hcju/isabelleps huggingface dataset:
induct: fixes P :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> bool"   and a0 :: "fm"   and a1 :: "poly"   and a2 :: "tm" assumes "\<And>p q c t. \<lbrakk>P p c t; P q c t\<rbrakk> \<Longrightarrow> P (And p q) c t"   and "\<And>p q c t. \<lbrakk>P p c t; P q c t\<rbrakk> \<Longrightarrow> P (Or p q) c t"   and "\<And>a r c t. P (Eq (CNP 0 a r)) c t"   and "\<And>a r c t. P (NEq (CNP 0 a r)) c t"   and "\<And>a r c t. P (Lt (CNP 0 a r)) c t"   and "\<And>a r c t. P (Le (CNP 0 a r)) c t"   and "\<And>c t. P T c t"   and "\<And>c t. P F c t"   and "\<And>va c t. P (Le (CP va)) c t"   and "\<And>va c t. P (Le (tm.Bound va)) c t"   and "\<And>va vb c t. P (Le (tm.Add va vb)) c t"   and "\<And>va vb c t. P (Le (tm.Mul va vb)) c t"   and "\<And>va c t. P (Le (tm.Neg va)) c t"   and "\<And>va vb c t. P (Le (tm.Sub va vb)) c t"   and "\<And>vd vb vc c t. P (Le (CNP (Suc vd) vb vc)) c t"   and "\<And>va c t. P (Lt (CP va)) c t"   and "\<And>va c t. P (Lt (tm.Bound va)) c t"   and "\<And>va vb c t. P (Lt (tm.Add va vb)) c t"   and "\<And>va vb c t. P (Lt (tm.Mul va vb)) c t"   and "\<And>va c t. P (Lt (tm.Neg va)) c t"   and "\<And>va vb c t. P (Lt (tm.Sub va vb)) c t"   and "\<And>vd vb vc c t. P (Lt (CNP (Suc vd) vb vc)) c t"   and "\<And>va c t. P (Eq (CP va)) c t"   and "\<And>va c t. P (Eq (tm.Bound va)) c t"   and "\<And>va vb c t. P (Eq (tm.Add va vb)) c t"   and "\<And>va vb c t. P (Eq (tm.Mul va vb)) c t"   and "\<And>va c t. P (Eq (tm.Neg va)) c t"   and "\<And>va vb c t. P (Eq (tm.Sub va vb)) c t"   and "\<And>vd vb vc c t. P (Eq (CNP (Suc vd) vb vc)) c t"   and "\<And>va c t. P (NEq (CP va)) c t"   and "\<And>va c t. P (NEq (tm.Bound va)) c t"   and "\<And>va vb c t. P (NEq (tm.Add va vb)) c t"   and "\<And>va vb c t. P (NEq (tm.Mul va vb)) c t"   and "\<And>va c t. P (NEq (tm.Neg va)) c t"   and "\<And>va vb c t. P (NEq (tm.Sub va vb)) c t"   and "\<And>vd vb vc c t. P (NEq (CNP (Suc vd) vb vc)) c t"   and "\<And>v c t. P (NOT v) c t"   and "\<And>v va c t. P (Imp v va) c t"   and "\<And>v va c t. P (Iff v va) c t"   and "\<And>v c t. P (E v) c t"   and "\<And>v c t. P (A v) c t" shows "P a0 a1 a2"
FV_Upd: fixes t :: "sterm"   and l :: "Label"   and s :: "sterm" shows "FV (Upd t l s) = FV t \<union> FV s"
uncountable_open_interval: fixes a :: "real"   and b :: "real" shows "uncountable {a<..<b} = (a < b)"
Rep_bit1: fixes x :: "'b bit1" shows "Rep_bit1 x \<in> {0..<1 + 2 * int CARD('b)}"
s'_def: shows "s' = take \<^bold>|p'\<^bold>| w"
fold_assoc_matches_first_step_not_None: fixes x :: "typ"   and xs :: "typ list"   and y :: "typ"   and ys :: "typ list"   and subs :: "((variable \<times> String.literal set) \<times> typ) list"   and subs' :: "((variable \<times> String.literal set) \<times> typ) list" assumes "fold (\<lambda>(T, U) subs. Option.bind subs (assoc_match T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'" obtains point :: "((variable \<times> String.literal set) \<times> typ) list" where "assoc_match x y subs = Some point" and "fold (\<lambda>(T, U) subs. Option.bind subs (assoc_match T U)) (zip xs ys) (Some point) = Some subs'"
dghm_ObjMap_vrange: shows "\<R>\<^sub>\<circ> (\<GG>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
lset_LCons: fixes x :: "'c"   and xs :: "'c llist" shows "lset (LCons x xs) = insert x (lset xs)"
rep_eq: fixes x :: "'d list" shows "pdevs_apply (pdevs_of_list x) = (\<lambda>i. if i < length x then x ! i else (0::'d))"
L4: shows "\<forall>z. (z \<in> V) = (\<exists>x y. z = c_pair x y \<and> (x \<in> A \<and> y = 0 \<or> x \<notin> A \<and> y = 1))"
closed_segment_same_Im: fixes a :: "complex"   and b :: "complex" assumes "Im a = Im b" shows "closed_segment a b = {z. Im z = Im a \<and> Re z \<in> closed_segment (Re a) (Re b)}"
ax5_d: fixes x :: "'a" shows "d (x \<sqinter> nc) \<cdot> U = (x \<sqinter> nc) \<cdot> U"
is_document_ptr_kind_obtains: fixes ptr :: "(_) object_ptr" assumes "is_document_ptr_kind ptr" obtains document_ptr :: "(_) document_ptr" where "ptr = 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"
opp_res_fold_A_endo_A: shows "ChamberComplexEndomorphism A (restrict1 opp_fold_A (\<Union> A))"
merge_stable_segments: fixes cmp :: "'b comparator"   and l :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "Sorting_Algorithms.merge cmp (filter (\<lambda>y. compare cmp l y = Equiv) xs) (filter (\<lambda>y. compare cmp l y = Equiv) ys) = filter (\<lambda>y. compare cmp l y = Equiv) xs @ filter (\<lambda>y. compare cmp l y = Equiv) ys"
lex_ord_pair_ko_def: fixes ko :: "'a key_order" shows "lex_ord_pair_ko ko \<equiv> comparator.lex_ord_pair (key_compare ko)"
IO: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and input :: "'f"   and \<Gamma> :: "('e, 'f) \<I>"   and x :: "'d" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out"   and "x \<in> results_gpv \<Gamma> (c input)" shows "x \<in> results_gpv \<Gamma> gpv"
is_locale: shows "index_locale index_class.less_eq index_class.less idx index_class.size"
Abs_ainj_all: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "good X"   and "good X'" shows "(Abs xs x X = Abs xs' x' X') = (xs = xs' \<and> (\<forall>y. (y = x \<or> fresh xs y X) \<and> (y = x' \<or> fresh xs y X') \<longrightarrow> X #[y \<and> x]_xs = X' #[y \<and> x']_xs))"
ex_has_least_nat: fixes k :: "'a ugraph"   and m :: "'a ugraph \<Rightarrow> nat" assumes "is_spanning_tree g k" shows "\<exists>x. is_spanning_tree g x \<and> (\<forall>y. is_spanning_tree g y \<longrightarrow> m x \<le> m y)"
surjI: fixes g :: "'i \<Rightarrow> 'h"   and f :: "'h \<Rightarrow> 'i" assumes "\<And>x. g (f x) = x" shows "surj g"
wf_dia: fixes E :: "(node fset \<times> command_gadget \<times> node fset) list"   and V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget" assumes "\<forall>e\<in>set E. wf_com (snd3 e)"   and "\<forall>v\<in>fset V. wf_ass (\<Lambda> v)"   and "acyclicity E"   and "linearity E"   and "\<forall>e\<in>set E. fst3 e |\<union>| thd3 e |\<subseteq>| V" shows "wf_dia (Graph V \<Lambda> E)"
count_it_ex: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (WHILE b DO c, s) \<Rightarrow> t" shows "\<exists>n. count_it \<pi> b c s n"
pelims: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b \<times> 'b \<times> 'b \<times> 'b"   and y :: "'a \<times> 'a \<times> 'a \<times> 'a" assumes "mat_map x xa = y"   and "mat_map_dom (x, xa)" obtains f :: "'b \<Rightarrow> 'a" and a :: "'b" and b :: "'b" and c :: "'b" and d :: "'b" where "x = f" and "xa = (a, b, c, d)" and "y = (f a, f b, f c, f d)" and "mat_map_dom (f, a, b, c, d)"
TESL_sem_decreases_tail: fixes \<Phi> :: "'b TESL_atomic list"   and \<phi> :: "'b TESL_atomic" shows "\<lbrakk>\<lbrakk> \<Phi> @ [\<phi>] \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<subseteq> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
adopt_node_locs_def: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   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 known_ptrs :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr" assumes "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs remove" shows "adopt_node_locs parent owner_document document_ptr = (if parent = None then {} else remove_child_locs (the parent) owner_document) \<union> set_disconnected_nodes_locs document_ptr \<union> set_disconnected_nodes_locs owner_document"
simple_imp_good_ruleset: fixes rs :: "'a rule list" assumes "simple_ruleset rs" shows "good_ruleset rs"
assms_4: shows "\<forall>i<k. f i = g i"
prj_simps_7: shows "C.cod \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>1 = \<mu>.apex"
SINVAR_LIB_BLPtrusted_def: shows "SINVAR_LIB_BLPtrusted \<equiv> \<lparr>nm_name = ''BLPtrusted'', nm_receiver_violation = SINVAR_BLPtrusted.receiver_violation, nm_default = SINVAR_BLPtrusted.default_node_properties, nm_sinvar = SINVAR_BLPtrusted_impl.sinvar, nm_offending_flows = SINVAR_BLPtrusted_impl.BLP_offending_list, nm_node_props = SINVAR_BLPtrusted_impl.NetModel_node_props, nm_eval = SINVAR_BLPtrusted_impl.BLP_eval\<rparr>"
DERIV_deriv_iff_has_field_derivative: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "(f has_field_derivative deriv f x) (at x) = (\<exists>f'. (f has_field_derivative f') (at x))"
reduced_GB_Polys: fixes X :: "'a option set"   and F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "extended_ord.punit.reduced_GB F \<subseteq> P[X]"
abs_summable_on_norm_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "((\<lambda>x. norm (f x)) abs_summable_on A) = (f abs_summable_on A)"
ex_psi_in_U: fixes s :: "nat \<Rightarrow> nat option" assumes "cons_lim s" shows "\<exists>j. cons_lim.psi s j \<in> U\<^bsub>LIM-CONS\<^esub>"
key_sets_crypts_subset: fixes U :: "msg set"   and X :: "msg"   and H :: "(agent \<times> msg) set"   and H' :: "(agent \<times> msg) set" assumes "U \<in> key_sets X (crypts (Log -` spied H))"   and "H \<subseteq> H'" shows "U \<in> key_sets X (crypts (Log -` spied H'))"
simps_8: fixes bs :: "real list"   and a :: "MIR.num" shows "Inum bs (Floor a) = real_of_int \<lfloor>Inum bs a\<rfloor>"
moebius_ocircline_comp_inv_left: fixes M :: "moebius"   and H :: "ocircline" shows "moebius_ocircline (moebius_inv M) (moebius_ocircline M H) = H"
psubset_insert_iff: fixes A :: "'d set"   and x :: "'d"   and B :: "'d set" shows "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_set_val type_wf set_val set_val_locs"   and "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_val create_character_data known_ptr" shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs type_wf create_character_data known_ptr"
subset_imp_guarantees_UNIV: fixes X :: "'b program set"   and Y :: "'b program set" assumes "X \<subseteq> Y" shows "X guarantees Y = UNIV"
insertI: fixes A :: "'a set"   and a :: "'a" assumes "finite A" shows "finite (insert a A)"
intros_6: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and f :: "'c" shows "\<Gamma>\<turnstile>c \<down> Fault f"
am6: fixes x :: "'a" shows "x \<cdot> ad (1::'a) = ad (1::'a)"
ladder_last_n_intro: fixes L :: "(nat \<times> nat \<times> nat) list" assumes "L \<noteq> []" shows "ladder_n L (length L - Suc 0) = ladder_last_n L"
th: shows "real_of_int \<lfloor>Inum bs e\<rfloor> = Inum bs e"
distinct_33: shows "CharactersOp \<noteq> string_unop.ToIntegerOp"
While2: fixes b :: "bexp"   and l :: "nat"   and c :: "com" assumes "sec b \<le> l"   and "\<turnstile> c : l" shows "\<turnstile> WHILE b DO c : l"
finfun_snd_const: fixes bc :: "'c \<times> 'b" shows "finfun_snd (finfun_const bc) = finfun_const (snd bc)"
atS_simps_3: fixes l :: "'a"   and s :: "('b, 'a, 'c, 'd, 'e) system_state"   and p :: "'c"   and ls :: "'a set" assumes "at p l s"   and "l \<in> ls" shows "atS p ls s"
Diff_Diff_Int: fixes A :: "'b set"   and B :: "'b set" shows "A - (A - B) = A \<inter> B"
Suc_times_gbinomial: fixes k :: "nat"   and a :: "'a" shows "of_nat (Suc k) * (a + (1::'a) gchoose Suc k) = (a + (1::'a)) * (a gchoose k)"
subset_ordLeq_strict: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "|A| <o |B|" shows "A \<subset> B"
sum_mset_delta: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and y :: "'b"   and c :: "'a"   and A :: "'b multiset" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum_mset plus zero {#if x = y then c else zero. x \<in># A#} = times c (semiring_1.of_nat one plus zero (count A y))"
all: shows "\<forall>x. \<beta> x \<le> \<alpha> x"
real_pp_np_pos_times: fixes a :: "real"   and f :: "'b \<Rightarrow> real" assumes "0 \<le> a" shows "pp (\<lambda>t. a * f t) = (\<lambda>t. a * pp f t) \<and> np (\<lambda>t. a * f t) = (\<lambda>t. a * np f t)"
dimension_one: fixes K :: "'a set" assumes "subfield K R" shows "dimension 1 K K"
valid_frac_one: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "Comm_Ring.quotient_ring S R addition multiplication zero unit" shows "quotient_ring.valid_frac S R addition multiplication zero (quotient_ring.one_rel S R addition multiplication zero unit)"
ccSUP_subset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable B"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
atoms_norm: fixes r :: "nat rexp" shows "atoms (norm r) \<subseteq> atoms r"
retraction_def: fixes g :: "'a \<times> 'a" shows "CC.retraction g \<equiv> \<exists>f. CC.ide (CC.comp g f)"
simps_1: fixes u :: "'a local.term" shows "\<^bold>\<I> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u = u"
l_set_mode_get_tag_name_axioms: shows "l_set_mode_get_tag_name type_wf set_mode set_mode_locs get_tag_name get_tag_name_locs"
eq_statesI: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "\<R>\<^sub>G \<alpha> \<beta> s = s'"   and "bij \<alpha>"   and "bij \<beta>" shows "s \<approx> s'"
cnj_inverse: fixes x :: "quat" shows "Quaternions.cnj (inverse x) = inverse (Quaternions.cnj x)"
assms_1: shows "netgmap sr (SubnetS s1 s2) = netmask (net_tree_ips (n1 \<parallel> n2)) (\<sigma>, \<zeta>)"
neg_mask_combine: fixes a :: "nat"   and b :: "nat" shows "NOT (mask a) AND NOT (mask b) = NOT (mask (max a b))"
c: shows "(P, s) -c\<rightarrow> (Q, t)"
fresh_set: fixes a :: "atom"   and xs :: "'a list" shows "a \<sharp> set xs = a \<sharp> xs"
map\<^sub>1_simps_5: fixes f :: "'c" assumes "C.ide f" shows "D.cod (\<rho>\<^sub>1 f) = \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f"
ns_s_mul_ext_trans: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "trans s"   and "trans ns"   and "compatible_l ns s"   and "compatible_r ns s"   and "refl ns"   and "(A, B) \<in> ns_mul_ext ns s"   and "(B, C) \<in> s_mul_ext ns s" shows "(A, C) \<in> s_mul_ext ns s"
inext_less_mono_rev: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "inext a I < inext b I"   and "a \<in> I"   and "b \<in> I" shows "a < b"
comp_reduce: fixes f :: "'c"   and g :: "'c"   and k :: "'c"   and h :: "'c" assumes "C f g = k"   and "C.seq f g"   and "C.seq g h" shows "C f (C g h) = C k h"
conjugate_scalar_prod_Im: fixes v :: "complex vec" shows "Im (inner_prod v v) = 0"
poly_set_iteratei_axioms: shows "poly_set_iteratei \<alpha> invar list_it"
assms_1: shows "S \<subseteq> U\<^sub>M\<^sub>_\<^sub>1"
cInf_le_finite: fixes X :: "'a multiset set"   and x :: "'a multiset" assumes "finite X"   and "x \<in> X" shows "Inf X \<subseteq># x"
le_1: shows "real_of_rat l \<le> real_of_1 (p, l, r)"
RTranCl_induct: fixes h :: "'c"   and h' :: "'c"   and succs :: "'c \<Rightarrow> 'c list"   and P :: "'c \<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'"
det_mul: fixes A :: "(('b, 'c) vec, 'c) vec"   and B :: "(('b, 'c) vec, 'c) vec" shows "det (A ** B) = det A * det B"
total_on_subset: fixes B :: "'b set"   and R :: "('b \<times> 'b) set"   and A :: "'b set" assumes "total_on B R"   and "A \<subseteq> B" shows "total_on A R"
GR: shows "(Gi, G) \<in> igbg_impl_rel_ext Re Rv"
local_ring_axioms_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "local_ring_axioms R addition multiplication zero unit \<equiv> (\<forall>I J. max_lideal I R addition multiplication zero unit \<longrightarrow> max_lideal J R addition multiplication zero unit \<longrightarrow> I = J) \<and> (\<exists>\<ww>. max_lideal \<ww> R addition multiplication zero unit)"
inj_uni_samp_add: fixes y :: "nat"   and q :: "nat" shows "inj_on (\<lambda>b. (y + b) mod q) {..<q}"
rep_eq: shows "pmf local.geometric_pmf = (\<lambda>n. (1 - p) ^ n * p)"
accimethds_simp: fixes G :: "prog"   and I :: "qtname"   and pack :: "pname" assumes "G \<turnstile> Iface I accessible_in pack" shows "accimethds G pack I = DeclConcepts.imethds G I"
adj_reachable_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<rightarrow> b"   and "b \<rightarrow>\<^sup>* c" shows "a \<rightarrow>\<^sup>* c"
binomial: fixes a :: "nat"   and b :: "nat"   and n :: "nat" shows "(a + b) ^ n = (\<Sum>k\<le>n. of_nat (n choose k) * a ^ k * b ^ (n - k))"
rel_interior_Times: fixes S :: "'b set"   and T :: "'c set" assumes "convex S"   and "convex T" shows "rel_interior (S \<times> T) = rel_interior S \<times> rel_interior T"
optimize_matches_append: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr"   and rs1 :: "'b rule list"   and rs2 :: "'b rule list" shows "optimize_matches f (rs1 @ rs2) = optimize_matches f rs1 @ optimize_matches f rs2"
prefix_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<le>p ys" shows "xs \<le>p ys \<cdot> zs"
is_process5_S1: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set"   and sa :: "'a event list"   and X :: "'a event set"   and Y :: "'a event set" assumes "is_process P"   and "(sa, X) \<in> FAILURES P"   and "(sa, X \<union> Y) \<notin> FAILURES P" shows "\<exists>c. c \<in> Y \<and> (sa @ [c], {}) \<in> FAILURES P"
t: shows "t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t S"
indRelST_symm: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "sym SRel"   and "sym TRel" shows "sym (indRelST SRel TRel)"
inf_bot_right: fixes x :: "'a \<times> 'b" shows "inf_prod x bot_prod = bot_prod"
inject_4: fixes x8 :: "nat"   and y8 :: "nat" shows "(Br x8 = Br y8) = (x8 = y8)"
has_field_idemp: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C has F,b:T in D" shows "P \<turnstile> D has F,b:T in D"
int_bv_int: fixes w :: "bit list" shows "int_to_bv (bv_to_int w) = norm_signed w"
welltyped_Abs_fresh: fixes x :: "char list"   and T :: "Unary_PCF.type"   and u :: "expr"   and U :: "Unary_PCF.type" assumes "fresh (x, T) u"   and "open0_Var (x, T) u ::: U" shows "\<Lambda>\<langle>T\<rangle> u ::: T \<rightarrow> U"
in_sat_d: shows "set_mset CC' \<subseteq> saturate DD"
preserves_reflects_arr: fixes f :: "'j" shows "C.arr (\<chi> f) = J.arr f"
select_result_I2: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog"   and x :: "'o" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = x"
normalize_square: fixes p :: "(real, 2) vec" shows "(normalize p $ 1)\<^sup>2 + (normalize p $ 2)\<^sup>2 = 1"
measure_const: shows "(strict {(x, y). x \<sqsubseteq> y})\<inverse> \<subseteq> measure m_const"
insert_def: fixes e :: "'a"   and a :: "'b"   and q :: "('a, 'b) SkewBinomialHeap" shows "SkewBinomialHeap_loc.insert e a q \<equiv> Abs_SkewBinomialHeap (BsSkewBinomialHeapStruc.bs_insert e a (Rep_SkewBinomialHeap q))"
to_map_def: fixes A :: "('b \<times> 'b) set"   and f :: "'b \<times> 'b \<Rightarrow> 'b \<times> 'b" shows "to_map A f = \<lparr>edge_rev = swap_in A, edge_succ = f\<rparr>"
get_dom_component_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_component ptr" shows "ptr |\<in>| object_ptr_kinds h"