hcju/isabelleps huggingface dataset:
parallel_constructive_security_obsf: fixes real1 :: "('m + 'n, 'o + 'p) resource"   and ideal1 :: "('q + 'n, 'r + 'p) resource"   and sim1 :: "('m, 'o, 'q, 'r) converter"   and \<I>_real1 :: "('m, 'o) \<I>"   and \<I>_inner1 :: "('q, 'r) \<I>"   and \<I>_common1 :: "('n, 'p) \<I>"   and \<A> :: "(('m + 's) + 'n + 't, ('o + 'u) + 'p + 'v) distinguisher_obsf"   and sim2 :: "('s, 'u, 'w, 'x) converter"   and ideal2 :: "('w + 't, 'x + 'v) resource"   and adv1 :: "real"   and real2 :: "('s + 't, 'u + 'v) resource"   and \<I>_real2 :: "('s, 'u) \<I>"   and \<I>_inner2 :: "('w, 'x) \<I>"   and \<I>_common2 :: "('t, 'v) \<I>"   and adv2 :: "real" assumes "constructive_security_obsf real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 (absorb \<A> (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2))))) adv1"   and "constructive_security_obsf real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 (absorb \<A> (obsf_converter (parallel_wiring \<odot> (converter_of_resource real1 |\<^sub>\<propto> 1\<^sub>C)))) adv2" shows "constructive_security_obsf (parallel_wiring \<rhd> real1 \<parallel> real2) (parallel_wiring \<rhd> ideal1 \<parallel> ideal2) (sim1 |\<^sub>= sim2) (\<I>_real1 \<oplus>\<^sub>\<I> \<I>_real2) (\<I>_inner1 \<oplus>\<^sub>\<I> \<I>_inner2) (\<I>_common1 \<oplus>\<^sub>\<I> \<I>_common2) \<A> (adv1 + adv2)"
perm1: fixes p :: "perm" assumes "supp p \<sharp>* c" shows "p \<bullet> f (as \<inter> supp x) x c = f (p \<bullet> (as \<inter> supp x)) (p \<bullet> x) c"
rel_frontier_convex_hull_cases: fixes S :: "'b set" assumes "\<not> affine_dependent S" shows "rel_frontier (convex hull S) = \<Union> {convex hull (S - {x}) |x. x \<in> S}"
rebalance_last_tree_bal: fixes ts :: "('a btree \<times> 'a) list"   and t :: "'a btree"   and k :: "nat" assumes "bal (Node ts t)"   and "ts \<noteq> []" shows "bal (rebalance_last_tree k ts t)"
prod_cases4: fixes y :: "'c \<times> 'd \<times> 'e \<times> 'f" obtains a :: "'c" and b :: "'d" and c :: "'e" and d :: "'f" where "y = (a, b, c, d)"
Real_leI: fixes X :: "nat \<Rightarrow> rat"   and y :: "real" assumes "Real.cauchy X"   and "\<forall>n. real_of_rat (X n) \<le> y" shows "Real.Real X \<le> y"
A_FactGroup_hom: shows "(\<lambda>X. the_elem (h ` X)) \<in> hom (G A_Mod a_kernel G H h) (add_monoid H)"
non_empty_backward_terminating_path_root_iff: fixes x :: "'a" shows "backward_terminating_path_root (start_points x) x = (backward_terminating_path x \<and> x \<noteq> 0)"
dvdE: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "dvd.dvd times b a" obtains k :: "'a" where "a = times b k"
fset_of_fmap_iff: fixes a :: "'e"   and b :: "'f"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "((a, b) |\<in>| fset_of_fmap m) = (fmlookup m a = Some b)"
Comm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "xvec \<sharp>* P" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'"
rep_eq: shows "fls_nth fls_X_inv = (\<lambda>n. if n = - 1 then 1::'a else (0::'a))"
CapN: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "N \<langle>\<phi>\<rangle> = \<langle>neg \<phi>\<rangle>"
Schema_is_CSP: fixes sc :: "'d \<times> 'd \<Rightarrow> bool" shows "is_CSP_process (R (\<lambda>(A, A'). Pre sc (alpha_rp.more A) \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A'))"
l8: shows "0 < q ^ 3 \<and> coprime (p ^ 3) (q ^ 3)"
det_lower_triangular: fixes n :: "nat"   and A :: "'a mat" assumes "\<And>i j. \<lbrakk>i < j; j < n\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'a)"   and "A \<in> carrier_mat n n" shows "Determinant.det A = prod_list (diag_mat A)"
simps_1: fixes x2 :: "'a"   and y2 :: "'a" shows "(Some x2 = Some y2) = (x2 = y2)"
4: shows "src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1"
listToFun_splice: fixes xs :: "'b list"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c list"   and h :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "set xs = UNIV"   and "distinct xs"   and "g \<in> set (listToFuns f xs)"   and "h \<in> set (listToFuns f xs)" shows "g(x := h x) \<in> set (listToFuns f xs)"
compute_mult_scalar_poly_mapping: fixes xs :: "('a \<times> 'c) list"   and q :: "'a \<times> 'b \<Rightarrow>\<^sub>0 'c" shows "pprod.mult_scalar (Pm_fmap (fmap_of_list xs)) q = (case xs of [] \<Rightarrow> Pm_fmap fmempty | (t, c) # ys \<Rightarrow> pprod.monom_mult c t q + pprod.mult_scalar (except (Pm_fmap (fmap_of_list ys)) {t}) q)"
mcont_results: shows "mcont resumption_lub resumption_ord \<Union> (\<subseteq>) results"
assms_2: shows "P \<turnstile> C sees_methods Mm"
axioms_1: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit" shows "CFGExit sourcenode targetnode kind valid_edge Entry Exit"
poly_fun_divide: fixes f :: "int \<Rightarrow> int"   and c :: "int" assumes "poly_fun f"   and "\<And>a. c dvd f a" shows "poly_fun (\<lambda>x. f x div c)"
iMOD_iprev_diff_const: fixes t :: "nat"   and r :: "nat"   and m :: "nat" assumes "t \<in> [ r, mod m ]"   and "r < t" shows "t - iprev t [ r, mod m ] = m"
span_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and S :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> ideal S" shows "sum f A \<in> ideal S"
characteriseSeq: fixes C :: "sequent" shows "\<exists>A B. C = ( A \<Rightarrow>* B)"
comp_intro: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c"   and Q :: "'c \<times> 'd \<Rightarrow> bool"   and c :: "'d" assumes "P (a, b)"   and "Q (b, c)" shows "(P ;; Q) (a, c)"
store_store_commute: fixes i1 :: "'c"   and mo2 :: "'c"   and i2 :: "'c"   and mo1 :: "'c"   and e1 :: "'d"   and e2 :: "'d" assumes "i1 \<noteq> mo2"   and "i2 \<noteq> mo1" shows "\<langle>Store e1 mo1 i1\<rangle> \<rhd> \<langle>Store e2 mo2 i2\<rangle> = \<langle>Store e2 mo2 i2\<rangle> \<rhd> \<langle>Store e1 mo1 i1\<rangle>"
sig_inv_setD_lt: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "r \<in> sig_inv_set' j"   and "r \<noteq> 0" shows "component_of_term (lt r) < j"
prems_10: shows "InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
flatten_pi_2_correct: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi>[\<F>\<G> \<phi> w]\<^sub>\<Pi>\<^sub>2 = w \<Turnstile>\<^sub>n \<phi>"
all_maps_correct: shows "all_maps (list_digraph G) = {M. digraph_map (with_proj (list_digraph G)) M}"
construct_witness_eq: fixes Q :: "'d \<Rightarrow> 'e tree option"   and r :: "('d, 'e) ta_rule"   and Q' :: "'d \<Rightarrow> 'e tree option" assumes "Q |` set (rhsq r) = Q' |` set (rhsq r)" shows "construct_witness Q r = construct_witness Q' r"
mult_minus1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and z :: "'c" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus one) z = uminus z"
intros: fixes DA :: "'a literal multiset"   and As :: "'a list" assumes "S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> maximal_wrt (As ! 0) DA" shows "eligible As DA"
intro: fixes \<V> :: "'b set"   and \<B> :: "'b set multiset"   and \<V>' :: "'b set"   and \<B>' :: "'b set multiset" assumes "incidence_system \<V> \<B>"   and "incidence_system \<V>' \<B>'" shows "two_set_systems \<V> \<B> \<V>' \<B>'"
Order_component: fixes E :: "'c Order" shows "E = \<lparr>carrier = carrier E, rel = rel E\<rparr>"
simps_13: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (Or p q) = (Ifm bbs bs p \<or> Ifm bbs bs q)"
wpt_Seq: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "wp\<^sub>Q (c\<^sub>1;; c\<^sub>2) Q = wp\<^sub>Q c\<^sub>1 (wp\<^sub>Q c\<^sub>2 Q)"
XQFXQ: fixes X :: "'a set" assumes "X \<in> Q" shows "F X \<in> Q"
suc_not_zero: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep a + 1 \<noteq> CARD('d)" shows "plus a one \<noteq> zero"
veval'_correct: fixes \<Gamma>\<^sub>2 :: "(name, value) fmap"   and t :: "sterm"   and v\<^sub>2 :: "value"   and \<Gamma>\<^sub>1 :: "(name, value) fmap" assumes "\<Gamma>\<^sub>2 \<turnstile>\<^sub>v t \<down> v\<^sub>2"   and "pre_strong_term_class.wellformed t"   and "wellformed_venv \<Gamma>\<^sub>2"   and "\<not> shadows_consts t"   and "not_shadows_vconsts_env \<Gamma>\<^sub>2"   and "consts t |\<subseteq>| all_consts"   and "fmpred (\<lambda>_. vwelldefined) \<Gamma>\<^sub>1"   and "fmrel_on_fset (frees t) vrelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2"   and "fmrel_on_fset (consts t) vrelated (fmap_of_list rs) \<Gamma>\<^sub>2" obtains v\<^sub>1 :: "value" where "rs, \<Gamma>\<^sub>1 \<turnstile>\<^sub>v t \<down> v\<^sub>1" and "\<turnstile>\<^sub>v v\<^sub>1 \<approx> v\<^sub>2"
assms_2: fixes x :: "'a" assumes "x \<in> B - A" shows "f x = (0::'b)"
limsup_MInfty: fixes X :: "nat \<Rightarrow> ereal" shows "X \<longlonglongrightarrow> - \<infinity> = (limsup X = - \<infinity>)"
tagged_fun_relD_both: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'f"   and A :: "('c \<times> 'e) set"   and B :: "('d \<times> 'f) set"   and x :: "'c"   and x' :: "'e" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, Let x' f') \<in> B"
less_eq_Dom_def: fixes d1 :: "Dom"   and d2 :: "Dom" shows "(d1 \<le> d2) = (if d1 = d2 then True else if d1 = low then True else False)"
sum_over_permutations_insert: fixes S :: "'b set"   and a :: "'b"   and f :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c" assumes "finite S"   and "a \<notin> S" shows "sum f {p. p permutes insert a S} = (\<Sum>b\<in>insert a S. \<Sum>q\<in>{p. p permutes S}. f (Fun.swap a b id \<circ> q))"
coprime_set_def: fixes N :: "nat" shows "coprime_set N \<equiv> {x. coprime x N \<and> 1 < x \<and> x < N}"
record_simps_1: fixes f :: "nat \<Rightarrow> nat"   and x :: "'a state" shows "clock (update_clock f x) = f (clock x)"
fresh_13: fixes a :: "atom"   and "term" :: "Syntax.term" shows "a \<sharp> Roll term = a \<sharp> term"
sum_Basis_prod_eq: fixes f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b" shows "sum f Basis = (\<Sum>i\<in>Basis. f (i, 0::'b)) + (\<Sum>i\<in>Basis. f (0::'a, i))"
RDirProd_list_carrier: fixes G :: "('a, 'b) ring_scheme"   and Gs :: "('a, 'b) ring_scheme list" shows "carrier (RDirProd_list (G # Gs)) = (\<lambda>(x, xs). x # xs) ` (carrier G \<times> carrier (RDirProd_list Gs))"
detAB_Znm_aux: fixes F :: "(nat \<Rightarrow> nat) set"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and A :: "'b mat" assumes "F = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i}" shows "(\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, i)) * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i)))) = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i)) * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i))))"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "domain R"   and "field_axioms R" shows "field R"
abs_eq: fixes x :: "'a" shows "Const x = MPoly (Const\<^sub>0 x)"
vsvE: fixes r :: "V" assumes "vsv r" obtains "vbrelation r" and "\<And>a b c. \<lbrakk>\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r; \<langle>a, c\<rangle> \<in>\<^sub>\<circ> r\<rbrakk> \<Longrightarrow> b = c"
assms_1: shows "p = {l}\<langle>fg\<rangle> qq"
elims: fixes x :: "'b list"   and xa :: "'c list"   and y :: "'a" assumes "foo x xa = y" obtains xb :: "'b" and xs :: "'b list" where "x = xb__ # xs__" and "xa = []" and "y = (0::'a)"   | xb :: "'b" and xs :: "'b list" and ya :: "'c" and ys :: "'c list" where "x = xb # xs" and "xa = ya__ # ys__" and "y = foo [] []"   | ya :: "'c" and ys :: "'c list" where "x = []" and "xa = ya # ys" and "y = (1::'a)"   | "x = []" and "xa = []" and "y = (3::'a)"
simps_1: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and x' :: "'v"   and y :: "'v" shows "split f (Vertex x, Edge x' y) = (if x' = x then f (x, y) else 0)"
hen_1: assumes "\<And>F. F \<in> set fs \<Longrightarrow> F \<in> set fs" shows "q.eq_m F (prod_list gs)"
unit_disc_fix_preserves_poincare_ray_meets_line: fixes M :: "moebius"   and a :: "complex_homo"   and b :: "complex_homo"   and c :: "complex_homo"   and d :: "complex_homo" assumes "unit_disc_fix M"   and "a \<in> unit_disc"   and "b \<in> unit_disc"   and "c \<in> unit_disc"   and "d \<in> unit_disc" shows "poincare_ray_meets_line (moebius_pt M a) (moebius_pt M b) (moebius_pt M c) (moebius_pt M d) = poincare_ray_meets_line a b c d"
field_def: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal" shows "field P C F \<equiv> THE (D, T, fm). P \<turnstile> C sees F:T (fm) in D"
trail_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "S.trail u p v \<equiv> S.awalk u p v \<and> distinct p"
twiddleHeap_compose: fixes \<beta> :: "(Location \<times> Location) set"   and h1 :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and h2 :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<gamma> :: "(Location \<times> Location) set"   and h3 :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "twiddleHeap \<beta> h1 h2"   and "twiddleHeap \<gamma> h2 h3"   and "\<beta> \<in> Pbij"   and "\<gamma> \<in> Pbij" shows "twiddleHeap (Pbij_compose \<beta> \<gamma>) h1 h3"
assm: shows "j \<noteq> bot \<and> k \<noteq> bot \<and> maximum j + 1 = minimum k"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_ring_hom hom" shows "map_poly_comm_ring_hom hom"
greatest: fixes k :: "nat" assumes "k \<in> length ` S" shows "k \<le> m"
pp_mapping_linorder_matrix_unique: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_mapping f = (\<forall>i. \<exists>!j. f (i, j) \<noteq> bot \<and> (\<forall>k. j \<noteq> k \<longrightarrow> f (i, k) = bot))"
simps_4: fixes \<phi> :: "'b formula" shows "FOV (FNot \<phi>) = FOV \<phi>"
pmat_extension_def: fixes m :: "'a mat" shows "st3.pmat_extension m = st3.ptensor_mat m (1\<^sub>m st4.d2)"
reachable_init: fixes s :: "'d"   and A :: "('d, 'e) automaton"   and I :: "'e \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> reachable A I"
L1: shows "r `` A = Range (r \<inter> A \<times> UNIV)"
OrdDom_hunion: fixes s1 :: "hf"   and s2 :: "hf" shows "OrdDom (s1 \<squnion> s2) = (OrdDom s1 \<and> OrdDom s2)"
bij_betw_imp_funcset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "f \<in> A \<rightarrow> B"
fresh_3: fixes a :: "name"   and x2 :: "ty"   and x1 :: "ty" shows "a \<sharp> x2\<rightarrow>x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
equality: fixes r :: "('e, 'f, 'g) pre_digraph_scheme"   and r' :: "('e, 'f, 'g) pre_digraph_scheme" assumes "verts r = verts r'"   and "arcs r = arcs r'"   and "tail r = tail r'"   and "head r = head r'"   and "pre_digraph.more r = pre_digraph.more r'" shows "r = r'"
eq_neutr_iff: fixes a :: "'a"   and b :: "'a" shows "(sup a b = bot) = (a = bot \<and> b = bot)"
inj_on_eq_iff: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e"   and y :: "'e" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x = f y) = (x = y)"
simps_1: fixes uu :: "'c"   and x :: "'c" shows "the_default uu (Some x) = x"
update_valuation_nonlhs: fixes s :: "('i, 'a) state"   and x :: "nat"   and x' :: "nat"   and v :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "x' \<notin> lvars (\<T> s) \<longrightarrow> Mapping.lookup (\<V> (update x v s)) x' = (if x = x' then Some v else Mapping.lookup (\<V> s) x')"
comp_arr_inv: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.inverse_arrows f g" shows "VV.comp f g = VV.dom g"
associatedI: fixes a :: "'a"   and b :: "'a" assumes "a dvd b"   and "b dvd a" shows "normalize a = normalize b"
AbGroup_inner_dirsum_el_decomp_eq: fixes Gs :: "'a set list"   and x :: "'a"   and gs :: "'a list" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "x \<in> (\<Oplus>G\<leftarrow>Gs. G)"   and "gs \<in> listset Gs"   and "x = sum_list gs" shows "\<Oplus>Gs\<leftarrow> x = gs"
transpose_column: fixes xs :: "'g list list"   and i :: "nat" assumes "sorted (rev (map length xs))"   and "i < length xs" shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs)) = xs ! i"
deg_pols_add1: fixes p :: "'a"   and q :: "'a" assumes "p \<in> carrier R"   and "q \<in> carrier R"   and "deg R S X p < deg R S X q" shows "deg R S X (p \<plusminus> q) = deg R S X q"
xor_self: fixes x :: "'c" shows "x XOR x = (0::'c)"
inverts_components: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.ide a" shows "inverse_arrows \<a>[fst a, fst (snd a), snd (snd a)] (\<alpha>' a)"
hyps: shows "fst (Gauss_Jordan_upt_k_PA A k) ** A = snd (Gauss_Jordan_upt_k_PA A k)"
tail_union: fixes G :: "('d, 'c) pre_digraph"   and H :: "('d, 'c) pre_digraph" shows "tail (Digraph_Component.union G H) = tail G"
open_Collect_const: fixes P :: "bool" shows "open {x. P}"
basis_card_eq_dim: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B :: "'e set"   and V :: "'e set" assumes "vector_space scale"   and "B \<subseteq> V"   and "V \<subseteq> module.span scale B"   and "\<not> module.dependent scale B" shows "card B = vector_space.dim scale V"
comp_fun_Pi: fixes g :: "'a \<Rightarrow> nat" shows "(\<lambda>x. f x ^^ g x) \<in> A \<rightarrow> A \<rightarrow> A"
is_renaming_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "Term.is_renaming \<sigma> = ((\<forall>x. is_Var (\<sigma> x)) \<and> inj_on \<sigma> (subst_domain \<sigma>))"
P_no_deadends: fixes n :: "nat" assumes "enat (Suc n) < llength P'" shows "\<not> deadend (P' $ n)"
ik\<^sub>s\<^sub>tD: fixes t :: "('d, 'e) Term.term"   and S :: "('d, 'e) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set S"
bal: shows "bal (dblock s' p) = mbal (dblock s p)"
assms: fixes k :: "'a\<cdot>'e writer\<cdot>'f" shows "P (WriterT\<cdot>k)"
apply_adaptation_def: fixes first :: "bool"   and mem :: "'c \<Rightarrow> 'd"   and A :: "'c \<Rightarrow> ('d \<times> 'd) option" shows "apply_adaptation first mem A = (\<lambda>x. case A x of None \<Rightarrow> mem x | Some (v\<^sub>1, v\<^sub>2) \<Rightarrow> if first then v\<^sub>1 else v\<^sub>2)"
indhyp: shows "f (\<Sum> F) = \<Sum> (f ` F)"
simps_2: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat" shows "delta_predict N der matur (Suc n) = delta_price N der matur n"
AE_finite_all: fixes S :: "'b set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c measure" assumes "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) = (\<forall>i\<in>S. almost_everywhere M (P i))"
coerce_ErrorT: fixes k :: "'h\<cdot>'j error\<cdot>'i" shows "COERCE('h\<cdot>('i, 'j) errorT, 'e\<cdot>('f, 'g) errorT)\<cdot>(ErrorT\<cdot>k) = ErrorT\<cdot>(COERCE('h\<cdot>'j error\<cdot>'i, 'e\<cdot>'g error\<cdot>'f)\<cdot>k)"
hs_get_work_locs_subseteq_hp_IdleMarkSweep_locs: shows "hs_get_work_locs \<subseteq> hp_IdleMarkSweep_locs"
counit_is_iso: shows "D.iso (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
uni: fixes M :: "'a set set" assumes "\<forall>t\<in>M. t open \<and> P t" shows "P (\<Union> M)"
emeasure_finite: fixes A :: "'a set" shows "emeasure M A \<noteq> \<top>"
MGFn_def2: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and t :: "Term.term" shows "G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>} = G,A\<turnstile>{\<doteq> \<and>. G\<turnstile>init\<le>n} t\<succ> {\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y, s')}"
cut_def: fixes f :: "'e \<Rightarrow> 'f"   and R :: "('e \<times> 'e) set"   and x :: "'e" shows "Wfrec.cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
assoc: fixes a :: "'e multiset"   and b :: "'e multiset"   and c :: "'e multiset" shows "a \<inter># b \<inter># c = a \<inter># (b \<inter># c)"
lookup_notin: fixes x :: "name"   and e :: "name list"   and n :: "nat" assumes "x \<notin> set e" shows "lookup e n x = DBVar x"
Cd_domain: fixes d :: "'b"   and X :: "'a set" shows "Singleton_for_d.Cd d (X \<inter> Field (Pd_singletons_for_ds X {Xd x} d)) = Singleton_for_d.Cd d X"
assms_1: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F G \<tau>"
ActualEquivE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>(\<phi> \<^bold>\<equiv> \<psi>) in v]" shows "[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<A>\<psi> in v]"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<alpha> \<prec> P'"   and "guarded P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) \<alpha> P'" shows "Prop C \<Psi> (!P) \<alpha> P'"
lookup_delete: fixes k :: "'a"   and t :: "('a, 'b) RBT.rbt" shows "RBT.lookup (RBT.delete k t) = (RBT.lookup t)(k := None)"
ang_circ_simp: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" assumes "E \<noteq> \<mu>1"   and "E \<noteq> \<mu>2" shows "ang_circ E \<mu>1 \<mu>2 p1 p2 = \<downharpoonright>arg (E - \<mu>2) - arg (E - \<mu>1) + sgn_bool p1 * pi / 2 - sgn_bool p2 * pi / 2\<downharpoonleft>"
exH_eq_H: shows "exH_k (n - 1) = H_k (n - 1)"
step_a_z: fixes l :: "'e"   and g :: "('f, 'g) cconstraint"   and a :: "'h"   and r :: "'f list"   and l' :: "'e"   and A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and Z :: "('f \<Rightarrow> 'g) set" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', (Z \<inter> {u. u \<turnstile> g})\<^bsub>r \<rightarrow> 0\<^esub> \<inter> {u. u \<turnstile> inv_of A l'}\<rangle>"