :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a list" assumes "\<And>i xs. (\<And>x. \<lbrakk>\<not> length xs \<le> i + 1; x \<in> set_pmf (pmf_of_set {i..<length xs})\<rbrakk> \<Longrightarrow> P (i + 1) (swap xs i x)) \<Longrightarrow> P i xs" shows "P a0 a1"
eq_iff: fixes a :: "'e"   and b :: "'e" shows "(a = b) = (a \<le> b \<and> b \<le> a)"
inducts: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "lam \<Rightarrow> lam \<Rightarrow> bool" assumes " x1 \<longrightarrow>\<^sub>1\<^sup>* x2"   and "\<And>M. P M M"   and "\<And>M1 M2 M3. \<lbrakk> M1 \<longrightarrow>\<^sub>1\<^sup>* M2; P M1 M2;  M2 \<longrightarrow>\<^sub>1 M3\<rbrakk> \<Longrightarrow> P M1 M3" shows "P x1 x2"
bound1: fixes z :: "complex"   and u :: "real" assumes "Re z \<in> {x..y}"   and "u \<in> {0<..<exp (- 1)}" shows "cmod (f z u) \<le> g1 u"
mask_AND_NOT_mask: fixes w :: "'b word"   and n :: "nat" shows "(w && ~~ mask n) && mask n = 0"
abs_eq_iff: fixes x :: "'a"   and y :: "'a" shows "(\<bar>x\<bar> = \<bar>y\<bar>) = (x = y \<or> x = - y)"
Inf_insert: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and a :: "'b::type"   and A :: "'b::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) (insert (a::'b::type) (A::'b::type set)) = (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) a (Inf A)"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')" shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
f: shows "\<forall>c. f c = (0::'b) \<longrightarrow> c = (0::'a)"
cr_rename_gba_refine: fixes nds :: "'a node set"   and nds' :: "'a node set"   and \<phi>' :: "'a ltlr"   and \<phi> :: "'a ltlr" assumes "nds_invars nds"   and "(nds', nds) \<in> Id"   and "(\<phi>', \<phi>) \<in> Id" shows "cr_rename_gba nds' \<phi>' \<le> \<Down> Id (RETURN (gba_rename name (create_gba_from_nodes \<phi> nds)))"
coprime_mod_left_iff: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "algebraic_semidom.coprime times one (modulo a b) b = algebraic_semidom.coprime times one a b"
ESem_fresh_cong: fixes e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value"   and \<rho>' :: "Vars.var \<Rightarrow> Value" assumes "\<rho> f|` fv e = \<rho>' f|` fv e" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>"
simps_3: fixes gi :: "int poly"   and gj :: "int poly"   and gs :: "int poly list" shows "choose_u (gi # gj # gs) = min_degree_poly gi (choose_u (gj # gs))"
op_sem_val_type: fixes oper :: "pdf_operator"   and v :: "val"   and t' :: "pdf_type" assumes "op_type oper (val_type v) = Some t'" shows "val_type (op_sem oper v) = t'"
valid_node_cases: fixes n :: "'b LDCFG_node" assumes "CFGExit_wf.valid_node n" obtains "n = NewEntry"   | "n = NewExit"   | "CFGExit_wf.inner_node n"
simps_2: fixes n :: "nat"   and s :: "'a stream" shows "stake (Suc n) s = shd s # stake n (stl s)"
mset_del_min: fixes t :: "'b tree" assumes "braun t"   and "t \<noteq> \<langle>\<rangle>" shows "mset_tree (del_min t) = mset_tree t - {#get_min t#}"
case_prodD: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
count_append: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" shows "count_list (xs @ ys) x = count_list xs x + count_list ys x"
simps_2: fixes x :: "'a set" shows "min_gallery [x] = chamber x"
power_le_imp_le_base: 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 zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat"   and b :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (power.power one times a (Suc n)) (power.power one times b (Suc n))"   and "less_eq zero b" shows "less_eq a b"
simps_3: fixes x :: "'b"   and y :: "'b"   and v :: "'b"   and va :: "'b list" shows "decompose x (y # v # va) = (case bezout_coefficients y (prod_list (v # va)) of (a, b) \<Rightarrow> b * x # decompose (a * x) (v # va))"
dvd_div_mult: 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 c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b" shows "times (divide b c) a = divide (times b a) c"
get_dom_component_root_node_same: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and root_ptr :: "(_) object_ptr"   and x :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root_ptr"   and "x \<in> set c" shows "h \<turnstile> get_root_node x \<rightarrow>\<^sub>r root_ptr"
comp_converter_parallel2'_1: fixes conv1l :: "('a, 'c, 'i, 'k) converter"   and conv1r :: "('b, 'd, 'j, 'l) converter"   and conv2l :: "('i, 'k, 'e, 'g) converter"   and conv2r :: "('j, 'l, 'f, 'h) converter" shows "comp_converter (parallel_converter2 conv1l conv1r) (parallel_converter2 conv2l conv2r) = parallel_converter2 (comp_converter conv1l conv2l) (comp_converter conv1r conv2r)"
continuous_on_eq: fixes s :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "continuous_on s f"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "continuous_on s g"
eqvt: fixes x1 :: "lam"   and x2 :: "lam"   and pi :: "(name \<times> name) list" assumes " x1 \<longrightarrow>\<^sub>\<beta> x2" shows " pi \<bullet> x1 \<longrightarrow>\<^sub>\<beta> pi \<bullet> x2"
strict_implies_not_eq: fixes a :: "'a"   and b :: "'a" assumes "lt a b" shows "a \<noteq> b"
subst_apply_img_var: fixes v :: "'a"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<in> fv (t \<cdot> \<delta>)"   and "v \<notin> fv t" obtains w :: "'a" where "w \<in> fv t" and "v \<in> fv (\<delta> w)"
Pi_pmf_def: fixes A :: "'b set"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" shows "Pi_pmf A dflt p = embed_pmf (\<lambda>f. if \<forall>x. x \<notin> A \<longrightarrow> f x = dflt then \<Prod>x\<in>A. pmf (p x) (f x) else 0)"
subset_dfs_bij: fixes q :: "bool list"   and q0 :: "bool list"   and v :: "nat" assumes "nfa_is_node A q"   and "nfa_is_node A q0" shows "(bdd_lookup (fst (subset_dfs A q0)) q = Some v) = (v < length (snd (subset_dfs A q0)) \<and> snd (subset_dfs A q0) ! v = q)"
BijGroup_def: fixes S :: "'c set" shows "BijGroup S = \<lparr>carrier = Bij S, mult = \<lambda>g\<in>Bij S. restrict (compose S g) (Bij S), one = \<lambda>x\<in>S. x\<rparr>"
IN_h'_B: fixes x :: "'v" assumes "x \<in> B \<Gamma>" shows "d_IN h' x = d_IN (h 0) x + g (SOURCE, \<langle>x\<rangle>)"
elims_2: fixes x :: "real_max_bot"   and xa :: "real_max_bot" assumes "x < xa" obtains uv_ :: "real" where "xa__ = MInfty" and "xa = R uv_"   | x :: "real" and y :: "real" where "xa__ = R x" and "xa = R y" and "x < y"
no_matching_Goto_take: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and n :: "nat" assumes "no_matching_Goto \<gamma> p rs" shows "no_matching_Goto \<gamma> p (take n rs)"
submodule_mOp_closed: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and h :: "'e" assumes "R module M"   and "submodule R M H"   and "h \<in> H" shows "-\<^sub>a\<^bsub>M\<^esub> h \<in> H"
terminationLess_def: fixes F0 :: "'a"   and decisionVars :: "nat set"   and learnL :: "(Literal list list \<times> Literal list list) set" shows "terminationLess F0 decisionVars learnL \<equiv> {(stateA, stateB). (stateA, stateB) \<in> lexLessState F0 decisionVars \<or> (stateA, stateB) \<in> learnLessState learnL}"
less_b_def: fixes a :: "'a"   and b :: "'a" shows "less_b a b = (lesseq_b a b \<and> \<not> lesseq_b b a)"
unitary_intros_2: shows "unitary (\<lambda>s. 1)"
inv_closed: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
distinct_2: fixes name' :: "name"   and frame' :: "'d frame"   and assertion :: "'d" shows "\<lparr>\<nu>name'\<rparr>frame' \<noteq> \<langle>\<epsilon>, assertion\<rangle>"
valid_in_equiv: fixes I :: "interp"   and \<phi> :: "fml"   and \<psi> :: "fml" shows "valid_in I (\<phi> \<leftrightarrow> \<psi>) = (fml_sem I \<phi> = fml_sem I \<psi>)"
wt_le_wt_sym: fixes f :: "'s"   and \<zeta> :: "('s, 'v) hd" assumes "f \<in> local.ground_heads \<zeta>" shows "wt (Hd \<zeta>) \<le> wt_sym f + the_enat (enat \<delta> * arity_sym f)"
cancel_to_1_append: fixes a :: "(bool \<times> 'c) list"   and b :: "(bool \<times> 'c) list"   and l :: "(bool \<times> 'c) list"   and l' :: "(bool \<times> 'c) list" assumes "cancels_to_1 a b" shows "cancels_to_1 (l @ a @ l') (l @ b @ l')"
is_Zero_def: fixes rexp :: "'a rexp" shows "is_Zero rexp = (rexp = Zero)"
sub: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> left.translation ` G" shows "a \<in> G \<rightarrow>\<^sub>E G"
simps_3: fixes a :: "tm" shows "bound0 (Lt a) = tmbound0 a"
normalize_uminus: fixes x :: "'b" shows "normalize (- x) = normalize x"
select_convs_1: fixes mstate_i :: "nat"   and mstate_m :: "('a, 'b) mformula"   and mstate_n :: "nat"   and more :: "'c" shows "mstate_i \<lparr>mstate_i = mstate_i, mstate_m = mstate_m, mstate_n = mstate_n, \<dots> = more\<rparr> = mstate_i"
projection_to_emptyset_is_empty_trace: fixes l :: "'b list" shows "l \<upharpoonleft> {} = []"
card_of_Times_mono2: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set" assumes "|A| \<le>o |B|" shows "|C \<times> A| \<le>o |C \<times> B|"
null_def: shows "C.null = (THE n. \<forall>f. C n f = n \<and> C f n = n)"
cont_discrete_cpo: fixes f :: "'g \<Rightarrow> 'h" shows "cont f"
UNION_eq: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "\<Union> (B ` A) = {y. \<exists>x\<in>A. y \<in> B x}"
small_image_nat: fixes g :: "nat \<Rightarrow> 'a"   and N :: "nat set" shows "small (g ` N)"
assms_2: shows "\<Gamma> = \<Gamma>1 ++ \<Gamma>2"
det_def': fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "det A = (\<Sum>p | p permutes {0..<n}. signof p * (\<Prod>i = 0..<n. A $$ (i, p i)))"
tail_invar_def: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.tail_invar bq = (SkewBinomialHeapStruc.queue_invar bq \<and> SkewBinomialHeapStruc.rank_invar bq)"
wqo_on_subset: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<subseteq> B"   and "wqo_on P B" shows "wqo_on P A"
closedI: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "Semilat A r f" shows "closed A f"
rep_eq: fixes x :: "ivl"   and xa :: "ivl" shows "(x \<le> xa) = le_rep (rep_ivl x) (rep_ivl xa)"
simps_2: fixes x :: "complex"   and y :: "complex" shows "Im (x * y) = Re x * Im y + Im x * Re y"
U1\<alpha>_def: shows "U1\<alpha> = \<Sigma> \<times> mrg_right"
distrib_imp1: fixes 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 x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice inf less_eq less sup"   and "\<And>x y z. inf x (sup y z) = sup (inf x y) (inf x z)" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
Chn_vcomp: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "V.seq \<nu> \<mu>" shows "Chn (\<nu> \<bullet> \<mu>) = Chn \<nu> \<cdot> Chn \<mu>"
Sigma_cong: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g \<Rightarrow> 'h set"   and D :: "'g \<Rightarrow> 'h set" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sigma A C = Sigma B D"
7: shows "e * (w \<sqinter> - d)\<^sup>\<star> * e = bot"
subsetCI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<notin> B\<rbrakk> \<Longrightarrow> False" shows "A \<subseteq> B"