hcju/isabelleps huggingface dataset:
is_process8: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>s X. s \<notin> DIVERGENCES P \<or> (s, X) \<in> FAILURES P"
x_def: shows "x = f\<lparr>i\<rparr>"
obj_ty_Arr1: fixes obj :: "obj"   and T :: "ty"   and i :: "int" assumes "tag obj = Arr T i" shows "obj_ty obj = T.[]"
OrderingSetIso_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "OrderingSetIso less_eq less less_eq' less' P f \<equiv> OrderingSetMap less_eq less less_eq' less' P f \<and> OrderingSetIso_axioms less_eq less less_eq' less' P f"
REC'_rec: shows "REC' = (mid`?`x \<rightarrow> right`!`x \<rightarrow> ack \<rightarrow> REC')"
psubset_imp_ex_mem: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subset> B" shows "\<exists>b. b \<in> B - A"
parCasesB: fixes P :: "pi"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and PQ' :: "pi"   and Prop :: "pi \<Rightarrow> bool" assumes "P \<parallel> Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> PQ'"   and "x \<sharp> P"   and "x \<sharp> Q"   and "\<And>P'. P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<Longrightarrow> Prop (P' \<parallel> Q)"   and "\<And>Q'. Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q' \<Longrightarrow> Prop (P \<parallel> Q')" shows "Prop PQ'"
open_Arg2pi2pi_less_Int: fixes s :: "real"   and t :: "real" assumes "0 \<le> s"   and "t \<le> 2 * pi" shows "open ({y. s < Arg2pi y} \<inter> {y. Arg2pi y < t})"
Joints_ConsD: fixes x :: "'a"   and xs :: "'a list"   and X :: "'a \<times> 'a pdevs"   and XS :: "('a \<times> 'a pdevs) list" assumes "x # xs \<in> Joints (X # XS)" shows "xs \<in> Joints XS"
def_body': fixes x :: "Set('d)"   and y :: "'e state \<times> 'e state \<Rightarrow> 'd"   and \<tau> :: "'e state \<times> 'e state" assumes "x \<noteq> \<bottom>"   and "x \<noteq> null"   and "(\<upsilon> y) \<tau> = true \<tau>" shows "\<lfloor>\<lfloor>y \<tau> \<notin> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>\<rfloor>\<rfloor> \<noteq> \<bottom> \<and> \<lfloor>\<lfloor>y \<tau> \<notin> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>\<rfloor>\<rfloor> \<noteq> null"
negligible_Union_nat: fixes S :: "nat \<Rightarrow> 'c set" assumes "\<And>n. negligible (S n)" shows "negligible (\<Union> (range S))"
splitFace_split_face: fixes oldF :: "face"   and g :: "graph"   and f\<^sub>1 :: "face"   and f\<^sub>2 :: "face"   and newGraph :: "graph"   and ram\<^sub>1 :: "nat"   and ram\<^sub>2 :: "nat"   and newVs :: "nat list" assumes "oldF \<in> \<F> g"   and "(f\<^sub>1, f\<^sub>2, newGraph) = splitFace g ram\<^sub>1 ram\<^sub>2 oldF newVs" shows "(f\<^sub>1, f\<^sub>2) = split_face oldF ram\<^sub>1 ram\<^sub>2 newVs"
op_ivl_Suc: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "prod g {m..<Suc n} = (if n < m then 1::'a else prod g {m..<n} * g n)"
assms_2: fixes "is" :: "nat list" assumes "is \<lhd> ds" shows "polyfun N (\<lambda>x. Tensor.lookup (T x) is)"
constrains_weaken: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A co A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B co B'"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "comm_monoid f z" shows "comm_monoid_fset f z"
snd_apsnd: fixes f :: "'e \<Rightarrow> 'c"   and x :: "'d \<times> 'e" shows "snd (apsnd f x) = f (snd x)"
cofactor_iarrays_def: fixes A :: "'c iarray iarray"   and i :: "nat"   and j :: "nat" shows "cofactor_iarrays A i j = det_iarrays_rings (minorM_iarrays A i j)"
an_neq_inf: fixes n :: "nat" shows "an n \<noteq> \<infinity>"
a_get_M_def: fixes ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u" shows "get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter = Heap_Error_Monad.bind get_heap (\<lambda>h. case get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h of None \<Rightarrow> error SegmentationFault | Some res \<Rightarrow> return (getter res))"
degree_coefficients: fixes p :: "'b poly" shows "Polynomial_Expression.degree p = length (coefficients p) - 1"
fls_times_base: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_nth (f * g) (fls_subdegree f + fls_subdegree g) = fls_nth f (fls_subdegree f) * fls_nth g (fls_subdegree g)"
that: fixes c3 :: "real" assumes "\<And>x i. \<lbrakk>x\<^sub>1 \<le> x; i < k\<rbrakk> \<Longrightarrow> c3 * g x \<le> g_approx i x"   and "0 < c3" shows "thesis"
nofail_SPEC_triv_refine: fixes m :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "nofail m"   and "\<And>x. \<Phi> x" shows "m \<le> SPEC \<Phi>"
A: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g" shows "((\<lambda>x. (1::'b) - (if f x = (0::'b) \<and> g x = (0::'b) then 1::'b else f x / g x)) \<longlongrightarrow> (0::'b)) F"
isnonconstant_coefficients_length: fixes p :: "poly" assumes "isnonconstant p" shows "1 < length (coefficients p)"
intros_6: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "Lt a \<prec> Lt b"
abs2_def: shows "abs2 \<equiv> \<mu> abs2. \<Lambda> v. Fixrec.run\<cdot>(match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>usvs))"
alice_out_def: fixes \<phi> :: "complex Matrix.mat"   and q :: "complex Matrix.mat" shows "alice_out \<phi> q \<equiv> if q = Tensor.mat_of_cols_list 8 [[\<phi> $$ (0, 0), \<phi> $$ (1, 0), 0, 0, 0, 0, 0, 0]] then (bit.zero, bit.zero) else if q = Tensor.mat_of_cols_list 8 [[0, 0, \<phi> $$ (1, 0), \<phi> $$ (0, 0), 0, 0, 0, 0]] then (bit.zero, bit.one) else if q = Tensor.mat_of_cols_list 8 [[0, 0, 0, 0, \<phi> $$ (0, 0), - \<phi> $$ (1, 0), 0, 0]] then (bit.one, bit.zero) else if q = Tensor.mat_of_cols_list 8 [[0, 0, 0, 0, 0, 0, - \<phi> $$ (1, 0), \<phi> $$ (0, 0)]] then (bit.one, bit.one) else undefined"
simps_2: fixes r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a fps"   and n :: "nat" shows "radical r 0 a (Suc n) = (0::'a)"
coeff: fixes x :: "'a poly" shows "coeff x \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'a)}"
ux_def: shows "ux = (SOME ux. ux \<in> \<lfloor>U, x\<rfloor>)"
assms_2: fixes x :: "'a" assumes "x \<in> cbox a b" shows "f x \<bullet> i \<le> g x \<bullet> i"
P3A: shows "Coplanar A B C T \<and> Bet P T R"
facet_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and z :: "'b set" assumes "ChamberComplexAutomorphism X f"   and "SimplicialComplex.maxsimp X C"   and "z \<lhd> C" shows "f ` z \<lhd> f ` C"
Hausdorff_space_injective_preimage: fixes Y :: "'c topology"   and X :: "'d topology"   and f :: "'d \<Rightarrow> 'c" assumes "Hausdorff_space Y"   and "continuous_map X Y f"   and "inj_on f (topspace X)" shows "Hausdorff_space X"
a_comm_monoid: fixes G :: "('a, 'b) ring_scheme" assumes "abelian_monoid G" shows "Group.comm_monoid (add_monoid G)"
propT_ok: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "wf_theory \<Theta>" shows "typ_ok \<Theta> propT"
sets_lebesgue_outer_open: fixes S :: "'c set"   and e :: "real" assumes "S \<in> sets lebesgue"   and "0 < e" obtains T :: "'c set" where "open T" and "S \<subseteq> T" and "T - S \<in> lmeasurable" and "emeasure lebesgue (T - S) < ennreal e"
mk_rtrancl_complete: fixes a :: "'a"   and init :: "'b" assumes "a \<in> R\<^sup>* `` set_of init" shows "\<exists>b. b \<in> mk_rtrancl init \<and> subsumes b a"
epsclo_def: fixes Q :: "hf set" shows "ST.epsclo Q \<equiv> nfa.states \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr> \<inter> (\<Union>q\<in>Q. {q'. (q, q') \<in> (eps \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr>)\<^sup>*})"
components_iff: fixes S :: "'c set"   and U :: "'c set" shows "(S \<in> components U) = (\<exists>x. x \<in> U \<and> S = connected_component_set U x)"
5: fixes x :: "int" assumes "x \<in> carrier (add_monoid (residue_ring (p ^ k))) \<and> x \<otimes>\<^bsub>add_monoid (residue_ring (p ^ k))\<^esub> y = \<one>\<^bsub>add_monoid (residue_ring (p ^ k))\<^esub> \<and> y \<otimes>\<^bsub>add_monoid (residue_ring (p ^ k))\<^esub> x = \<one>\<^bsub>add_monoid (residue_ring (p ^ k))\<^esub>" shows "x = \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> (y mod p ^ k)"
lem_ge_sum: fixes x :: "real"   and y :: "real"   and l :: "real" assumes "x + y < l" shows "\<exists>lx ly. l = lx + ly \<and> x < lx \<and> y < ly"
3: shows "\<forall>S\<in>\<S>. wf\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (\<A> @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])) (dual\<^sub>s\<^sub>t S)"
relcomp_trans_1: fixes f :: "'d"   and g :: "'e"   and A\<^sub>1 :: "('d \<times> 'e) set"   and h :: "'f"   and A\<^sub>2 :: "('e \<times> 'f) set" assumes "(f, g) \<in> A\<^sub>1"   and "(g, h) \<in> A\<^sub>2" shows "(f, h) \<in> A\<^sub>1 O A\<^sub>2"
YU: shows "(\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (uset (rlfm (simpfm p))))))) = (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (set (uset (rlfm (simpfm p))) \<times> set (uset (rlfm (simpfm p))))"
case_1: fixes f1 :: "'d"   and f2 :: "dag \<Rightarrow> ref \<Rightarrow> dag \<Rightarrow> 'd" shows "(case Tip of Tip \<Rightarrow> f1 | Node x xa xb \<Rightarrow> f2 x xa xb) = f1"
prod_rel_sv: fixes R1 :: "('d \<times> 'e) set"   and R2 :: "('f \<times> 'g) set" assumes "single_valued R1"   and "single_valued R2" shows "single_valued (R1 \<times>\<^sub>r R2)"
while_SE_unfold: fixes b :: "'d \<Rightarrow> bool"   and c :: "'d \<Rightarrow> (unit \<times> 'd) option" shows "_while_SE b c = _if_SE b (c ;- _while_SE b c) (unit\<^sub>S\<^sub>E ())"
Codt: fixes f :: "'f" assumes "f \<in> Fn" shows "sCod f \<in> Ty"
FunctorM_axioms_def: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" shows "FunctorM_axioms F \<equiv> \<forall>f X Y. f maps\<^bsub>CatDom F\<^esub> X to Y \<longrightarrow> F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y"
cases: fixes a1 :: "'a literal multiset set"   and a2 :: "'a literal multiset set" assumes "a1 \<triangleright> a2" obtains N :: "'a literal multiset set" and M :: "'a literal multiset set" where "a1 = M" and "a2 = N" and "N - M \<subseteq> concls_of (inferences_from M)" and "M - N \<subseteq> Rf N"
terminates_replicate_raw: fixes a :: "'e" shows "terminates (replicate_raw a)"
intro: fixes alphabet1 :: "'b list"   and init1 :: "'c \<Rightarrow> 'd"   and delta1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and accept1 :: "'d \<Rightarrow> bool"   and wellformed1 :: "'d \<Rightarrow> bool"   and Language1 :: "'d \<Rightarrow> 'b language"   and wf1 :: "'c \<Rightarrow> bool"   and Lang1 :: "'c \<Rightarrow> 'b language"   and alphabet2 :: "'e list"   and init2 :: "'f \<Rightarrow> 'g"   and delta2 :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and accept2 :: "'g \<Rightarrow> bool"   and wellformed2 :: "'g \<Rightarrow> bool"   and Language2 :: "'g \<Rightarrow> 'e language"   and wf2 :: "'f \<Rightarrow> bool"   and Lang2 :: "'f \<Rightarrow> 'e language"   and letter_eq :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "DFA alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1"   and "DFA alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2"   and "DFAs_axioms alphabet1 alphabet2 letter_eq" shows "DFAs alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1 alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2 letter_eq"
lfp_Kleene_iter: fixes f :: "'a \<Rightarrow> 'a"   and k :: "nat" assumes "mono f"   and "(f ^^ Suc k) bot = (f ^^ k) bot" shows "lfp f = (f ^^ k) bot"
assms_1: shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p s"
that: fixes x :: "'a"   and s :: "'b" assumes "s' = (x, s)"   and "x = x'" shows "thesis"
prefixI: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "ys = xs @ zs" shows "prefix xs ys"
rep_eq: fixes x :: "'e['g, 'f]"   and xa :: "'e['h, 'f]" shows "fromQ (x \<^bold>\<cdot> xa) = fromQ x \<cdot> fromQ xa"
simps_5: fixes wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp"   and s :: "'c rexp" shows "alphabet.wf wf_atom n (Plus r s) = (alphabet.wf wf_atom n r \<and> alphabet.wf wf_atom n s)"
foldD_insert: fixes A :: "'b set"   and x :: "'b"   and e :: "'a" assumes "finite A"   and "x \<notin> A"   and "x \<in> B"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D (\<cdot>) e (insert x A) = x \<cdot> foldD D (\<cdot>) e A"
simps_3: fixes c1 :: "'a acom"   and c2 :: "'a acom"   and c1' :: "'a acom"   and c2' :: "'a acom" shows "(c1;; c2 \<sqsubseteq> c1';; c2') = (c1 \<sqsubseteq> c1' \<and> c2 \<sqsubseteq> c2')"
swapEnv_involutive: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "goodEnv rho" shows "rho &[x \<and> y]_zs &[x \<and> y]_zs = rho"
simps_1: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c :: "('a, 'c, 'd) Stmt" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "sifum_lang.cxt_to_stmt [] c = c"
simps: fixes xs :: "'varSort"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm" shows "qGoodAbs (qAbs xs x X) = qGood X"
edge'E: fixes xo :: "'v vertex"   and yo :: "'v vertex" assumes "edge' xo yo" obtains x :: "'v" and y :: "'v" where "xo = \<langle>x__\<rangle>" and "yo = \<langle>y__\<rangle>" and "edge \<Gamma> x__ y__ \<or> edge \<Gamma> y__ x__"   | x :: "'v" where "xo = \<langle>xa__\<rangle>" and "yo = vertex.SINK" and "xa__ \<in> A \<Gamma>"   | x :: "'a" where "xo = SOURCE" and "yo = \<langle>b\<rangle>"   | y :: "'v" where "xo = vertex.SINK" and "yo = \<langle>y\<rangle>" and "y \<in> A \<Gamma>"
next_non_root_interval: fixes p :: "real poly"   and lb :: "real" assumes "p \<noteq> 0" obtains ub :: "real" where "lb < ub" and "\<forall>z. lb < z \<and> z \<le> ub \<longrightarrow> poly p z \<noteq> 0"
stream_all2_SCons1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a stream"   and ys :: "'b stream" shows "stream_all2 P (x ## xs) ys = (\<exists>z zs. ys = z ## zs \<and> P x z \<and> stream_all2 P xs zs)"
min_galleryI_betw: fixes x :: "'a set"   and y :: "'a set"   and xs :: "'a set list" assumes "x \<noteq> y"   and "gallery (x # xs @ [y])"   and "\<And>ys. gallery (x # ys @ [y]) \<Longrightarrow> order.greater_eq (length ys) (length xs)" shows "min_gallery (x # xs @ [y])"
fpxs_compose_power_eq_0_iff: fixes r :: "rat"   and f :: "'b fpxs" assumes "0 < r" shows "(fpxs_compose_power f r = 0) = (f = 0)"
push_down_state_rank_oldest_token: fixes q :: "'b"   and n :: "nat"   and i :: "nat" assumes "state_rank q n = Some i" shows "\<exists>x. oldest_token q n = Some x"
max_less_iff_conj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.linorder less_eq less" shows "less (ord.max less_eq x y) z = (less x z \<and> less y z)"
mult_1_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.monoid_mult one times" shows "times a one = a"
higher_differentiable_on_add: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and g :: "'b \<Rightarrow> 'c" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x + g x) n"
mr_cospOpTr: fixes H :: "'a set"   and m :: "'a"   and n :: "'a" assumes "submodule R M H"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "mr_cospOp M H (m \<uplus>\<^bsub>M\<^esub> H) (n \<uplus>\<^bsub>M\<^esub> H) = (m \<plusminus> n) \<uplus>\<^bsub>M\<^esub> H"
trimmingD_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "trimming \<Gamma> f g" shows "current \<Gamma> g"
quotient_is_domain: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "primeideal I R" shows "domain (R Quot I)"
cong_mod_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b (modulo c a) a = unique_euclidean_semiring.cong modulo b c a"
nth_rotate1: fixes ls :: "'c list"   and n :: "nat" assumes "0 < |ls|" shows "ls ! (Suc n mod |ls|) = rotate1 ls ! (n mod |ls|)"
is_in: fixes x :: "'a"   and t :: "'a tree" shows "is_in x t = is_in\<^sub>0 x (erase t)"
simps_1: fixes \<zeta> :: "('s, 'v) hd" shows "wt (Hd \<zeta>) = (LEAST w. \<exists>f\<in>local.ground_heads \<zeta>. w = wt_sym f + the_enat (enat \<delta> * arity_sym f))"
measurable_on_iff_borel_measurable: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue" shows "(f measurable_on S) = (f \<in> borel_measurable (lebesgue_on S))"
closed_segment_real_eq: fixes u :: "real"   and v :: "real" shows "closed_segment u v = (\<lambda>x. (v - u) * x + u) ` {0..1}"
chain_boundary_diff: fixes p :: "nat"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p (a - b) = chain_boundary p a - chain_boundary p b"
is_OK_returns_heap_E: fixes h :: "'aq"   and f :: "('aq, 'ar, 'as) prog" assumes "h \<turnstile> ok f" obtains h' :: "'aq" where "h \<turnstile> f \<rightarrow>\<^sub>h h'"
col_polys_to_mat: fixes j :: "nat"   and ts :: "'c list"   and ps :: "('c \<Rightarrow>\<^sub>0 'd) list" assumes "j < length ts" shows "col (polys_to_mat ts ps) j = vec_of_list (map (\<lambda>p. lookup p (ts ! j)) ps)"
FP: fixes n :: "nat" assumes "n < length dl" shows "part {..<brn (dl ! n)} (F n ` P n)"
insertCI: fixes a :: "'f"   and B :: "'f set"   and b :: "'f" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
in_set_enumerate_eq: fixes p :: "nat \<times> 'a"   and n :: "nat"   and xs :: "'a list" shows "(p \<in> set (enumerate n xs)) = (n \<le> fst p \<and> fst p < length xs + n \<and> xs ! (fst p - n) = snd p)"
closed_subsets_def: fixes \<aa> :: "'a set" shows "\<V> \<aa> \<equiv> {I. pr_ideal R I (+) (\<cdot>) \<zero> \<one> \<and> \<aa> \<subseteq> I}"
i_set_infinite_as_iMOD: fixes I :: "nat set" assumes "I \<in> i_set"   and "infinite I" shows "\<exists>r m. I = [ r, mod m ]"
prime_power_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and n :: "nat" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.prime zero times one normalize (power.power one times p n) = (normalization_semidom.prime zero times one normalize p \<and> n = 1)"
comm_ring_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 "comm_ring R addition multiplication zero unit \<equiv> ring R addition multiplication zero unit \<and> comm_ring_axioms R multiplication"
MtransC_induct_temp: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x3a :: "('a, 'c) com"   and x3b :: "'b"   and x4a :: "('a, 'c) com"   and x4b :: "'b"   and P1 :: "('a, 'c) com \<Rightarrow> 'b \<Rightarrow> ('a, 'c) com \<Rightarrow> 'b \<Rightarrow> bool"   and P2 :: "('a, 'c) com \<Rightarrow> 'b \<Rightarrow> ('a, 'c) com \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL.MtransC tval aval (x3a, x3b) (x4a, x4b)"   and "\<And>c1 s c1' s' c2. \<lbrakk>PL.transC tval aval (c1, s) (c1', s'); P1 c1 s c1' s'\<rbrakk> \<Longrightarrow> P1 (c1 ;; c2) s (c1' ;; c2) s'"   and "\<And>c1 s s' c2. PL.transT tval aval (c1, s) s' \<Longrightarrow> P1 (c1 ;; c2) s c2 s'"   and "\<And>tst s c1 c2. tval tst s \<Longrightarrow> P1 (if tst then c1 else c2) s c1 s"   and "\<And>tst s c1 c2. \<not> tval tst s \<Longrightarrow> P1 (if tst then c1 else c2) s c2 s"   and "\<And>tst s c. tval tst s \<Longrightarrow> P1 (while tst do c) s (c ;; while tst do c) s"   and "\<And>c1 s c1' s' c2. \<lbrakk>PL.transC tval aval (c1, s) (c1', s'); P1 c1 s c1' s'\<rbrakk> \<Longrightarrow> P1 (c1 | c2) s (c1' | c2) s'"   and "\<And>c2 s c2' s' c1. \<lbrakk>PL.transC tval aval (c2, s) (c2', s'); P1 c2 s c2' s'\<rbrakk> \<Longrightarrow> P1 (c1 | c2) s (c1 | c2') s'"   and "\<And>c1 s s' c2. PL.transT tval aval (c1, s) s' \<Longrightarrow> P1 (c1 | c2) s c2 s'"   and "\<And>c2 s s' c1. PL.transT tval aval (c2, s) s' \<Longrightarrow> P1 (c1 | c2) s c1 s'"   and "\<And>c s. P2 c s c s"   and "\<And>c s c' s' c'' s''. \<lbrakk>PL.MtransC tval aval (c, s) (c', s'); P2 c s c' s'; PL.transC tval aval (c', s') (c'', s''); P1 c' s' c'' s''\<rbrakk> \<Longrightarrow> P2 c s c'' s''" shows "P2 x3a x3b x4a x4b"
g_def: shows "g \<equiv> \<lambda>x. if x \<le> 0 then 0 else g0 x"
gcd_dvd1: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) a"
reduce_rules_mono: fixes P :: "'a set"   and P' :: "'a set"   and \<delta> :: "('a, 'b) ta_rule set" assumes "P \<subseteq> P'" shows "reduce_rules \<delta> P \<subseteq> reduce_rules \<delta> P'"
sig_red_single_tail_lt_in_keys_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg (\<prec>) p q f t" shows "punit.lt (rep_list p) \<in> keys (rep_list q)"
less_option_def: fixes x :: "'a option"   and y :: "'a option" shows "(x < y) = strict (\<le>) x y"
interval_of_plus: fixes a :: "'a"   and b :: "'a" shows "interval_of (a + b) = interval_of a + interval_of b"