hcju/isabelleps huggingface dataset:
group_subgroup_generated: fixes G :: "('f, 'g) monoid_scheme"   and S :: "'f set" assumes "Group.group G" shows "Group.group (subgroup_generated G S)"
sum_divide_distrib: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and r :: "'a" shows "sum f A / r = (\<Sum>n\<in>A. f n / r)"
subgraph: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow> b" shows "E a b"
extensional_arb: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e" assumes "f \<in> extensional A"   and "x \<notin> A" shows "f x = undefined"
preserves_iso: fixes f :: "'d" assumes "local.iso f" shows "local.iso (\<ll>' f)"
of_nat_mod_real: fixes a :: "nat"   and b :: "nat" shows "real (a mod b) = rnatmod (real a) (real b)"
Loop: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and c :: "stmt"   and l :: "label" assumes "E,dt\<Turnstile>e\<Colon>-PrimT Boolean"   and "E,dt\<Turnstile>c\<Colon>\<surd>" shows "E,dt\<Turnstile>l\<bullet> While(e) c\<Colon>\<surd>"
method_nameI: fixes cl :: "cl"   and meth :: "char list"   and vds :: "vd list"   and meth_body :: "meth_body" shows "method_name (meth_def_def (meth_sig_def cl meth vds) meth_body) meth"
fresh_1: fixes a :: "name"   and x1 :: "nat" shows "a \<sharp> Atom x1 = a \<sharp> x1"
negligible_imp_measurable: fixes S :: "'b set" assumes "negligible S" shows "S \<in> lmeasurable"
map_upds_twist: fixes a :: "'b"   and as :: "'b list"   and m :: "'b \<Rightarrow> 'c option"   and b :: "'c"   and bs :: "'c list" assumes "a \<notin> set as" shows "m(a \<mapsto> b, as [\<mapsto>] bs) = m(as [\<mapsto>] bs, a \<mapsto> b)"
star_ref: fixes x :: "'a" shows "(1::'a) \<le> qstar x"
lm046: fixes a :: "('c \<times> 'd set) set"   and X :: "'c set"   and i :: "'c" assumes "Domain a \<inter> X \<noteq> {}"   and "a \<in> allocationsUniverse" shows "{\<Union> (a `` (X \<union> {i}))} - {{}} = {\<Union> (a `` (X \<union> {i}))}"
strict_mono_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x = f y) = (x = y)"
P3: shows "Bet Q' Q Y \<and> Cong Q Y P A"
continuous_on_op_minus: fixes s :: "'a set"   and x :: "'a" shows "continuous_on s ((-) x)"
fold_maybe_or_cons: fixes a :: "trilean"   and l :: "trilean list"   and b :: "trilean" shows "fold (\<or>?) (a # l) b = a \<or>? fold (\<or>?) l b"
drop_append: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'a list" shows "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
mem_tangent_space: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" shows "(X \<in> manifold_eucl.dest.tangent_space p) = (manifold_eucl.dest.is_derivation X p \<and> extensional0 manifold_eucl.dest.diff_fun_space X)"
plus_stream_stl: fixes x :: "'a stream"   and y :: "'a stream" shows "stl (x + y) = stl x + stl y"
dom_cod: fixes f :: "'a2" assumes "A2.arr f" shows "A2.dom (A2.cod f) = A2.cod f"
elims_2: fixes x :: "nat"   and xa :: "(nat \<times> nat) list"   and xb :: "nat" assumes "isPath x xa xb" obtains u :: "nat" and v :: "nat" where "x = ua__" and "xa = []" and "xb = va__" and "ua__ = va__"   | u :: "nat" and xc :: "nat" and y :: "nat" and p :: "(nat \<times> nat) list" and v :: "nat" where "x = u" and "xa = (xc, y) # p" and "xb = v" and "u = xc \<and> (xc, y) \<in> E \<and> isPath y p v"
fmap_writer_def: shows "fmap = writer_map\<cdot>ID"
rep_eq: fixes x :: "'b fset fset" shows "\<langle>ffUnion x\<rangle>\<^sub>f = \<Union> (fset ` \<langle>x\<rangle>\<^sub>f)"
nres_more_simps_6: fixes x :: "'j"   and y :: "'j" shows "(RETURN x = RETURN y) = (x = y)"
CSP2_Sup: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "\<lambda>A. A \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` A} is CSP2 healthy"
ih_9: shows "InvariantWatchListsUniq (getWatchList state')"
insert_def: fixes a :: "'e"   and B :: "'e set" shows "Set.insert a B = {x. x = a} \<union> B"
in_GSMP_in_proj: fixes t :: "('fun, 'var) Term.term"   and A :: "('a strand_label \<times> ('fun, 'var) strand_step) list" assumes "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t A)" shows "\<exists>n. t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A))"
ops2: shows "insert_ops (before @ (oid, ref) # op_list)"
image_image: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and A :: "'h set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
image_mset_subseteq_mono: fixes A :: "'k multiset"   and B :: "'k multiset"   and f :: "'k \<Rightarrow> 'l" assumes "A \<subseteq># B" shows "image_mset f A \<subseteq># image_mset f B"
eulerian_cons: fixes v0 :: "'v"   and w :: "'w"   and v1 :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v2 :: "'v" assumes "valid_unMultigraph.is_Eulerian_trail (del_unEdge v0 w v1 G) v1 ps v2"   and "(v0, w, v1) \<in> E" shows "is_Eulerian_trail v0 ((v0, w, v1) # ps) v2"
the_ext_def: fixes y' :: "'c"   and g :: "'d" shows "the_ext y' g = (THE f. is_ext y' g f)"
pt_name1: fixes x :: "'b" shows "[] \<bullet> x = x"
hyps: shows "(xs, zs) \<in> \<langle>R O S\<rangle>list_wset_rel"
chain_map_add: fixes p :: "nat"   and g :: "'c \<Rightarrow> 'b"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "chain_map p g (a + b) = chain_map p g a + chain_map p g b"
interaction_bound_fixp_induct: fixes P :: "(('a, 'b, 'c) gpv \<Rightarrow> enat) \<Rightarrow> bool"   and "consider" :: "'b \<Rightarrow> bool" assumes "lfp.admissible P"   and "P (\<lambda>_. 0)"   and "\<And>interaction_bound'. \<lbrakk>P interaction_bound'; \<And>gpv. interaction_bound' gpv \<le> interaction_bound consider gpv; \<And>gpv. interaction_bound' gpv \<le> (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))\<rbrakk> \<Longrightarrow> P (\<lambda>gpv. \<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))" shows "P (interaction_bound consider)"
transl_rules_def: fixes T :: "('b allegorical_term \<times> 'b allegorical_term) set" shows "transl_rules T = (\<Union>(x, y)\<in>T. {(translation x, translation (A_Int x y)), (translation y, translation (A_Int y x))})"
strategyproof_sds_axioms_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "strategyproof_sds_axioms agents alts sds \<equiv> \<forall>R i Ri'. pref_profile_wf agents alts R \<longrightarrow> i \<in> agents \<longrightarrow> total_preorder_on alts Ri' \<longrightarrow> \<not> social_decision_scheme.manipulable_profile sds R i Ri'"
eq_\<I>_gpv_1: shows "gen_lossless_gpv b \<I> gpv"
mojmir_accept_iff_token_set_accept: shows "accept\<^sub>R' = (\<exists>i<max_rank\<^sub>R. finite fail \<and> finite (local.merge i) \<and> infinite (succeed i))"
quadratic_poly_eval: fixes c :: "real"   and b :: "real"   and a :: "real"   and x :: "real" shows "poly [:c, b, a:] x = a * x\<^sup>2 + b * x + c"
PREFER_tagI: fixes P :: "bool" assumes "P" shows "PREFER_tag P"
simps_1: fixes c :: "'b poly"   and p :: "'b poly" shows "Polynomial_Expression.degree (CN c 0 p) = 1 + Polynomial_Expression.degree p"
arden_l: fixes y :: "'a list set"   and x :: "'a list set"   and z :: "'a list set" assumes "\<not> l_ewp y"   and "x = y \<cdot> x + z" shows "x = y\<^sup>\<star> \<cdot> z"
sum_indicator_eq_card: fixes A :: "'b set"   and B :: "'b set" assumes "finite A" shows "sum (chi B) A = card (A \<inter> B)"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "TE.int \<xi> (Fn f Tl) = TE_intF f (map (TE.int \<xi>) Tl)"
conf_hext: fixes h :: "heap"   and h' :: "heap"   and P :: "'b prog"   and v :: "val"   and T :: "ty" assumes "h \<unlhd> h'"   and "P,h \<turnstile> v :\<le> T" shows "P,h' \<turnstile> v :\<le> T"
gjk: fixes x :: "'a" assumes "x \<in> carrier B" shows "g (j x) = k x"
intros_7: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and i :: "name" assumes "H \<turnstile> A IMP B"   and "atom i \<sharp> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> exi i A IMP B"
fminus_disjoint: fixes A :: "'c fset"   and B :: "'c fset" shows "A |\<inter>| (B |-| A) = {||}"
simps_1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and pc :: "nat" shows "match_ex_table P C pc [] = None"
a2: fixes r :: "'d rexp"   and s :: "'d rexp"   and t :: "'d rexp" shows "Alt r (Alt s t) ~ Alt (Alt r s) t"
fact_1: fixes q :: "int poly"   and p :: "int poly" assumes "q \<in> set (factors_of_int_poly p)" shows "irreducible q \<and> 0 < lead_coeff q \<and> degree q \<le> degree p \<and> degree q \<noteq> 0"
set_integral_norm_bound: fixes M :: "'b measure"   and k :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "set_integrable M k f" shows "norm (set_lebesgue_integral M k f) \<le> \<integral>x\<in>k. norm (f x)\<partial>M"
le_less: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't" shows "(x\<le>\<^sub>py) = (x<\<^sub>py \<or> x = y)"
liftrT_set: fixes a :: "char list"   and T :: "POPLmarkRecord.type"   and fs :: "(char list \<times> POPLmarkRecord.type) list"   and n :: "nat"   and k :: "nat" assumes "(a, T) \<in> set fs" shows "(a, \<up>\<^sub>\<tau> n k T) \<in> set (\<up>\<^sub>r\<^sub>\<tau> n k fs)"
subgroup_is_submonoid: shows "submonoid {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}} (product_group I G)"
produces_imp_Pos_in_lits: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C = {A}" shows "Pos A \<in># C"
of_int_of_bool: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and P :: "bool" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (of_bool P) = zero_neq_one.of_bool one zero P"
Suc_le_lessD: fixes m :: "nat"   and n :: "nat" assumes "Suc m \<le> n" shows "m < n"
lens_create_def: fixes X :: "'d \<Longrightarrow> 'c"   and v :: "'d" shows "create\<^bsub>X\<^esub> v = put\<^bsub>X\<^esub> src\<^bsub>X\<^esub> v"
f13: shows "\<forall>x0 x1. (x1 < x0) = (\<not> 0 \<le> x1 + - 1 * x0)"
code_lemmas_6: fixes c :: "(nat, 'a \<times> QDelta) mapping \<times> (nat, 'a \<times> QDelta) mapping"   and cs :: "QDelta ns_constraint list"   and asi_tv :: "('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list"   and s :: "('a, QDelta) state" shows "backtrack_simplex' c (cs, asi_tv, s) = (cs, asi_tv, backtrack_s c s)"
Collect_imp_eq: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
set_attribute_get_attribute: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and k :: "char list"   and v :: "char list option"   and h' :: "(_) heap" assumes "h \<turnstile> set_attribute ptr k v \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_attribute ptr k \<rightarrow>\<^sub>r v"
freeIn_elimVar_eq: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and p :: "real mpoly" shows "freeIn var (elimVar var L F (Eq p))"
ResRel: fixes R :: "pi"   and S :: "pi"   and y :: "name" assumes "(R, S) \<in> Rel" shows "(<\<nu>y>R, <\<nu>y>S) \<in> Rel'"
present_def: fixes h :: "heap"   and r :: "'b ref" shows "Ref.present h r = (addr_of_ref r < lim h)"
61229: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<sqinter> (- - y \<sqinter> - (x \<sqinter> z)) = - x \<sqinter> - - y"
msetext_huet_imp_count_gt: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "msetext_huet gt ys xs" shows "\<exists>x. count (mset xs) x < count (mset ys) x"
stoch_proc_filt_triv_init: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> real" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc (infinite_coin_toss_space.nat_filtration M) X" shows "init_triv_filt M (stoch_proc_filt M X borel)"
sf_mono: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "sf x \<le> sf y"
assms_2: shows "is_nth_power_upto_unit n (x * y)"
that: shows "map_\<I> g id \<I> \<turnstile>g gpv \<surd>"
emeasure_exponential_Ioi: fixes l :: "real"   and x :: "real" assumes "0 < l"   and "0 \<le> x" shows "emeasure (exponential l) {x<..} = ennreal (exp (- x * l))"
neq: shows "(\<noteq>) x = (\<lambda>y. \<not> x \<le> y \<or> \<not> y \<le> x)"
induct: fixes x1 :: "bexp"   and x2 :: "com"   and x3 :: "char list \<Rightarrow> int"   and x4 :: "nat"   and P :: "bexp \<Rightarrow> com \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> bool" assumes "Its x1 x2 x3 x4"   and "\<And>b s c. \<not> bval b s \<Longrightarrow> P b c s 0"   and "\<And>b s c s' n. \<lbrakk>bval b s; (c, s) \<Rightarrow> s'; Its b c s' n; P b c s' n\<rbrakk> \<Longrightarrow> P b c s (Suc n)" shows "P x1 x2 x3 x4"
ltl_Until_WeakUntil_con_2: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> W\<^sub>c \<psi> = \<xi> \<Turnstile>\<^sub>c \<phi> U\<^sub>c \<psi> or\<^sub>c G\<^sub>c \<phi>"
of_nat_numeral: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (numeral n) = numeral.numeral one plus n"
totalizer_def: fixes d :: "nat"   and i :: "nat"   and x :: "nat" shows "totalizer d i x \<equiv> if the (delay i x) = 0 then Some d else \<phi> i (e_take (the (delay i x)) x)"
outside: shows "x \<notin> S \<or> x \<bullet> i \<noteq> c"
fstI: fixes x :: "'a \<times> 'b"   and y :: "'a"   and z :: "'b" assumes "x = (y, z)" shows "cc x = y"
l': shows "l' \<equiv> LEAST l'. \<pi>' l' = \<pi> l \<and> k' < l' \<and> l' < m'"
gcd_dvd_1: fixes a :: "'a"   and b :: "'a" shows "(some_gcd a b dvd (1::'a)) = comm_monoid_mult_class.coprime a b"
max_word_less_eq_iff: fixes w :: "'b word" shows "(- 1 \<le> w) = (w = - 1)"
continuous_open_preimage: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "open S"   and "open T" shows "open (S \<inter> f -` T)"
intros_6: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and a :: "nat"   and Cs :: "char list list"   and Cs' :: "char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>ref (a, Cs @ [C] @ Cs'),s\<rangle> \<rightarrow> \<langle>ref (a, Cs @ [C]),s\<rangle>"
induct: fixes P :: "'b hp option \<Rightarrow> bool"   and a0 :: "'b hp option" assumes "\<And>x uu_. P (Some (Hp x uu_))"   and "P None" shows "P a0"
assms_1: fixes v :: "i" shows "[\<psi> \<^bold>\<equiv> \<chi> in v]"
ord_equiv: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Order D"   and "Order E"   and "ord_isom D E f" shows "ord_equiv D E"
canc_unitl: fixes e :: "'a"   and x :: "'a" assumes "D e x"   and "e \<oplus> x = x" shows "e \<in> E"
step_z_complete: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and u :: "'f \<Rightarrow> 'g"   and l' :: "'e"   and u' :: "'f \<Rightarrow> 'g"   and Z :: "('f \<Rightarrow> 'g) set" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"   and "u \<in> Z" shows "\<exists>Z'. A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle> \<and> u' \<in> Z'"
S6: shows "(u + 2 ^ x) div 2 ^ y = 2 ^ (x - y) + u div 2 ^ y"
eConseq1: fixes A :: "triple set"   and P' :: "assn"   and e :: "expr"   and Q :: "vassn"   and P :: "assn" assumes "A \<turnstile>\<^sub>e {P'} e {Q}"   and "\<forall>s. P s \<longrightarrow> P' s" shows "A \<turnstile>\<^sub>e {P} e {Q}"
has_domain_char: fixes \<mu> :: "'a arrow_of_spans_data" shows "(V.domains \<mu> \<noteq> {}) = arrow_of_spans (\<cdot>) \<mu>"
rel_sel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a fundef"   and b :: "'b fundef" shows "rel_fundef R a b = (list_all2 R (body a) (body b) \<and> arity a = arity b)"
source: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "(SourceTerm S, SourceTerm S) \<in> encoding.indRelRTPO Enc TRel"
simps: fixes f :: "'h \<Rightarrow> 'f"   and as :: "'h list"   and bs :: "'g"   and cs :: "('h \<times> 'h + nat) list"   and n :: "nat" shows "app_args f ((as, bs, cs), n) = ((map f as, bs, map (app_pair f) cs), n)"
switch_unique: fixes v :: "view"   and c :: "cars"   and u :: "view"   and w :: "view" shows "v = c > u \<and> v = c > w \<longrightarrow> u = w"
is_subseq_ind: fixes s :: "nat \<Rightarrow> 'a"   and s' :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "is_subseq_of s s'" shows "\<exists>l. s' k = s l"
mult_unit_dvd_iff: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b dvd (1::'a)" shows "(a * b dvd c) = (a dvd c)"
simps_1: fixes uu :: "'c"   and B :: "'c"   and uv :: "'d" shows "A_B.Cod (A_B.MkArr uu B uv) = B"
many_max: fixes a :: "one\<^sub>\<bottom>" shows "a \<sqsubseteq> many"
IH: assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p S" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (dual\<^sub>s\<^sub>t S)"
regularCard_def: fixes r :: "('c \<times> 'c) set" shows "regularCard r \<equiv> \<forall>K. K \<subseteq> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
distinct_9: fixes x2 :: "fresh_t"   and x4 :: "nat" shows "aNon x2 \<noteq> aNum x4"
eq_mset: shows "filter_mset ((\<subseteq>) ps) (mset_set \<G>) = mset_set {b \<in> \<G>. ps \<subseteq> b}"
finite_Un: fixes F :: "'f set"   and G :: "'f set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
assms_2: shows "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (mat_leqb_eqc A b c)"
adjust_assign_before_partial: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and y :: "char list" assumes "HT_partial \<pi> P c Q" shows "HT_partial \<pi> (\<lambda>s. P (s(x := s y))) (x[] ::= y;; c) (\<lambda>s\<^sub>0. Q (s\<^sub>0(x := s\<^sub>0 y)))"
update_convs_1: fixes xc' :: "int \<Rightarrow> int"   and xc :: "int"   and yc :: "int"   and more :: "'a" shows "xc_update xc' \<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = \<lparr>xc = xc' xc, yc = yc, \<dots> = more\<rparr>"
ipurge_tr_rev_aux_nil_sources: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list" assumes "ipurge_tr_rev_aux I D U xs = []" shows "sources_aux I D U xs = U"
kD_nsqn_quality_increases: fixes i :: "nat"   and \<xi> :: "state"   and \<xi>' :: "state" assumes "i \<in> kD (rt \<xi>)"   and "quality_increases \<xi> \<xi>'" shows "i \<in> kD (rt \<xi>') \<and> nsqn (rt \<xi>) i \<le> nsqn (rt \<xi>') i"
weak_least_unique: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" assumes "least L x A"   and "least L y A" shows "x .= y"
ass: shows "\<forall>p<n. Sorted_Less.sorted (rev (a ! p)) \<and> (a ! p \<noteq> rev [0..<n] \<longrightarrow> (\<forall>x\<in>set (a ! p). x < p)) \<and> (a ! p = rev [0..<n] \<longrightarrow> (\<exists>x\<in>set b. (x, p) \<in> g_E G \<and> x < p)) \<and> (p \<notin> set b \<longrightarrow> stable r step a p)"
pf_def: shows "pf \<equiv> \<lambda>t. q t + (g 0 - q 0) + t *\<^sub>R (g 1 - q 1 - (g 0 - q 0))"
assms: shows "saturated_binary_rule (ordered_sel_resolvent Sel) S"
assms_5: fixes t :: "'c \<times> 'b \<times> 'c" assumes "t \<in> reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0'" shows "(f t \<in> F) = (t \<in> F')"
invar_def: shows "invar \<equiv> \<lambda>(ts2, ws, r). \<exists>ts1. mset ts = ts1 + {#the (ws i). i \<in># mset_set (dom ws \<inter> {0..<n})#} + mset ts2 \<and> r = fold_mset f start ts1 \<and> set ts2 \<subseteq> set ts \<and> ran ws \<subseteq> set ts \<and> dom ws \<subseteq> {0..<n}"
union_single_eq_diff: fixes x :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" assumes "add_mset x M = N" shows "M = N - {#x#}"
select_convs_3: fixes obj_mark :: "bool"   and obj_fields :: "'a \<Rightarrow> 'c option"   and obj_payload :: "'a \<Rightarrow> 'b option"   and more :: "'d" shows "obj_payload \<lparr>obj_mark = obj_mark, obj_fields = obj_fields, obj_payload = obj_payload, \<dots> = more\<rparr> = obj_payload"
simps: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c llist"   and a2 :: "'c llist" shows "llexord r a1 a2 = ((\<exists>xs ys x. a1 = LCons x xs \<and> a2 = LCons x ys \<and> llexord r xs ys) \<or> (\<exists>x y xs ys. a1 = LCons x xs \<and> a2 = LCons y ys \<and> r x y) \<or> (\<exists>ys. a1 = LNil \<and> a2 = ys))"
lunit_eqI: fixes f :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : C.tensor \<I> a \<rightarrow>\<^sub>C a\<guillemotright>"   and "C.tensor \<I> f = C.tensor \<iota> a \<cdot> C.assoc' \<I> \<I> a" shows "f = C.lunit a"
nqmq: shows "2 ^ sq * nq \<le> mq"
dvd_smult_iff: fixes a :: "'c"   and p :: "'c poly"   and q :: "'c poly" assumes "a \<noteq> (0::'c)" shows "(p dvd smult a q) = (p dvd q)"
bij_lens_cong: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "X \<approx>\<^sub>L Y" shows "bij_lens X = bij_lens Y"
FH: shows "rel_mset (ddvd) F (image_mset normalize F)"
hq_less: shows "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
assms: fixes lm :: "nat list" assumes "P lm" shows "\<exists>n. abc_final (abc_steps_l (0, lm) p n) p \<and> Q abc_holds_for abc_steps_l (0, lm) p n"
1: shows "is_vector ((x\<^sup>+ \<cdot> 1') ; 1)"
diff_add: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "b - a + a = b"
continuous_divide: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous F f"   and "continuous F g"   and "g (netlimit F) \<noteq> (0::'b)" shows "continuous F (\<lambda>x. f x / g x)"
blinding_of_tree: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "blinding_of_on UNIV h bo" shows "blinding_of_on UNIV (hash_tree h) (blinding_of_tree h bo)"
funpow_0: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "(f ^^ 0) x = x"