hcju/isabelleps huggingface dataset:
subprob_space_axioms: fixes M :: "'c pmf" shows "subprob_space (measure_pmf M)"
P_def: shows "P \<equiv> (\<lambda>_. [])(Abs a := [Abs b])"
RTranCl_induct: fixes h :: "'b"   and h' :: "'b"   and succs :: "'b \<Rightarrow> 'b list"   and P :: "'b \<Rightarrow> bool" assumes "h [succs]\<rightarrow>* h'"   and "P h"   and "\<And>g g'. \<lbrakk>g' \<in> set (succs g); P g\<rbrakk> \<Longrightarrow> P g'" shows "P h'"
the_inv_f_f: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "inj f" shows "the_inv f (f x) = x"
per_rootD': fixes u :: "'b list"   and r :: "'b list" assumes "u \<le>p r\<^sup>\<omega>" shows "r \<noteq> \<epsilon>"
cont_nec_fact2_3: fixes v :: "i" shows "[\<^bold>\<not>WeaklyContingent E! in v]"
INT_D: fixes b :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and a :: "'c" assumes "b \<in> \<Inter> (B ` A)"   and "a \<in> A" shows "b \<in> B a"
simps_3: fixes \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "convert_multiway (formula.And \<phi> \<psi>) = (if safe_assignment (fv \<phi>) \<psi> then formula.And (convert_multiway \<phi>) \<psi> else if safe_formula \<psi> then formula.Ands (get_and_list (convert_multiway \<phi>) @ get_and_list (convert_multiway \<psi>)) else if is_constraint \<psi> then formula.And (convert_multiway \<phi>) \<psi> else formula.Ands (convert_multiway \<psi> # get_and_list (convert_multiway \<phi>)))"
rep_eq: fixes x :: "float" shows "real_of_float (floor_fl x) = real_of_int \<lfloor>real_of_float x\<rfloor>"
postdominate_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and n' :: "'b"   and n :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "Postdomination.postdominate sourcenode targetnode kind valid_edge Exit n' n \<equiv> CFG.valid_node sourcenode targetnode valid_edge n \<and> CFG.valid_node sourcenode targetnode valid_edge n' \<and> (\<forall>as pex. CFG.intra_path sourcenode targetnode kind valid_edge n as pex \<and> CFGExit.method_exit sourcenode kind valid_edge Exit pex \<longrightarrow> n' \<in> set (CFG.sourcenodes sourcenode as))"
the_elem_inj: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> carrier (R Quot a_kernel R (S Quot P) ((+>\<^bsub>S\<^esub>) P \<circ> h))"   and "Y \<in> carrier (R Quot a_kernel R (S Quot P) ((+>\<^bsub>S\<^esub>) P \<circ> h))"   and "the_elem (((+>\<^bsub>S\<^esub>) P \<circ> h) ` X) = the_elem (((+>\<^bsub>S\<^esub>) P \<circ> h) ` Y)" shows "X = Y"
simps_6: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "is_\<nu>LTL (\<phi> or\<^sub>n \<psi>) = (is_\<nu>LTL \<phi> \<and> is_\<nu>LTL \<psi>)"
weak_get_put: fixes \<sigma> :: "'b" assumes "\<sigma> \<in> \<S>" shows "put \<sigma> (get \<sigma>) = \<sigma>"
pr_comp2_3': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "pr_conv_2_to_1 f \<in> PrimRec1'"   and "pr_conv_3_to_1 g \<in> PrimRec1'"   and "pr_conv_3_to_1 h \<in> PrimRec1'" shows "pr_conv_3_to_1 (\<lambda>x y z. f (g x y z) (h x y z)) \<in> PrimRec1'"
nonempty_int: fixes A :: "'c set"   and B :: "'c set" assumes "A \<inter> B \<noteq> {}" shows "\<exists>x. x \<in> A \<inter> B"
ft: shows "ft = max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
assms_1: fixes x :: "'a" assumes "x \<in> results_gpv (\<I>_uniform UNIV {x. Domainp R x}) gpv" shows "A x x"
vsubst_preserves_wls: fixes s :: "'sort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x :: "'var" assumes "wls s Y" shows "wls s (Y #[x1 // x]_xs)"
\<theta>_simps_2: shows "src \<theta>' = src (src r)"
bounded_cone_is_zero: fixes C :: "'a vec set"   and bnd :: "'a" assumes "C \<subseteq> carrier_vec n"   and "cone C \<subseteq> Bounded_vec bnd" shows "cone C = {0\<^sub>v n}"
mpoly_eqI: fixes p :: "'b mpoly"   and q :: "'b mpoly" assumes "\<And>mon. MPoly_Type.coeff p mon = MPoly_Type.coeff q mon" shows "p = q"
value_CValue_cases: fixes x :: "Value"   and y :: "CValue" obtains "x = \<bottom>" and "y = \<bottom>"   | f :: "Value \<rightarrow> Value" where "x = Fn\<cdot>f__" and "y = \<bottom>"   | g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" where "x = \<bottom>" and "y = CFn\<cdot>g__"   | f :: "Value \<rightarrow> Value" and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" where "x = Fn\<cdot>fa__" and "y = CFn\<cdot>ga__"   | b\<^sub>1 :: "bool" where "x = B\<cdot>(Discr b\<^sub>1__)" and "y = \<bottom>"   | b\<^sub>1 :: "bool" and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" where "x = B\<cdot>(Discr b\<^sub>1'__)" and "y = CFn\<cdot>g"   | b\<^sub>1 :: "bool" and b\<^sub>2 :: "bool" where "x = B\<cdot>(Discr b\<^sub>1)" and "y = CB\<cdot>(Discr b\<^sub>2__)"   | f :: "Value \<rightarrow> Value" and b\<^sub>2 :: "bool" where "x = Fn\<cdot>f" and "y = CB\<cdot>(Discr b\<^sub>2'__)"   | b\<^sub>2 :: "bool" where "x = \<bottom>" and "y = CB\<cdot>(Discr b\<^sub>2)"
mHom_add: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i"   and m :: "'e"   and n :: "'e" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "f (m \<plusminus>\<^bsub>M\<^esub> n) = f m \<plusminus>\<^bsub>N\<^esub> f n"
assms_1: shows "list_all2 (rel_prod (=) R) cs\<^sub>1 cs\<^sub>2"
simps_3: fixes uw :: "Formula.trm"   and ux :: "Formula.trm" shows "Formula.future_bounded (formula.Eq uw ux) = True"
bin_elem: fixes x :: "item"   and I :: "item set"   and k :: "nat" assumes "x \<in> bin I k" shows "x \<in> I"
assms_2: assumes "f = (\<lambda>u. True)" shows "P f"
Field_osum: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" shows "Field (r +o r') = Inl ` Field r \<union> Inr ` Field r'"
listsum_all_0_but_one: fixes j :: "nat"   and As :: "'b tensor list"   and ds :: "nat list" assumes "\<And>i. \<lbrakk>i \<noteq> j; i < length As\<rbrakk> \<Longrightarrow> As ! i = tensor0 ds"   and "dims (As ! j) = ds"   and "j < length As" shows "listsum ds As = As ! j"
char_poly_def: fixes A :: "'c mat" shows "char_poly A = det (char_poly_matrix A)"
enum_ex_finite_2_def: fixes P :: "finite_2 \<Rightarrow> bool" shows "enum_ex P = (P a\<^sub>1 \<or> P a\<^sub>2)"
Or\<^sub>n_prop_semantics: fixes \<Phi> :: "'b ltln set"   and \<A> :: "'b ltln set" assumes "finite \<Phi>" shows "\<A> \<Turnstile>\<^sub>P Or\<^sub>n \<Phi> = (\<exists>\<phi>\<in>\<Phi>. \<A> \<Turnstile>\<^sub>P \<phi>)"
in_set_assert_spmf: fixes x :: "unit"   and p :: "bool" shows "(x \<in> set_spmf (assert_spmf p)) = p"
dom_dom: fixes f :: "'d" assumes "D.arr f" shows "D.dom (D.dom f) = D.dom f"
prim_triv_root: fixes u :: "'b list"   and t :: "'b list" assumes "primitive u"   and "u \<in> t*" shows "t = u"
region_not_empty_aux: fixes f :: "real"   and g :: "real"   and d :: "nat"   and d' :: "nat" assumes "0 < f"   and "f < 1"   and "0 < g"   and "g < 1" shows "(frac (get_intv_val (Intv d) f) \<le> frac (get_intv_val (Intv d') g)) = (f \<le> g)"
not_has_disc_normalize_match: fixes disc :: "'c \<Rightarrow> bool"   and neg :: "bool"   and m :: "'c match_expr"   and m' :: "'c match_expr" assumes "\<not> has_disc_negated disc neg m"   and "m' \<in> set (normalize_match m)" shows "\<not> has_disc_negated disc neg m'"
msteps_psnoc: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and tdb :: "(String.literal \<times> event_data list) set \<times> nat"   and st :: "('a, 'b) mstate" assumes "last_ts \<pi> \<le> snd tdb" shows "msteps (psnoc \<pi> tdb) st = (let (V', st') = msteps \<pi> st; (V'', st'') = mstep (map_prod mk_db id tdb) st' in (V' @ V'', st''))"
discr_cont: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd"   and i :: "nat"   and s :: "'b" assumes "PL_Indis cval indis"   and "PL_Indis.discr aval tval cval indis c"   and "i < PL.brn c" shows "PL_Indis.discr aval tval cval indis (PL.cont aval tval cval c s i)"
mult_ac_1: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * b * c = a * (b * c)"
simps_9: fixes x101 :: "Term.var"   and x102 :: "expr"   and y101 :: "Term.var"   and y102 :: "expr" shows "(x101:=x102 = y101:=y102) = (x101 = y101 \<and> x102 = y102)"
induct: fixes P :: "('a, 'b, 'c, 'd) Tree \<Rightarrow> ('a, 'b, 'c, 'd) Tree \<Rightarrow> bool"   and a0 :: "('a, 'b, 'c, 'd) Tree"   and a1 :: "('a, 'b, 'c, 'd) Tree" assumes "\<And>tset1 tset2. (\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2\<rbrakk> \<Longrightarrow> P a b) \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. P t1 t2 \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>f1 \<phi>1 f2 \<phi>2. P (tPred f1 \<phi>1) (tPred f2 \<phi>2)"   and "\<And>f1 \<alpha>1 t1 f2 \<alpha>2 t2. \<lbrakk>\<And>x. P (x \<bullet> t1) t2; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t2) t2\<rbrakk> \<Longrightarrow> P (tAct f1 \<alpha>1 t1) (tAct f2 \<alpha>2 t2)"   and "\<And>v va. P (tNot v) (tConj va)"   and "\<And>v va vb. P (tNot v) (tPred va vb)"   and "\<And>v va vb vc. P (tNot v) (tAct va vb vc)"   and "\<And>v va vb. P (tPred v va) (tConj vb)"   and "\<And>v va vb. P (tPred v va) (tNot vb)"   and "\<And>v va vb vc vd. P (tPred v va) (tAct vb vc vd)"   and "\<And>v va vb vc. P (tAct v va vb) (tConj vc)"   and "\<And>v va vb vc. P (tAct v va vb) (tNot vc)"   and "\<And>v va vb vc vd. P (tAct v va vb) (tPred vc vd)"   and "\<And>va v. P (tConj va) (tNot v)"   and "\<And>va vb v. P (tPred va vb) (tNot v)"   and "\<And>va vb vc v. P (tAct va vb vc) (tNot v)"   and "\<And>vb v va. P (tConj vb) (tPred v va)"   and "\<And>vb v va. P (tNot vb) (tPred v va)"   and "\<And>vb vc vd v va. P (tAct vb vc vd) (tPred v va)"   and "\<And>vc v va vb. P (tConj vc) (tAct v va vb)"   and "\<And>vc v va vb. P (tNot vc) (tAct v va vb)"   and "\<And>vc vd v va vb. P (tPred vc vd) (tAct v va vb)" shows "P a0 a1"
simps_1: fixes p :: "char list list" shows "mult_monoms p [] = p"
gcd_dvdI1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c" shows "dvd.dvd times (gcd a b) c"
mkt_def: fixes x :: "'b"   and l :: "'b tree"   and r :: "'b tree" shows "mkt x l r = MKT x l r (max (ht l) (ht r) + 1)"
reconstruct_prod: fixes a :: "complex"   and as :: "complex list"   and b :: "complex"   and bs :: "complex list" shows "reconstruct_poly a as * reconstruct_poly b bs = reconstruct_poly (a * b) (as @ bs)"
eq_foldr: fixes xs :: "('c \<Rightarrow> ('d, 'c) Term.term) list" shows "subst_monoid_mult.prod_list xs = foldr (\<circ>\<^sub>s) xs Var"
s3: shows "\<langle>f x, sum f F\<rangle> = (\<Sum>a\<in>F. \<langle>f x, f a\<rangle>)"
triangle_in_hom_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "partial_magma.in_hom V (V (H \<epsilon> f) (V (bicategory.\<a>' V H \<a> src trg f g f) (H f \<eta>))) (H f (src f)) (H (trg f) f)"
h_def: fixes t :: "real" shows "h t = f (2 - t) / (f (2 - t) + f (t - 1))"
rGamma_complex_plus1: fixes z :: "complex" shows "z * rGamma (z + 1) = rGamma z"
le_multiset_plus_right_nonempty: fixes N :: "'b multiset"   and M :: "'b multiset" assumes "N \<noteq> {#}" shows "M < M + N"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "dlex_pp x xa = dlex_pm (pp.mapping_of x) (pp.mapping_of xa)"
cs_not_not_started_stable: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "snd (cs c cid) \<noteq> NotStarted"   and "channel cid = Some (p, q)" shows "snd (cs c' cid) \<noteq> NotStarted"
stone: fixes x :: "'a" shows "- x \<squnion> - - x = top"
Pz: shows "distributed M P Z (\<lambda>x. ennreal (Pz x))"
rep_list_monomial': fixes j :: "nat" shows "rep_list (monomial (1::'b) (term_of_pair (0::'a, j))) = (fs ! j when j < length fs)"
demorgans1: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) = - x \<squnion> - y"
simps: fixes h :: "heap"   and as :: "nat set" shows "one_assn_raw (h, as) = (as = {})"
le_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<preceq> b"   and "b \<preceq> c" shows "a \<preceq> c"
submatrix_def: fixes A :: "'a mat"   and I :: "nat set"   and J :: "nat set" shows "submatrix A I J = mat (card {i. i < dim_row A \<and> i \<in> I}) (card {j. j < dim_col A \<and> j \<in> J}) (\<lambda>(i, j). A $$ (pick I i, pick J j))"
f1: shows "\<exists>t1 t2. Some t1 = NEXT s1 \<and> Some t2 = NEXT s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0"
Ik_intT_int: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wt T"   and "I.wtE \<xi>"   and "infTp (tpOf T) \<or> (\<forall>x\<in>nv2T T. tpOfV x \<noteq> tpOf T)" shows "Ik_intT (tpOf T) (I.int \<xi> T)"
l_get_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and create_document :: "((_) heap, exception, (_) document_ptr) prog" shows "l_get_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_tag_name create_document \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name \<and> l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document"
normalize_positive_ports_step_normalized_n_primitive: fixes m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc_negated disc False m" shows "\<forall>m'\<in>set (normalize_positive_ports_step (disc, sel) C m). normalized_n_primitive (disc, sel) (\<lambda>ps. case ps of L4Ports x pts \<Rightarrow> length pts \<le> 1) m'"
simps_12: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "bexp"   and vc :: "'a acom"   and vd :: "'a acom"   and ve :: "'a" shows "(v;; va \<sqsubseteq> IF vb THEN vc ELSE vd {ve}) = False"
bigthetaD1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "f \<in> O[F](g)"
cases: fixes a1 :: "'b"   and a2 :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a1, a2) \<in> r\<^sup>*" obtains a :: "'b" where "a1 = a__" and "a2 = a__"   | a :: "'b" and b :: "'b" and c :: "'b" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>*" and "(b, c) \<in> r"
imp_disj_not1: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "(P \<longrightarrow> Q \<or> R) = (\<not> Q \<longrightarrow> P \<longrightarrow> R)"
NSLIMSEQ_add: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a"   and Y :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a"   and "Y \<longlonglongrightarrow>\<^sub>N\<^sub>S b" shows "(\<lambda>n. X n + Y n) \<longlonglongrightarrow>\<^sub>N\<^sub>S a + b"
composites_are_arrows_1: shows "hseq g (f \<star> g)"
ncrit_spec_chain_ncrit: shows "punit.ncrit_spec punit.chain_ncrit"
abs_eq: fixes xa :: "'b"   and x :: "'b" shows "Upair xa x = abs_uprod (xa, x)"
posaux3: fixes a :: "'b"   and l :: "'b list"   and b :: "'b" assumes "a \<in> set l"   and "b \<in> set l"   and "a \<noteq> b" shows "position a l \<noteq> position b l"
lock_thread_okE: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option" assumes "lock_thread_ok ls ts" obtains "\<forall>l t. has_lock (ls $ l) t \<longrightarrow> (\<exists>xw. ts t = \<lfloor>xw\<rfloor>)"
simps_6: fixes r :: "mregex" shows "LPD (MStar r) = MTimesR (LPD r) (MStar r)"
PAC_Format_subset_ideal: fixes \<V> :: "nat set"   and A :: "int mpoly multiset"   and \<V>' :: "nat set"   and B :: "int mpoly multiset" assumes "PAC_Format (\<V>, A) (\<V>', B)"   and "\<Union> (vars ` set_mset A) \<subseteq> \<V>" shows "restricted_ideal_to\<^sub>I \<V> B \<subseteq> restricted_ideal_to\<^sub>I \<V> A \<and> \<V> \<subseteq> \<V>' \<and> \<Union> (vars ` set_mset B) \<subseteq> \<V>'"
t\<^sub>p_def: fixes s :: "'b list"   and q :: "'b"   and a :: "nat \<times> nat list" shows "t\<^sub>p s q a = (let (mf, sws) = a in index (swaps sws s) q + length sws)"
pair_term: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<times> 'c" assumes "term_powerprod pair_of_term term_of_pair" shows "pair_of_term (term_of_pair p) = p"
intro: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "\<alpha> \<in> S \<rightarrow>\<^sub>E T" shows "Set_Theory.map \<alpha> S T"
Inf_lower: fixes x :: "'a \<times> 'b"   and A :: "('a \<times> 'b) set" assumes "x \<in> A" shows "less_eq_prod (Inf_prod A) x"
lappend_inf: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "\<not> lfinite xs" shows "lappend xs ys = xs"
adm_subset: fixes u :: "'a \<Rightarrow> 'b set"   and v :: "'a \<Rightarrow> 'b set" assumes "cont u"   and "cont v" shows "adm (\<lambda>x. u x \<subseteq> v x)"
weak_ranking_altdef: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite_total_preorder_on carrier le" shows "weak_ranking le = finite_total_preorder_on.weak_ranking_aux carrier le carrier"
flow_initial_time: fixes x0 :: "'a" assumes "t\<^sub>0 \<in> T"   and "x0 \<in> S" shows "flow t\<^sub>0 x0 t\<^sub>0 = x0"
inj_on_arev: fixes S :: "'b set" shows "inj_on rev_H S"
awalk_ends: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "G.awalk u p v"   and "G.awalk u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
R\<^sub>C_of_closed_glob_consistent: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" assumes "\<And>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mdsa mem\<^sub>2. (\<langle>c\<^sub>1, mdsa, mem\<^sub>1\<rangle>\<^sub>A, \<langle>c\<^sub>2, mdsa, mem\<^sub>2\<rangle>\<^sub>A) \<in> \<R>\<^sub>A \<Longrightarrow> abs.low_mds_eq mdsa mem\<^sub>1 mem\<^sub>2"   and "abs.closed_glob_consistent \<R>\<^sub>A"   and "secure_refinement \<R>\<^sub>A \<R> P" shows "conc.closed_glob_consistent (R\<^sub>C_of \<R>\<^sub>A \<R> P)"
record_call_below_arg: fixes x :: "'a"   and f :: "'a \<Rightarrow> one\<^sub>\<bottom>" shows "record_call x\<cdot>f \<sqsubseteq> f"
add_closed: fixes g :: "'f"   and h :: "'f" assumes "g \<in> UNIV"   and "h \<in> UNIV" shows "g + h \<in> UNIV"
nsubst_fml: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'a \<Rightarrow> ('sf, 'sz) trm"   and \<phi> :: "('sf + 'a, 'sc, 'sz) formula"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "NFadmit \<sigma> \<phi>"   and "fsafe \<phi>"   and "\<forall>i. dsafe (\<sigma> i)" shows "(\<nu> \<in> fml_sem I (FsubstFO \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (adjointFO I \<sigma> \<nu>) \<phi>)"
simps_6: fixes a :: "Arity"   and scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp" shows "ccTransform a (scrut ? e1 : e2) = (ccTransform 0 scrut ? ccTransform a e1 : ccTransform a e2)"