hcju/isabelleps huggingface dataset:
fresh_5: fixes a :: "name"   and x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "a \<sharp> [x3\<frown>x2]x1 = (a \<sharp> x3 \<and> a \<sharp> x2 \<and> a \<sharp> x1)"
bit_of_int_iff: fixes k :: "int"   and n :: "nat" shows "bit (of_int k) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> bit k n)"
left_def: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.left unapp t = (case unapp t of Some (l, x) \<Rightarrow> l)"
isUnitClauseAppendValuation: fixes uClause :: "Literal list"   and uLiteral :: "Literal"   and valuation :: "Literal list"   and l :: "Literal" assumes "isUnitClause uClause uLiteral valuation"   and "l \<noteq> uLiteral"   and "l \<noteq> opposite uLiteral" shows "isUnitClause uClause uLiteral (valuation @ [l])"
Gromov_boundary_abs_converging: fixes x :: "'c Gromov_completion"   and u :: "nat \<Rightarrow> 'c" assumes "x \<in> Gromov_boundary"   and "abs_Gromov_completion u = x"   and "Gromov_completion_rel u u" shows "Gromov_converging_at_boundary u"
le_less_linear: fixes y :: "'a list"   and x :: "'a list" shows "y \<le>lex x \<or> x <lex y"
cod_MkArr: fixes A :: "'e set"   and B :: "'e set"   and F :: "'e \<Rightarrow> 'e" assumes "arr (MkArr A B F)" shows "cod (MkArr A B F) = MkIde B"
bind_advantage_def: fixes \<A> :: "'pub_input \<Rightarrow> ('msg \<times> 'challenge \<times> 'response \<times> 'challenge \<times> 'response) spmf" shows "abstract_com.bind_advantage \<A> \<equiv> spmf (abstract_com.bind_game \<A>) True"
eql: shows "(\<pi> @\<^bsup>k\<^esup> \<pi>l) =\<^bsub>k\<^esub> \<pi>"
kruskal_injective_inv_3: fixes e :: "'b"   and f :: "'b" assumes "stone_algebra_class.comp_inf.arc e"   and "stone_algebra_class.comp_inf.forest f" shows "bounded_distrib_lattice_class.comp_inf.coreflexive (stone_algebra_class.comp_inf.rel_access (bounded_distrib_lattice_class.comp_inf.tc (top \<sqinter> e)) (bounded_distrib_lattice_class.comp_inf.tc (top \<sqinter> e)) \<sqinter> stone_algebra_class.comp_inf.rel_access f f)"
linord_of_list_range: fixes xs :: "'b list" shows "linord_of_list xs \<subseteq> set xs \<times> set xs"
simps: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and xs :: "'varSort"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm" shows "qAbs xs x X $[[rho]] = (let x' = pickQFreshEnv xs {x} {X} {rho} in qAbs xs x' X #[[x' \<and> x]]_xs #[[rho]])"
split_list: fixes x :: "'g"   and xs :: "'g list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
sgn_1: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn one = one"
mod_mult_right_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a (modulo b c)) c = modulo (times a b) c"
simps: fixes rhs :: "'b trm set" shows "lang_rhs rhs = \<Union> (lang_trm ` rhs)"
geometric_sums_times_norm: fixes c :: "'b" assumes "norm c < 1" shows "(\<lambda>n. norm (c ^ n * of_nat n)) sums (norm c / (1 - norm c)\<^sup>2)"
arctan_less_pi4: fixes x :: "real" assumes "\<bar>x\<bar> < 1" shows "\<bar>arctan x\<bar> < pi / 4"
assms_1: shows "c \<equiv> ASSERT \<Phi> \<bind> (\<lambda>_. m')"
occurs_in_def: fixes t :: "'b trm"   and s :: "'b trm" shows "occurs_in t s = (\<exists>p. subterm s p t)"
strict_order_order: fixes x :: "'a" assumes "strict_order x" shows "order (x \<squnion> (1::'a))"
distinct_11: shows "GuarNoReadOrWrite \<noteq> GuarNoWrite"
assms: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm vs bs (qe p) = Ifm vs bs (E p)"
ideal_ring_multiple: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and x :: "'c"   and r :: "'c" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "r \<in> carrier R" shows "r \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<in> I"
contg: shows "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of (topspace Z - U)}) Y g"
thetaWhileW0_def: shows "thetaWhileW0 \<equiv> {(while tst do c, while tst do d) |tst c d. compatTst tst \<and> c \<approx>T d} \<union> {(c1 ;; while tst do c, d1 ;; while tst do d) |tst c1 d1 c d. compatTst tst \<and> c1 \<approx>T d1 \<and> c \<approx>T d}"
splits_at_\<alpha>: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and s :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "Derives1 \<delta> i r s"   and "splits_at \<delta> i \<alpha> N \<beta>" shows "\<alpha> = take i \<delta> \<and> \<alpha> = take i s \<and> length \<alpha> = i"
ide_in_hom: fixes a :: "'a \<times> 'a" shows "VoV.ide a = VoV.in_hom a a a"
iT_drule_6: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]" shows "r \<le> x"
fresh_cart_prod: fixes Xs :: "'a set"   and Ys :: "'c set"   and x :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "fs TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "finite Xs"   and "finite Ys"   and "Xs \<noteq> {}"   and "Ys \<noteq> {}" shows "x \<sharp> Xs \<times> Ys = (x \<sharp> Xs \<and> x \<sharp> Ys)"
that: fixes n :: "nat"   and m :: "int" assumes "0 < n"   and "x = of_int m / of_nat n"   and "coprime m (int n)" shows "thesis"
zip_eq_ConsE: fixes xs :: "'d list"   and ys :: "'e list"   and xy :: "'d \<times> 'e"   and xys :: "('d \<times> 'e) list" assumes "xs || ys = xy # xys" obtains x :: "'d" and xs' :: "'d list" and y :: "'e" and ys' :: "'e list" where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = xs' || ys'"
intra_SDG_path_intra_CFG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n'" obtains as :: "'edge list" where "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'"
transitivity2_L: fixes Cn :: "'b set \<Rightarrow> 'b set"   and B :: "'b set"   and A :: "'b set"   and \<psi> :: "'b" assumes "Tarskian_logic Cn"   and "\<forall>\<phi>\<in>B. \<phi> \<in> Cn A"   and "\<psi> \<in> Cn B" shows "\<psi> \<in> Cn A"
composite_cell_in_hom: fixes w :: "'a"   and u :: "'a"   and \<theta> :: "'a" assumes "ide w"   and "\<guillemotleft>w : src u \<rightarrow> src s\<^sub>0\<guillemotright>"   and "\<guillemotleft>\<theta> : s\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>" shows "\<guillemotleft>(s \<star> \<theta>) \<cdot> \<a>[s, s\<^sub>0, w] \<cdot> (\<sigma> \<star> w) : s\<^sub>1 \<star> w \<Rightarrow> s \<star> u\<guillemotright>"
Tree\<^sub>\<alpha>_free_6: fixes \<phi> :: "'f"   and \<alpha> :: "'g"   and t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "Pred\<^sub>\<alpha> \<phi> \<noteq> Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>"
finprod_multf: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "f \<in> A \<rightarrow> carrier G"   and "g \<in> A \<rightarrow> carrier G" shows "(\<Otimes>x\<in>A. f x \<otimes> g x) = finprod G f A \<otimes> finprod G g A"
preserves_inv: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "functor A B F"   and "category.iso A f" shows "F (category.inv A f) = category.inv B (F f)"
point_add_def: fixes A :: "point"   and B :: "point" shows "A + B = Point (abscissa A + abscissa B) (ordinate A + ordinate B)"
vreal_uminus_closed: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "-\<^sub>\<real> x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
nat_term_compare_DEG: fixes to :: "'a nat_term_order" shows "nat_term_compare (DEG to) = deg_comp (nat_term_compare to)"
quat_mult_cnj_commute: fixes q :: "quat" shows "Quaternions.cnj q * q = q * Quaternions.cnj q"
st_mat: fixes x :: "'a st_mat" shows "st_mat x \<in> {a. non_neg_mat a \<and> right_stoch_mat a}"
step_valid: fixes I' :: "nat set"   and I :: "nat set" assumes "(I', I) \<in> step_rel" shows "I' \<in> valid_I"
all_s: shows "\<forall>s. s \<le>s w \<and> Lyndon s \<longrightarrow> s \<le>s q"
to_sas_plus_list_of_transformed_sas_plus_problem_operators_structure: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops' :: "('c \<times> 'd) strips_operator list"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "op \<in> set (map (strips_op_to_sasp \<Psi>) ops')" shows "op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+) \<and> (\<exists>op'\<in>set ops'. op' = \<phi>\<^sub>O \<Psi> op)"
codomains_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C" shows "partial_magma.codomains C f \<equiv> {b. partial_magma.ide C b \<and> C b f \<noteq> partial_magma.null C}"
self_financingE: fixes Mkt :: "('b, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real" assumes "self_financing Mkt p" shows "\<forall>n. val_process Mkt p n = cls_val_process Mkt p n"
degeneralize_nodes_finite: fixes A :: "('d, 'c) dgca" shows "finite (DCA.nodes (degeneralize A)) = finite (DGCA.nodes A)"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> 'univ"   and xa :: "('fsym, 'psym) lit" assumes "\<not> satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "\<not> satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> satA \<xi> a"
compactE_image: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and C :: "'c set"   and f :: "'c \<Rightarrow> 'b set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "\<And>T. T \<in> C \<Longrightarrow> open (f T)"   and "S \<subseteq> \<Union> (f ` C)" obtains C' :: "'c set" where "C' \<subseteq> C" and "finite C'" and "S \<subseteq> \<Union> (f ` C')"
is_moebius_cross_ratio: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo" assumes "z1 \<noteq> z2"   and "z2 \<noteq> z3"   and "z1 \<noteq> z3" shows "is_moebius (\<lambda>z. cross_ratio z z1 z2 z3)"
not_Nil_step1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "\<not> step1 r [] xs"
oguardT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and g :: "'e \<Rightarrow> 'e set"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i \<in> g (\<sigma> i)" shows "((\<sigma>, {l}\<langle>g\<rangle> p), \<tau>, \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
1: shows "\<pi>' (Suc (k' + l'')) = \<pi> l"
learn_cons_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" shows "learn_cons \<psi> U s \<equiv> learn_lim \<psi> U s \<and> (\<forall>f\<in>U. \<forall>n k. k \<le> n \<longrightarrow> \<psi> (the (s (f \<triangleright> n))) k = f k)"
functor_axioms: shows "functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a)"
forall_4: fixes P :: "4 \<Rightarrow> bool" shows "(\<forall>i. P i) = (P 1 \<and> P 2 \<and> P 3 \<and> P 4)"
funpow_Suc_inside: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "(f ^^ Suc n) x = (f ^^ n) (f x)"
Ball_subset_right: fixes T :: "'a set"   and S :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "T \<subseteq> S"   and "\<forall>x\<in>S - T. P x" shows "(\<forall>x\<in>S. P x) = (\<forall>x\<in>T. P x)"
arr_cod_iff_arr: fixes f :: "'a" shows "C.arr (C.cod f) = C.arr f"
simps_2: fixes x :: "'c ltl"   and xs :: "'c ltl list" shows "Or (x # xs) = foldl LTLOr x xs"
tr_f_to_rel: fixes T :: "x \<Rightarrow> x option"   and s :: "s" shows "\<forall>s'. tr_s_f T s = s' \<longrightarrow> tr_s T s s'"
smooth_compat_restrict_chartI: fixes k :: "enat"   and c :: "('c, 'd) chart"   and c' :: "('c, 'd) chart"   and S :: "'c set" assumes "k-smooth_compat c c'" shows "k-smooth_compat (restrict_chart S c) c'"
assms: shows "\<guillemotleft>\<tau> : p\<^sub>1 \<Rightarrow> (k\<^sup>* \<star> h) \<star> p\<^sub>0\<guillemotright>"
uadmit_ode_ntadjoint': fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and ODE :: "('sf + 'b, 'sz) ODE" assumes "\<And>i. dsafe (\<sigma> i)"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO ODE}. FVT (\<sigma> y))"   and "osafe ODE" shows "ODE_sem (adjointFO I \<sigma> \<nu>) ODE = ODE_sem (adjointFO I \<sigma> \<omega>) ODE"
cmp_simps'_1: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (\<Phi>\<^sub>F (\<mu>, \<nu>))"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and xa :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (x - xa) = (\<lambda>k. lookup x k - lookup xa k)"
conc_fun_SPEC: fixes R :: "('e \<times> 'f) set"   and \<Phi> :: "'f \<Rightarrow> bool" shows "\<Down> R (SPEC \<Phi>) = SPEC (\<lambda>y. \<exists>x. (y, x) \<in> R \<and> \<Phi> x)"
tab_simps_4: shows "local.dom \<sigma> = s\<^sub>1"
rel_ad_def: fixes R :: "('a \<times> 'a) set" shows "rel_ad R = \<lceil>\<lambda>x. \<nexists>y. (x, y) \<in> R\<rceil>"
exhaust: fixes y :: "('a, 'b) Expression" obtains "y = Vx"   | x21 :: "'b" and x22 :: "('a, 'b) Expression" where "y = x21 E@ x22"
ctr_transfer_2: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun R1 (rel_fun (tllist_all2 R1 R2) (tllist_all2 R1 R2)) TCons TCons"
hyps_2: assumes "\<And>i. i < length xs \<Longrightarrow> length (xs ! i) = length (ys ! i)"   and "\<And>i j. \<lbrakk>i < length xs; j < length (xs ! i)\<rbrakk> \<Longrightarrow> P (xs ! i ! j) (ys ! i ! j)" shows "\<forall>k<length (concat xs). P (concat xs ! k) (concat ys ! k)"
I_1: shows "SRCTransforms.mset C \<equiv> \<LM> Compound F Fs  \<RM>"
prelim_element_match: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "matrix_match A1 A2 B1 B2" shows "\<forall>i j. i < row_length A1 * row_length B1 \<and> j < length A2 * length B2 \<longrightarrow> (A1 \<circ> A2 \<otimes> B1 \<circ> B2) ! j ! i = ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2)) ! j ! i"
subset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "(A \<subseteq> x \<triangleright> B) = (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
transpI: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
sum_mono: fixes K :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset"   and g :: "'a \<Rightarrow> 'b multiset" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<subseteq># g i" shows "comm_monoid_add.sum (+) {#} f K \<subseteq># comm_monoid_add.sum (+) {#} g K"
reachable_refl_iff: fixes u :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> u) = (u \<in> verts H)"
Sent_def: shows "Sent \<equiv> {x. \<exists>y z w. Form y \<and> Form z \<and> Form w \<and> Sent_axioms x y z w}"
I: fixes u :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "f (u 0) (u n) \<le> (\<Sum>i = 0..<n. f (u i) (u (Suc i))) * 2"
node_rreq_rrep_nsqn_fresh_any_step: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg (\<lambda>_. rreq_rrep_sn) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (msg_fresh \<sigma>) a)"
fst_proj_mem: fixes A :: "'a"   and M :: "('a \<times> 'b) set" shows "(A \<in> fst_proj M) = (\<exists>B. (A, B) \<in> M)"
a: fixes a :: "'a iarray" shows "IArray (IArray.list_of a) = a"
lleq_refl: fixes x :: "char list" shows "lleq_charlist x x"
Top_intv_not_empty: fixes x :: "'a" assumes "x \<in> A" shows "interval r x (Top cl) \<noteq> {}"
disjoint: shows "{0..<Suc (Suc k)} \<inter> {Suc (Suc k)..<n} = {}"
rotate2: fixes A2 :: "fm"   and A1 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A2 (insert A1 H) \<turnstile> B" shows "insert A1 (insert A2 H) \<turnstile> B"
no_nonce_YM1_YM2: fixes B' :: "agent"   and A' :: "agent"   and NB :: "nat"   and nb' :: "msg"   and evs :: "event list"   and B :: "agent"   and A :: "agent"   and na :: "msg" assumes "Crypt (shrK B') \<lbrace>Agent A', Nonce NB, nb'\<rbrace> \<in> parts (knows Spy evs)"   and "Nonce NB \<notin> analz (knows Spy evs)"   and "evs \<in> yahalom" shows "Crypt (shrK B) \<lbrace>Agent A, na, Nonce NB\<rbrace> \<notin> parts (knows Spy evs)"
sup_impr_inf: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<squnion> b) r\<rightarrow> c = (a r\<rightarrow> c) \<sqinter> (b r\<rightarrow> c)"
strict_subtuple_trans'': fixes zm :: "'d \<rightharpoonup>\<^sub>f 'c"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ym :: "'d \<rightharpoonup>\<^sub>f 'c"   and xm :: "'d \<rightharpoonup>\<^sub>f 'c" assumes "acyclicP_on (fmran' zm) R"   and "strict_subtuple R xm ym"   and "strict_subtuple R\<^sup>*\<^sup>* ym zm" shows "strict_subtuple R\<^sup>*\<^sup>* xm zm"
lens_plus_right_sublens: fixes Y :: "'d \<Longrightarrow> 'e"   and Z :: "'f \<Longrightarrow> 'e"   and X :: "'g \<Longrightarrow> 'e" assumes "vwb_lens Y"   and "Y \<bowtie> Z"   and "X \<subseteq>\<^sub>L Z" shows "X \<subseteq>\<^sub>L Y +\<^sub>L Z"
IH2: assumes "wf_net_tree p2" shows "{(\<sigma>, snd (netgmap sr s)) |\<sigma> s. s \<in> init (pnet np p2) \<and> (\<forall>i. if i \<in> net_tree_ips p2 then \<sigma> i = the (fst (netgmap sr s) i) else \<sigma> i \<in> (fst \<circ> sr) ` init (np i))} \<subseteq> init (opnet onp p2)"
nth_strict_mono_on: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n"   and "strict_mono_on f {0..<n}"   and "i < n" shows "f i = sorted_list_of_set (f ` {0..<n}) ! i"
preservation_of_barbs_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_preserves_binary_pred (Rel\<^sup>*) (\<lambda>a. hasBarb a CWB)"
swap_cols_rows_block_index: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and low :: "nat"   and high :: "nat" assumes "i < dim_row A"   and "i < dim_col A"   and "j < dim_row A"   and "j < dim_col A"   and "low \<le> high"   and "high < dim_row A"   and "high < dim_col A" shows "??.Jordan_Normal_Form_Existence.swap_cols_rows_block low high A $$ (i, j) = A $$ (if i = low then high else if low < i \<and> i \<le> high then i - 1 else i, if j = low then high else if low < j \<and> j \<le> high then j - 1 else j)"
adjoint_dim: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "adjoint A \<in> carrier_mat n n"
ps0_sub: shows "set ps0 \<subseteq> set bs0 \<times> (set gs \<union> set bs0)"
int_pow_mult_distrib: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and i :: "int" assumes "Group.group G"   and "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> y) [^]\<^bsub>G\<^esub> i = x [^]\<^bsub>G\<^esub> i \<otimes>\<^bsub>G\<^esub> y [^]\<^bsub>G\<^esub> i"
setdist_eq_0I: fixes x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "x \<in> S"   and "x \<in> T" shows "setdist S T = 0"
simps_8: fixes f :: "'c \<Rightarrow> 'b" shows "map f [] = []"
IH: fixes i :: "nat"   and j :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (0::'a) \<le> len m i i xs"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "\<exists>xs. set xs \<subseteq> {0..k} \<and> fw m n k i j = len m i j xs"
ex_extension: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and H1 :: "'a set"   and h :: "'a"   and g :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "H1 \<subseteq> H"   and "h \<in> H - H1"   and "(H1, g) \<in> fsps R M N f H" shows "\<exists>k. (H1 \<union> {h}, k) \<in> fsps R M N f H"
urel64_def: fixes x :: "uint64"   and y :: "int" shows "urel64 x y = (y = int_of_uint64 x \<and> y < p)"
equals0D: fixes A :: "'f set"   and a :: "'f" assumes "A = {}" shows "a \<notin> A"
access_new: fixes l :: "Location"   and s :: "Store"   and t :: "New" assumes "ref l = new s t"   and "isNewArr t \<longrightarrow> l \<noteq> arr_len (new s t)" shows "s\<langle>t\<rangle>@@l = init (ltype l)"
t_twice_append: fixes ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)] @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp = (Suc t_twice_len + length t_wcode_main_first_part div 2, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (2 * rs) @ Bk \<up> rn)"
ereal_eq_minus: fixes y :: "ereal"   and x :: "ereal"   and z :: "ereal" assumes "\<bar>y\<bar> \<noteq> \<infinity>" shows "(x = z - y) = (x + y = z)"
intro: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool" assumes "personal_quorums quorum_of" shows "with_w quorum_of"
Seq_decomp_relpowp_Stuck: fixes n :: "nat"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma>\<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>nn (Skip, Stuck)" shows "(\<exists>n1. \<Gamma>\<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Skip, Stuck)) \<or> (\<exists>t n1 n2. \<Gamma>\<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Skip, Normal t) \<and> n1 < n \<and> n2 < n \<and> \<Gamma>\<turnstile> (p\<^sub>2, Normal t) \<rightarrow>\<^sup>nn2 (Skip, Stuck))"
Next_def: fixes s :: "state"   and s' :: "state" shows "Next s s' = (\<exists>p. StartBallot s s' p \<or> (\<exists>d. Phase0Read s s' p d \<or> Phase1or2Write s s' p d \<or> (\<exists>q. q \<noteq> p \<and> Phase1or2Read s s' p d q)) \<or> EndPhase1or2 s s' p \<or> Fail s s' p \<or> EndPhase0 s s' p)"
psubst_eq_rawpsubst2: fixes \<phi> :: "'fmla"   and x1 :: "'var"   and x2 :: "'var"   and t1 :: "'trm"   and t2 :: "'trm" assumes "\<phi> \<in> fmla"   and "x1 \<in> var"   and "x2 \<in> var"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x1 \<noteq> x2"   and "x2 \<notin> FvarsT t1" shows "psubst \<phi> [(t1, x1), (t2, x2)] = rawpsubst \<phi> [(t1, x1), (t2, x2)]"