hcju/isabelleps huggingface dataset:
uint64_sshiftr_def: fixes x :: "uint64"   and n :: "integer" shows "uint64_sshiftr x n = (if n < 0 \<or> 64 \<le> n then undefined (>>>) x n else x >>> nat_of_integer n)"
fresh_trm_dom: fixes x :: "vrs"   and \<Gamma> :: "binding list" shows "x \<sharp> trm_dom \<Gamma> = x \<sharp> \<Gamma>"
measure_pmf_of_set: fixes S :: "'c set"   and A :: "'c set" assumes "S \<noteq> {}"   and "finite S" shows "measure_pmf.prob (pmf_of_set S) A = real (card (S \<inter> A)) / real (card S)"
nat_pow_eone: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> 1 = x"
finsert_eq_iff: fixes a :: "'a"   and A :: "'a fset"   and b :: "'a"   and B :: "'a fset" assumes "a |\<notin>| A"   and "b |\<notin>| B" shows "(finsert a A = finsert b B) = (if a = b then A = B else \<exists>C. A = finsert b C \<and> b |\<notin>| C \<and> B = finsert a C \<and> a |\<notin>| C)"
generate_is_subgroup: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "subgroup (generate G H) G"
subgraph_induce_subgraphI: fixes G :: "('c, 'd) pre_digraph"   and V :: "'c set" assumes "wf_digraph G"   and "V \<subseteq> verts G" shows "subgraph (G \<restriction> V) G"
assms: shows "length xs\<^sub>2 = length ys\<^sub>2"
invertible_inverse_closed: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "local.inverse u \<in> G"
emeasure_measure_of: fixes M :: "'b measure"   and \<Omega> :: "'b set"   and A :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal"   and X :: "'b set" assumes "M = measure_of \<Omega> A \<mu>"   and "A \<subseteq> Pow \<Omega>"   and "positive (sets M) \<mu>"   and "countably_additive (sets M) \<mu>"   and "X \<in> sets M" shows "emeasure M X = \<mu> X"
insert_queue_invar: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and e :: "'a"   and a :: "'b" assumes "SkewBinomialHeapStruc.queue_invar q" shows "SkewBinomialHeapStruc.queue_invar (SkewBinomialHeapStruc.insert e a q)"
steps_or: fixes L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and q :: "bool list"   and w :: "'b list" shows "((start (or L R), q) \<in> NAe.steps (or L R) w) = (w = [] \<and> q = start (or L R) \<or> (\<exists>p. q = True # p \<and> (start L, p) \<in> NAe.steps L w \<or> q = False # p \<and> (start R, p) \<in> NAe.steps R w))"
aff_dim_UNIV: shows "aff_dim UNIV = int DIM('a)"
proj_subset_if_no_label_2: fixes l :: "'d"   and A :: "('d strand_label \<times> 'e) list"   and l' :: "'d" assumes "\<not> list_ex (is_LabelN l) A" shows "set (proj_unl l A) \<subseteq> set (proj_unl l' A)"
adv: fixes \<eta> :: "nat" assumes "exception_\<I> (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) \<turnstile>g \<A> \<eta> \<surd>" shows "advantage (\<A> \<eta>) (obsf_resource (sim \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal_resource \<eta>)) (obsf_resource (real_resource \<eta>)) \<le> adv \<eta>"
simps_2: fixes x :: "'a"   and ys :: "'a list" shows "local.strict_sorted (x # ys) = (Ball (set ys) (lt x) \<and> local.strict_sorted ys)"
hd_bl_sign_sint: fixes w :: "'b word" shows "hd (to_bl w) = (bin_sign (sint w) = - 1)"
unique_factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "unique_factorization_m f cfs = (Mf ` Collect (factorization_m f) = {Mf cfs})"
elims_3: fixes x :: "'a ext"   and xa :: "'a ext" assumes "\<not> x \<le> xa" obtains uv_ :: "'a" where "xa__ = Val uv_" and "xa = Bot"   | x :: "'a" and y :: "'a" where "xa__ = Val x" and "xa = Val y" and "\<not> x \<le> y"   | "xa__ = Top" and "xa = Bot"   | ux_ :: "'a" where "xa__ = Top" and "xa = Val ux_"
I: fixes t :: "real"   and s :: "real" assumes "0 \<le> t"   and "t \<le> s"   and "s \<le> dist x z" shows "dist (f s) (f t) = s - t"
nil: fixes root :: "(att \<times> char list, att, nat) env" shows "root \<Midarrow>[]\<Rightarrow> root"
ncrit_specI: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool" assumes "\<And>d m data gs bs hs ps B q_in_bs p q. \<lbrakk>dickson_grading d; set gs \<union> set bs \<union> set hs \<subseteq> B; fst ` B \<subseteq> dgrad_p_set d m; snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs); unique_idx (gs @ bs @ hs) data; is_Groebner_basis (fst ` set gs); q_in_bs \<longrightarrow> q \<in> set gs \<union> set bs; \<And>p' q'. \<lbrakk>(p', q') \<in>\<^sub>p snd ` set ps; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q'); \<And>p' q'. \<lbrakk>p' \<in> set gs \<union> set bs; q' \<in> set gs \<union> set bs; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q'); p \<in> set hs; q \<in> set gs \<union> set bs \<union> set hs; fst p \<noteq> 0; fst q \<noteq> 0; crit data gs bs hs q_in_bs ps p q\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)" shows "ncrit_spec crit"
complex_cnj_1: fixes x :: "complex"   and y :: "complex" shows "cnj (x - y) = cnj x - cnj y"
triangle_left': shows "\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[f] = f"
until_monoR: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi>1 :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi>2 :: "'a stream \<Rightarrow> bool" assumes "(\<phi> until \<psi>1) xs"   and "\<And>xs. \<psi>1 xs \<Longrightarrow> \<psi>2 xs" shows "(\<phi> until \<psi>2) xs"
outer_measure_of_def: fixes M :: "'c measure"   and A :: "'c set" shows "outer_measure_of M A = Inf (emeasure M ` {B \<in> sets M. A \<subseteq> B})"
lexord_append: fixes xs\<^sub>1 :: "'b list"   and ys\<^sub>1 :: "'b list"   and xs\<^sub>2 :: "'b list"   and ys\<^sub>2 :: "'b list"   and R :: "('b \<times> 'b) set" assumes "(xs\<^sub>1 @ ys\<^sub>1, xs\<^sub>2 @ ys\<^sub>2) \<in> lexord R"   and "length xs\<^sub>1 = length xs\<^sub>2" shows "(xs\<^sub>1, xs\<^sub>2) \<in> lexord R \<or> xs\<^sub>1 = xs\<^sub>2 \<and> (ys\<^sub>1, ys\<^sub>2) \<in> lexord R"
\<F>\<G>_subformulas\<^sub>\<nu>: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> \<phi> w \<subseteq> subformulas\<^sub>\<nu> \<phi>"
IH_1: fixes va :: "nat"   and vd :: "'a list \<times> 'a list"   and x :: "'a list"   and y :: "'a list" assumes "va = Time_Monad.val (length_tm (x__ # y__ # xs'))"   and "vd = Time_Monad.val (split_at_tm (va div 2) (x__ # y__ # xs'))"   and "(x, y) = vd" shows "real (time (mergesort_tm f x)) \<le> mergesort_recurrence (length x)"
contract_vacuity: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<phi> \<notin> K" shows "K \<div> \<phi> = K"
weak_action_modality_def: fixes \<alpha> :: "'act"   and x :: "('idx, 'pred, 'act) formula" shows "\<langle>\<langle>\<alpha>\<rangle>\<rangle>x \<equiv> if \<alpha> = \<tau> then weak_tau_modality x else weak_tau_modality (Act \<alpha> (weak_tau_modality x))"
isomorphic_def: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a' :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.isomorphic a a' = (\<exists>f. Span.in_hom f a a' \<and> Span.iso f)"
step_5: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
that: fixes g :: "real \<Rightarrow> 'a" assumes "arc g"   and "g ` {0..<1} \<subseteq> S"   and "pathstart g = x"   and "pathfinish g \<in> V" shows "thesis"
good_fun_def2: fixes f :: "(val \<times> val) fset"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> bval) list" shows "good_fun f x e \<rho> = ffold (good_entry x e \<rho> \<circ> good_prod) True f"
lipschitz: fixes s :: "real" assumes "s \<in> cball t v" shows "B-lipschitz_on (cball x (min u v) \<inter> X) (f s)"
basis_exists: fixes V :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "V \<subseteq> sub.tangent_space p" obtains B :: "(('a \<Rightarrow> real) \<Rightarrow> real) set" where "B \<subseteq> sub.tangent_space p" and "B \<subseteq> V" and "independent B" and "V \<subseteq> span B" and "card B = sub.tangent_space.dim p V"
SD_strict_singleton: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier"   and "y \<in> carrier" shows "return_pmf x \<prec>[SD le] return_pmf y = x \<prec>[le] y"
succ_2: fixes s :: "hf"   and y :: "hf"   and s' :: "hf"   and y' :: "hf" assumes "k \<noteq> 0"   and "ord_rec_Seq T G s k y"   and "ord_rec_Seq T G s' k y'" shows "y' = y"
p_fusion_last: fixes ps :: "'a list"   and qs :: "'a list" assumes "List.last ps = hd qs"   and "ps \<noteq> []"   and "qs \<noteq> []" shows "List.last (p_fusion ps qs) = List.last qs"
map\<^sub>0_def: fixes a :: "'b" shows "H.map\<^sub>0 a \<equiv> src\<^sub>C (H a)"
alive_alloc_propagation: fixes x :: "Value"   and s :: "Store"   and t :: "New" assumes "alive x s" shows "alive x (s\<langle>t\<rangle>)"
L2_set_def: fixes f :: "'b \<Rightarrow> real"   and A :: "'b set" shows "L2_set f A = sqrt (\<Sum>i\<in>A. (f i)\<^sup>2)"
hyps_1: shows "realizable n\<^sub>0 ns n\<^sub>1"
lower_bound_left: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y \<le> - x"
skolemize2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x. \<exists>y. P x y" shows "\<exists>f. \<forall>x. P x (f x)"
leP: shows "P \<le>\<^sub>L 1\<^sub>m d"
comp_null_2: fixes f :: "nat" shows "J.comp f J.null = J.null"
B_3: fixes a :: "'b set" assumes "a \<in> B" shows "open a"
Cs_def: shows "Cs = (ARG_MIN length Cs. SimplicialComplex.maxsimpchain A (C # Cs @ [D]))"
mkChannels_cl_inv: fixes g :: "'a gState_scheme"   and p :: "pState"   and cs :: "channel list" assumes "cl_inv (g, p)" shows "cl_inv (mkChannels g p cs)"
Abs_fps_inverse: fixes y :: "'c list \<Rightarrow> 'd" assumes "y \<in> {f. True}" shows "($) (Abs_fps y) = y"
line_in_open_segment_iff: fixes a :: "'b"   and b :: "'b"   and i :: "real" shows "({a -- b}\<^bsub>i\<^esub> \<in> {a<--<b}) = (a \<noteq> b \<and> 0 < i \<and> i < 1)"
components_full_gb: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "component_of_term ` Keys (fst ` set (full_gb bs)) = component_of_term ` Keys (fst ` set bs)"
subtensor_induct: fixes P :: "'b tensor \<Rightarrow> bool"   and B :: "'b tensor" assumes "\<And>A. dims A = [] \<Longrightarrow> P A"   and "\<And>A. \<lbrakk>dims A \<noteq> []; \<And>i. i < hd (dims A) \<Longrightarrow> P (subtensor A i)\<rbrakk> \<Longrightarrow> P A" shows "P B"
fv_at_base: fixes a :: "'c" shows "fv a = {a}"
measurable_compose_rev: fixes f :: "'d \<Rightarrow> 'e"   and L :: "'d measure"   and N :: "'e measure"   and g :: "'f \<Rightarrow> 'd"   and M :: "'f measure" assumes "f \<in> L \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M L" shows "(\<lambda>x. f (g x)) \<in> M \<rightarrow>\<^sub>M N"
exec_appendR: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and P' :: "instr list" assumes "P \<turnstile> c \<rightarrow>* c'" shows "P @ P' \<turnstile> c \<rightarrow>* c'"
nn_integral_PInf: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "integral\<^sup>N M f \<noteq> \<infinity>" shows "emeasure M (f -` {\<infinity>} \<inter> space M) = 0"
take_hd_drop: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
6: shows "sa_iso_inv p = Rep_regular x \<sqinter> Rep_dense (stone_phi.rho_pair (x, y))"
assms_1: shows "convert_poly var p (xs' @ x # xs) = Some X"
poly2_by_map: fixes p :: "'c poly poly"   and x :: "'c" shows "poly2 p x = poly (map_poly (\<lambda>c. poly c x) p)"
one_le_numeral: fixes n :: "num" shows "(1::'b) \<le> numeral n"
rel_F_neg_distr: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co3 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co3' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra1 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and Contra3 :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra3' :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and uu :: "('af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'aj \<times> 'ak \<times> 'al \<times> 'am \<times> 'an \<times> 'ao) itself"   and L1 :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and L1' :: "'ag \<Rightarrow> 'ah \<Rightarrow> bool"   and L2 :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and L2' :: "'aj \<Rightarrow> 'ak \<Rightarrow> bool"   and L3 :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and L3' :: "'am \<Rightarrow> 'an \<Rightarrow> bool" assumes "rel_F_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' uu" shows "rel_F (L1 OO L1') (L2 OO L2') (L3 OO L3') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') \<le> rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 OO rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3'"
simps_1: fixes e :: "'a" shows "insert_body [] e = [e]"
replicate_in_nlists: fixes x :: "'c"   and A :: "'c set"   and n :: "nat" assumes "x \<in> A" shows "replicate n x \<in> nlists A n"
lm049: fixes a :: "('d \<times> 'c set) set" shows "Range (pseudoAllocation a) = \<Union> (finestpart ` Range a)"
assms_2: shows "psi 1 \<le> 3 / 2 * ln 2 * real 1"
Zero_Disjoint_I1: fixes H :: "fm set"   and t :: "tm" shows "H \<turnstile> Disjoint Zero t"
prems: shows "\<forall>i\<in>A. \<forall>a\<in>B. i < a"
fds_g: shows "fds g = fds_ind squarefree * fds moebius_mu"
hyps_1: fixes i :: "nat" assumes "n = j - i"   and "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "ps (S t i) p \<noteq> None"   and "Marker \<notin> set (msgs (S t i) cid)"   and "channel cid = Some (p, q)" shows "Marker \<notin> set (msgs (S t j) cid)"
simps_1: fixes uu :: "'b extN" shows "(extN.Bot \<le> uu) = True"
FnCall: fixes F :: "nat"   and Us :: "freeExp list" shows "FnCall F (Abs_ExpList Us) = Abs_Exp (exprel `` {FNCALL F Us})"
submonoid_invertible: fixes u :: "'a \<Rightarrow> 'a" assumes "transformation.sub.invertible u"   and "u \<in> Translations" shows "transformation.invertible u"
distinct_21: shows "ToUpperCaseOp \<noteq> ToRealOp"
blinding_of_F_mono: fixes boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and boa' :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and bob :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and bob' :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and rha :: "'e \<Rightarrow> 'g"   and rhb :: "'f \<Rightarrow> 'h" assumes "boa \<le> boa'"   and "bob \<le> bob'" shows "blinding_of_F rha boa rhb bob \<le> blinding_of_F rha boa' rhb bob'"
select_min: fixes measure :: "'a \<Rightarrow> nat" shows "det_selection_fun (select_min measure)"
elims_3: fixes x :: "16 word \<times> 16 word"   and xa :: "16 word" assumes "\<not> simple_match_port x xa" obtains s :: "16 word" and e :: "16 word" and p_p :: "16 word" where "x = (s, e)" and "xa = p_p" and "p_p \<notin> {s..e}"
inject_5: fixes x6 :: "'b val list"   and y6 :: "'b val list" shows "(Stars x6 = Stars y6) = (x6 = y6)"
intros_1: fixes Sl :: "'node set"   and p :: "'node"   and R :: "'node set"   and q :: "'node" assumes "Sl \<in> slices p"   and "\<forall>q\<in>Sl \<inter> W. q \<notin> R \<and> \<not> blocking_min R q"   and "q \<in> Sl" shows "not_blocked p R q"
axioms_1: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool"   and edges_list_it :: "'d \<Rightarrow> (('e \<times> 'f \<times> 'e) list \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<times> 'e \<Rightarrow> ('e \<times> 'f \<times> 'e) list \<Rightarrow> ('e \<times> 'f \<times> 'e) list) \<Rightarrow> ('e \<times> 'f \<times> 'e) list \<Rightarrow> ('e \<times> 'f \<times> 'e) list" assumes "graph_edges_it \<alpha> invar edges_list_it" shows "graph \<alpha> invar"
f_def: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "f x y n = T.prob x {\<omega> \<in> space (T x). ev_at (HLD {y}) n (x ## \<omega>)}"
assms_6: shows "l icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
calc_x_axis_intersection_cvec_def: fixes A :: "complex"   and B :: "complex" shows "calc_x_axis_intersection_cvec A B = (case let discr = (Re B)\<^sup>2 - (Re A)\<^sup>2 in (- Re B + sgn (Re B) * sqrt discr, A) of (x, y) \<Rightarrow> (cor x, y))"
CR1_def: fixes \<tau> :: "ty" shows "CR1 \<tau> \<equiv> \<forall>t. t \<in> RED \<tau> \<longrightarrow> SN t"
pick_psubstAbs_qPsubstAbs: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodAbs A"   and "goodEnv rho" shows "pick A $[rho] $= pick A $[[pickE rho]]"
zero_not_in_prime_factors: fixes x :: "'a" shows "(0::'a) \<notin> prime_factors x"
le_bot: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.order_bot bot less_eq less"   and "less_eq a bot" shows "a = bot"
induct: fixes P :: "'a \<Rightarrow> 'a acom \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a acom" assumes "\<And>S Q. P S (SKIP {Q})"   and "\<And>S x e Q. P S (x ::= e {Q})"   and "\<And>S C1 C2. \<lbrakk>P S C1; P (post C1) C2\<rbrakk> \<Longrightarrow> P S (C1;; C2)"   and "\<And>S b P1 C1 P2 C2 Q. \<lbrakk>P P1 C1; P P2 C2\<rbrakk> \<Longrightarrow> P S (IF b THEN {P1} C1 ELSE {P2} C2 {Q})"   and "\<And>S I b Pa C Q. P Pa C \<Longrightarrow> P S ({I} WHILE b DO {Pa} C {Q})" shows "P a0 a1"
hpair_inject: fixes a :: "hf"   and b :: "hf"   and a' :: "hf"   and b' :: "hf" assumes "\<langle>a, b\<rangle> = \<langle>a', b'\<rangle>" obtains "a = a'" and "b = b'"
inverse_transformations_inverse_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b" assumes "inverse_transformations A B F G \<sigma> \<sigma>'" shows "vertical_composite.map A B \<sigma>' \<sigma> = G"
run_step_basic_binop_relop_sound: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and b_e :: "b_e"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and t :: "t"   and iop :: "binop_i"   and fop :: "binop_f"   and irop :: "relop_i"   and frop :: "relop_f" assumes "run_one_step d i (s, vs, ves, $b_e) = (s', vs', RSNormal es')"   and "b_e = Binop_i t iop \<or> b_e = Binop_f t fop \<or> b_e = Relop_i t irop \<or> b_e = Relop_f t frop" shows "\<lparr>s;vs;vs_to_es ves @ [$b_e]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
tautology_deletion: fixes A :: "'a"   and C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" assumes "Neg A \<in># C"   and "Pos A \<in># C" shows "(N \<union> {C}, P, Q) \<leadsto>RP (N, P, Q)"
B: shows "{v1} \<times> {v2} \<in> sets (stock_measure (val_type v1) \<Otimes>\<^sub>M stock_measure (val_type v2))"
case_option_collapse: fixes x :: "'e" shows "case_option x (\<lambda>_. x) = (\<lambda>_. x)"
viewnres_cases: fixes r :: "(('a, 'b) Node \<times> ('a, 'b) FingerTreeStruc) option" obtains "r = None"   | a :: "('a, 'b) Node" and t :: "('a, 'b) FingerTreeStruc" where "r = Some (a, t)"
Var_apps_neq_Abs_apps: fixes n :: "name"   and ts :: "lam list"   and x :: "name"   and r :: "lam"   and ss :: "lam list" shows "Var n \<degree>\<degree> ts \<noteq> (Lam [x].r) \<degree>\<degree> ss"
tm_append_second_fetch_eq: fixes A :: "(action \<times> nat) list"   and off :: "nat"   and B :: "(action \<times> nat) list"   and s :: "nat"   and b :: "cell"   and ac :: "action"   and ns :: "nat" assumes "is_even (length A)"   and "off = length A div 2"   and "fetch B s b = (ac, ns)"   and "ns \<noteq> 0" shows "fetch (A @ shift B off) (s + off) b = (ac, ns + off)"
nth_transpose: fixes i :: "nat"   and xs :: "'g list list" assumes "i < length (transpose xs)" shows "transpose xs ! i = map (\<lambda>xs. xs ! i) (filter (\<lambda>ys. i < length ys) xs)"
genPrefix_trans_O: fixes x :: "'a list"   and y :: "'a list"   and r :: "('a \<times> 'a) set"   and z :: "'a list"   and s :: "('a \<times> 'a) set" assumes "(x, y) \<in> genPrefix r"   and "(y, z) \<in> genPrefix s" shows "(x, z) \<in> genPrefix (r O s)"
IH: fixes x :: "'a" assumes "RETURN x \<le> lfp (\<lambda>x. sup m (x \<bind> a))" shows "f x \<le> lfp (\<lambda>x. sup m (x \<bind> a)) \<bind> b"
top_le: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and top :: "'g"   and a :: "'g" assumes "class.order_top less_eq less top"   and "less_eq top a" shows "a = top"
cring_simprules_13: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
lin_trans_l: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset"   and D :: "'b multiset" assumes "locally_included R1 R2 (A + B) (C + D)" shows "locally_included R1 R2 A C"
continuous_closedin_preimage: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "closed T" shows "closedin (top_of_set S) (S \<inter> f -` T)"
restrict_Pi_cancel: fixes x :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" shows "(restrict x I \<in> Pi I A) = (x \<in> Pi I A)"
r_result1: fixes t :: "nat"   and i :: "nat"   and x :: "nat" shows "eval r_result1 [t, i, x] = eval r_result [t, i, singleton_encode x]"
par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Secrets :: "('fun, 'var) Term.term set" shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> Secrets \<equiv> (\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l1 \<A>) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l1 \<A>)) (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l2 \<A>) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l2 \<A>)) Secrets) \<and> ground Secrets \<and> (\<forall>s\<in>Secrets. \<forall>s'\<in>subterms s. {} \<turnstile>\<^sub>c s' \<or> s' \<in> Secrets) \<and> (\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> i = j)"
hyps: fixes lm2 :: "nat list"   and lm3 :: "nat list"   and lm4 :: "nat list" assumes "aa + n \<le> ba"   and "aa < ba"   and "length lm1 = aa"   and "length lm2 = n"   and "length lm3 = ba - aa - n" shows "{\<lambda>nl. nl = lm1 @ lm2 @ lm3 @ 0 \<up> n @ lm4} mv_boxes aa ba n {\<lambda>nl. nl = lm1 @ 0 \<up> n @ lm3 @ lm2 @ lm4}"