hcju/isabelleps huggingface dataset:
sq_norm_vec_eq_0: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(\<parallel>v\<parallel>\<^sup>2 = (0::'a)) = (v = 0\<^sub>v n)"
reduced_calc_is_calc: shows "calculus Bot Inf (\<Turnstile>) reduc_calc.Red_Red_I Red_F"
qSwapAll_igSwapAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "igSwapClsSTR MOD" shows "qInit MOD (qX #[[z1 \<and> z2]]_zs) = igSwap MOD zs z1 z2 (qInit MOD qX) \<and> qInitAbs MOD (qA $[[z1 \<and> z2]]_zs) = igSwapAbs MOD zs z1 z2 (qInitAbs MOD qA)"
append_subset_1: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "set xs = set (ys @ zs)" shows "set ys \<subseteq> set xs"
aform_err_lemma2: fixes err :: "real"   and x :: "real"   and X1 :: "real"   and A :: "real"   and e :: "'a \<Rightarrow> real"   and d :: "'a"   and B :: "real"   and e1 :: "real"   and p :: "nat" assumes "- 1 \<le> err"   and "err \<le> 1"   and "x \<le> X1 + (A - e d * B + err * B) + e1" shows "x \<le> X1 + (A - e d * B) + truncate_up p (\<bar>B\<bar> + e1)"
imp: fixes s :: "complex" assumes "a < Re s" shows "(I s \<longlongrightarrow> 0) at_top"
init_nsc_def: fixes nsc :: "('i \<times> 'a ns_constraint) list" shows "init_nsc nsc = (case preprocess nsc of (t, as, trans_v, ui) \<Rightarrow> ((create_map as, trans_v, remdups ui), init_s t))"
SeqDynCom: fixes r :: "'a set"   and a :: "('a, 'b, 'c) ann"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set"   and P :: "'a set"   and c :: "'a \<Rightarrow> ('a, 'b, 'c) com"   and Q :: "'a set"   and A :: "'a set" assumes "r \<subseteq> pre a"   and "\<forall>s\<in>r. \<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P a (c s) Q, A"   and "P \<subseteq> r" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnRec r a) (DynCom c) Q, A"
eval_app_inv: fixes \<rho> :: "(nat \<times> bval) list"   and e1 :: "exp"   and e2 :: "exp"   and v :: "bval" assumes "\<rho> \<turnstile> EApp e1 e2 \<Down> v" obtains x :: "nat" and e :: "exp" and \<rho>' :: "(nat \<times> bval) list" and arg :: "bval" where "\<rho> \<turnstile> e1 \<Down> BClos x e \<rho>'" and "\<rho> \<turnstile> e2 \<Down> arg" and "(x, arg) # \<rho>' \<turnstile> e \<Down> v"
worst_def: shows "local.worst = {x \<in> local.\<P>. \<forall>y\<in>local.\<P>. y \<succeq>[\<R>] x}"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "('test, 'atom) com \<times> 'state" assumes "a1 \<rightarrow>c a2" obtains c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1__ ;; c2__, sa__)" and "a2 = (c1'__ ;; c2__, s'__)" and "(c1__, sa__) \<rightarrow>c (c1'__, s'__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1a__ ;; c2a__, sb__)" and "a2 = (c2a__, s'a__)" and "(c1a__, sb__) \<rightarrow>t s'a__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tst__ then c1b__ else c2b__, sc__)" and "a2 = (c1b__, sc__)" and "tval tst__ sc__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tsta__ then c1c__ else c2c__, sd__)" and "a2 = (c2c__, sd__)" and "\<not> tval tsta__ sd__"   | tst :: "'test" and s :: "'state" and c :: "('test, 'atom) com" where "a1 = (while tst do c, se__)" and "a2 = (c ;; while tst do c, se__)" and "tval tst se__"   | c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1d__ | c2d__, sf__)" and "a2 = (c1' | c2d__, s'b__)" and "(c1d__, sf__) \<rightarrow>c (c1', s'b__)"   | c2 :: "('test, 'atom) com" and s :: "'state" and c2' :: "('test, 'atom) com" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1e__ | c2e__, sg__)" and "a2 = (c1e__ | c2', s'c__)" and "(c2e__, sg__) \<rightarrow>c (c2', s'c__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1f__ | c2f__, sh__)" and "a2 = (c2f__, s'd__)" and "(c1f__, sh__) \<rightarrow>t s'd__"   | c2 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1 | c2, s)" and "a2 = (c1, s')" and "(c2, s) \<rightarrow>t s'"
nat_add_1_add_1: fixes n :: "nat" shows "n + 1 + 1 = n + 2"
update_append_none: fixes env :: "('f, 'g, 'h) env"   and xs :: "'h list"   and y :: "'h"   and ys :: "'h list"   and opt :: "('f, 'g, 'h) env option" assumes "lookup env xs = None" shows "update (xs @ y # ys) opt env = env"
eventually_False: fixes F :: "'c filter" shows "(\<forall>\<^sub>F x in F. False) = (F = bot)"
M: fixes z :: "'a" assumes "M \<le> norm z" shows "norm (\<Sum>i\<le>n. c i * z ^ i) \<le> e * norm z ^ Suc n"
assms_1: shows "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"
iso_is_section: fixes f :: "'a" assumes "S.iso f" shows "S.section f"
errMOD_igWlsAbsIsInBar: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igWlsAbsIsInBar Delta MOD" shows "FixSyn.igWlsAbsIsInBar Delta (FixSyn.errMOD Delta MOD)"
rel_cset_iff: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c cset"   and b :: "'d cset" shows "rel_cset R a b = ((\<forall>t. cin t a \<longrightarrow> (\<exists>u. cin u b \<and> R t u)) \<and> (\<forall>t. cin t b \<longrightarrow> (\<exists>u. cin u a \<and> R u t)))"
h: fixes x :: "'a" assumes "x \<in> sphere (0::'a) 1 \<inter> S" shows "h x \<in> sphere (0::'a) 1 \<inter> T"
HSem_reorder: fixes \<Gamma> :: "('b \<times> 'a) list"   and \<Delta> :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "map_of \<Gamma> = map_of \<Delta>" shows "\<lbrace> \<Gamma> \<rbrace>\<rho> = \<lbrace> \<Delta> \<rbrace>\<rho>"
lens_comp_left_id: fixes X :: "'b \<Longrightarrow> 'c" shows "1\<^sub>L ;\<^sub>L X = X"
alpha'': fixes x :: "name"   and t :: "trm"   and y :: "name" assumes "x \<sharp> t" shows "[y].t = [x].([(y, x)] \<bullet> t)"
remove_child_heap_is_wellformed_preserved_3: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h'"
proots_cball_card_def: fixes p :: "complex poly"   and z0 :: "complex"   and r :: "real" shows "proots_cball_card p z0 r = card (proots_within p (cball z0 r))"
prems_2: shows "\<Gamma>\<^sub>1 \<subseteq> \<Gamma>\<^sub>2"
continuous_on_TimesI: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "local_lipschitz T X f"   and "\<And>x. x \<in> X \<Longrightarrow> continuous_on T (\<lambda>t. f t x)" shows "continuous_on (T \<times> X) (\<lambda>(t, x). f t x)"
\<i>_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (local.map \<i>\<^sub>C[a] \<cdot>\<^sub>D cmp (a, a)) = F.map\<^sub>0 a"
when_True: fixes X :: "'d \<Rightarrow> (unit \<times> 'd) \<times> bool" shows "when1 True X = X"
comm_neutral: fixes a :: "'a" shows "a \<sqinter> \<top> = a"
IH: fixes s :: "('a, 'b) fmap"   and vs :: "'a set"   and s' :: "('a, 'b) fmap" assumes "sat_precond_as s as"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "sat_precond_as s' (as_proj as vs)"
chain_homotopic_simplicial_subdivision2: fixes p :: "nat"   and q :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p (standard_simplex q) c" shows "simplicial_chain (Suc p) (standard_simplex q) (subd p c)"
1: shows "\<forall>j\<in>{0..<i}. \<phi> (map (sdrop (Suc j)) \<pi>l)"
fmdom'_fmap_of_list: fixes xs :: "('a \<times> 'b) list" shows "fmdom' (fmap_of_list xs) = set (map fst xs)"
init: fixes \<sigma> :: "'a"   and p :: "'b" assumes "(\<sigma>, p) \<in> init A" shows "P (\<sigma>, p)"
ads_d_def: fixes x :: "('b \<times> 'b) set" shows "rel_antirange_kleene_algebra.ars_r x = rel_ar (rel_ar x)"
separatedNC': fixes p :: "('b set set, 'c) Combinators list" assumes "OnlyTwoNets p"   and "NetsCollected2 p"   and "NetsCollected p"   and "noDenyAll1 p"   and "allNetsDistinct p" shows "separated p"
matches_ex_entry_def: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and xcp :: "nat \<times> nat \<times> String.literal option \<times> nat \<times> nat" shows "matches_ex_entry P C pc xcp \<equiv> let (s, e, C', h, d) = xcp in s \<le> pc \<and> pc < e \<and> (case C' of None \<Rightarrow> True | \<lfloor>C''\<rfloor> \<Rightarrow> P \<turnstile> C \<preceq>\<^sup>* C'')"
Inv_preserves_Can_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Dom C (monoidal_language.Inv C t) = monoidal_language.Cod C t"
induct: fixes x1 :: "trm"   and x2 :: "name"   and P :: "trm \<Rightarrow> name \<Rightarrow> bool" assumes "fin x1 x2"   and "\<And>x a. P (Ax x a) x"   and "\<And>x M a. x \<sharp> M \<Longrightarrow> P (NotL <a>.M x) x"   and "\<And>y x M. y \<sharp> [x].M \<Longrightarrow> P (AndL1 x.M y) y"   and "\<And>y x M. y \<sharp> [x].M \<Longrightarrow> P (AndL2 x.M y) y"   and "\<And>z x M y N. \<lbrakk>z \<sharp> [x].M; z \<sharp> [y].N\<rbrakk> \<Longrightarrow> P (OrL x.M y.N z) z"   and "\<And>y M x N a. \<lbrakk>y \<sharp> M; y \<sharp> [x].N\<rbrakk> \<Longrightarrow> P (ImpL <a>.M x.N y) y" shows "P x1 x2"
base_a_2: shows "s = (\<lambda>t. \<Sum>i\<in>S. a i * \<chi> (A i) t)"
vec_index_vCons_Suc: fixes a :: "'b"   and v :: "'b vec"   and n :: "nat" shows "vCons a v $ Suc n = v $ n"
Sigma_mset_Diff_distrib1: fixes I :: "'a multiset"   and J :: "'a multiset"   and C :: "'a \<Rightarrow> 'b multiset" shows "Sigma_mset (I - J) C = Sigma_mset I C - Sigma_mset J C"
the_Transaction\<^sub>m_def: fixes transaction\<^sub>m :: "transaction\<^sub>m" shows "the_Transaction\<^sub>m transaction\<^sub>m = (case transaction\<^sub>m of Transaction\<^sub>m x \<Rightarrow> x)"
snth_sconst: fixes s :: "'a stream"   and h :: "'a" shows "(\<forall>i. s !! i = h) = (s = sconst h)"
cyclic_on_obtain_list_succ': fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "cyclic_on f S"   and "f permutes S" obtains xs :: "'b list" where "f = list_succ xs" and "set xs = S" and "distinct xs"
decomps\<^sub>e\<^sub>s\<^sub>t_append: fixes A :: "('fun, 'var) extstrand_step list"   and S :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and B :: "('fun, 'var) extstrand_step list" assumes "A \<in> decomps\<^sub>e\<^sub>s\<^sub>t S N \<I>"   and "B \<in> decomps\<^sub>e\<^sub>s\<^sub>t S N \<I>" shows "A @ B \<in> decomps\<^sub>e\<^sub>s\<^sub>t S N \<I>"
DomainI: fixes a :: "'e"   and b :: "'f"   and r :: "('e \<times> 'f) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
less_not_sym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less"   and "less x y" shows "\<not> less y x"
spr_ii_sim_def: shows "spr_ii_sim \<equiv> \<lambda>t. (tObsC_ii_abs t, tLast t)"
closed_neg_Im_slit: shows "closed {z. Re z = 0 \<and> Im z \<le> 0}"
iT_not_empty_4: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m ] \<noteq> {}"
proots_upper_cindex_eq': fixes p :: "complex poly" assumes "lead_coeff p = 1" shows "real (proots_upper p) = real_of_int (int (degree p - proots_count p {x. Im x = 0}) - cindex_poly_ubd (map_poly Im p) (map_poly Re p)) / 2"
assms_4: shows "card {q. prime q \<and> q < p'} = n"
my_trancl_step_induct3: fixes ax :: "'a"   and ay :: "'b"   and az :: "'c"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and r :: "((('a \<times> 'b) \<times> 'c) \<times> ('a \<times> 'b) \<times> 'c) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "(((ax, ay), az), (bx, by), bz) \<in> r\<^sup>+"   and "\<And>a b ba aa bb bc. (((a, b), ba), (aa, bb), bc) \<in> r \<Longrightarrow> P a b ba aa bb bc"   and "\<And>a b ba aa bb bc ab bd be. \<lbrakk>P a b ba aa bb bc; (((aa, bb), bc), (ab, bd), be) \<in> r\<rbrakk> \<Longrightarrow> P a b ba ab bd be" shows "P ax ay az bx by bz"
individually_rational_on_Cd: fixes ds :: "'d set"   and X :: "'x set"   and d :: "'d" assumes "individually_rational_on ds X" shows "Cd d X = dX X d"
zero_aux: shows "fds_nth f 0 = (0::'a)"
cipher_queries_def: fixes ose :: "'b \<Rightarrow> ('c \<times> 'd \<times> 'a list) option" shows "ind_cca2.cipher_queries ose = (\<Union>(uu_, uu_, ciphers)\<in>ran ose. set ciphers)"
trancl_reflcl: fixes r :: "('c \<times> 'c) set" shows "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
Comm1: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto> a<b> \<prec> P'"   and "Q \<longmapsto> a[b] \<prec> Q'" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> P' \<parallel> Q'"
sorted_wrt_quicksort_by_rel: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "\<And>x y. R x y \<or> R y x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" shows "sorted_wrt R (quicksort_by_rel R [] xs)"
positions_border: fixes l :: "nat"   and w :: "'a list" assumes "\<forall>j<l. w ! j = w ! (length w - l + j)" shows "border (take l w) w"
generated_topology_is_topology: fixes S :: "'b set"   and B :: "'b set set" shows "topological_space S (generated_topology S B)"
that: fixes B :: "real" assumes "0 < B"   and "\<And>x. B * norm x \<le> norm (f x)" shows "thesis"
rep_eq: fixes x :: "'b word \<Rightarrow> 'c" shows "source_size x = LENGTH('b)"
finfun_upd_apply_same: fixes f :: "'d \<Rightarrow>f 'c"   and a :: "'d"   and b :: "'c" shows "f(a $:= b) $ a = b"
proper_it_mono_dres_flat: fixes it :: "'f \<Rightarrow> ('g \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and it' :: "'f \<Rightarrow> ('i dres \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'i dres \<Rightarrow> 'i dres) \<Rightarrow> 'i dres \<Rightarrow> 'i dres"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'i dres"   and f' :: "'h \<Rightarrow> 'i \<Rightarrow> 'i dres"   and s :: "'f"   and c :: "'i \<Rightarrow> bool"   and \<sigma> :: "'i dres" assumes "proper_it' it it'"   and "\<And>kv x. flat_ge (f kv x) (f' kv x)" shows "flat_ge (it' s (case_dres False False c) (\<lambda>kv s. s \<bind> f kv) \<sigma>) (it' s (case_dres False False c) (\<lambda>kv s. s \<bind> f' kv) \<sigma>)"
SeqSeq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and R :: "'e set"   and P\<^sub>2 :: "('e, 'd, 'f) ann"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and P\<^sub>1 :: "('e, 'd, 'f) ann"   and c\<^sub>1 :: "('e, 'd, 'f) com" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> R P\<^sub>2 c\<^sub>2 Q, A"   and "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P P\<^sub>1 c\<^sub>1 R, A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnComp P\<^sub>1 P\<^sub>2) (Seq c\<^sub>1 c\<^sub>2) Q, A"
left_total_conversep: fixes A :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "left_total A\<inverse>\<inverse> = right_total A"
relfun_eq1: fixes s :: "'w"   and t :: "'w" assumes "s \<in> S"   and "add_order (s + t) = 1" shows "t = s"
prod_mOp_mem: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and X :: "'c \<Rightarrow> 'd" assumes "\<forall>j\<in>I. aGroup (A j)"   and "X \<in> carr_prodag I A" shows "prod_mOp I A X \<in> carr_prodag I A"
cong_add_lcancel_0_nat: fixes a :: "nat"   and x :: "nat"   and n :: "nat" shows "[a + x = a] (mod n) = [x = 0] (mod n)"
bounded_check_eqv_def: shows "bounded_check_eqv \<equiv> WS1S_Alt.bounded.check_eqv"
is_root_def: fixes x :: "'b"   and t :: "'b tree" shows "is_root x t = (case t of \<langle>\<rangle> \<Rightarrow> False | \<langle>l, a, r\<rangle> \<Rightarrow> x = a)"
rel_interior_eq_closure: fixes S :: "'a set" shows "(rel_interior S = closure S) = affine S"
8: shows "\<forall>\<^sub>F k in sequentially. \<forall>s. ((aa k / bb k) powr (1 / (1 + \<epsilon>)) + real s) powr (1 + \<epsilon>) \<le> aa (k + s) / bb (k + s)"
existential: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<alpha> :: "'a" assumes "[\<phi> \<alpha> in v]" shows "[\<^bold>\<exists>\<alpha>. \<phi> \<alpha> in v]"
pquvd: shows "(case bezout (B $ from_nat i $ from_nat i) (B $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> p, case bezout (B $ from_nat i $ from_nat i) (B $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> q, case bezout (B $ from_nat i $ from_nat i) (B $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> u, case bezout (B $ from_nat i $ from_nat i) (B $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> v, case bezout (B $ from_nat i $ from_nat i) (B $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> d) = bezout (B $ from_nat i $ from_nat i) (B $ from_nat j $ from_nat j)"
deduct_hom_if_synth: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and m :: "('a, 'b) Term.term"   and \<A> :: "('d strand_label \<times> ('a, 'b) strand_step) list"   and Sec :: "('a, 'b) Term.term set"   and M :: "('a, 'b) Term.term set" assumes "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2"   and "\<not> labeled_typed_model.heterogeneous\<^sub>l\<^sub>s\<^sub>t arity public Ana \<Gamma> m \<A> Sec"   and "m \<in> labeled_typed_model.GSMP arity Ana \<Gamma> (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"   and "intruder_model.intruder_synth arity public M m" shows "labeled_typed_model.intruder_deduct_hom arity public Ana \<Gamma> M \<A> Sec m"
is_represented_in_state_def: fixes fst_snd :: "'d \<Rightarrow> ('e, 'f) state_scheme"   and x :: "'d \<Rightarrow> \<langle>'g\<rangle>\<^sub>\<bottom>"   and H :: "'e \<Rightarrow> 'g"   and \<tau> :: "'d" shows "is_represented_in_state fst_snd x H \<tau> = (x \<tau> \<in> (Some \<circ> H) ` ran (heap (fst_snd \<tau>)))"
gcd_dvd1: fixes a :: "'a"   and b :: "'a" shows "normalization_euclidean_semiring_class.gcd a b dvd a"
scan_nth: fixes i :: "nat"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b" assumes "i < length (scan f xs a)" shows "scan f xs a ! i = fold f (take (Suc i) xs) a"
lattice_of_def: fixes fs :: "rat vec list" shows "gs.lattice_of fs = range (\<lambda>c. gs.M.sumlist (map (\<lambda>i. rat_of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs]))"
pred_spmf_bind: fixes P :: "'c \<Rightarrow> bool"   and p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "pred_spmf P (p \<bind> f) = pred_spmf (pred_spmf P \<circ> f) p"
nat_of_ord_def: fixes x :: "hf" shows "nat_of_ord x = (THE n. x = ord_of n)"
special_coset_singleton: fixes S :: "'b set"   and w :: "'b" assumes "PreCoxeterSystem S"   and "w \<in> \<langle>S\<rangle>" shows "{w} \<in> PreCoxeterSystem.special_cosets S"
simps_7: fixes f1 :: "T1 \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x1 :: "T1" shows "rec_T2 f1 f2 (T2.C x1) = f1 x1"
bisimilarity_implies_equivalence_Act: fixes x :: "('a, 'pred, 'act) formula"   and P :: "'state"   and Q :: "'state"   and \<alpha> :: "'act" assumes "\<And>P Q. P \<sim>\<cdot> Q \<Longrightarrow> P \<Turnstile> x = Q \<Turnstile> x"   and "P \<sim>\<cdot> Q"   and "P \<Turnstile> Act \<alpha> x" shows "Q \<Turnstile> Act \<alpha> x"
\<phi>_simps_4: shows "local.dom r\<^sub>0s\<^sub>1.\<phi> = r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1"
simps: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and asi :: "'b"   and s :: "'c" shows "sum_wrap f (asi, s) = (case f asi s of Inl x \<Rightarrow> Inl x | Inr s' \<Rightarrow> Inr (asi, s'))"
oSuc_le_oSuc: fixes x :: "ordinal"   and y :: "ordinal" shows "(oSuc x \<le> oSuc y) = (x \<le> y)"
rel_flip: fixes R :: "'h \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g list"   and b :: "'h list" shows "list_all2 R\<inverse>\<inverse> a b = list_all2 R b a"
funcset_to_empty_iff: fixes A :: "'c set" shows "A \<rightarrow> {} = (if A = {} then UNIV else {})"
transpose_iarray_def: fixes A :: "'d iarray iarray" shows "transpose_iarray A = tabulate2 (ncols_iarray A) (nrows_iarray A) (\<lambda>a b. A !! b !! a)"
simps_2: fixes a :: "'a"   and ts :: "'a FOL_Fitting.term list" shows "paramst (App a ts) = {a} \<union> paramsts ts"
wf_mheadI: fixes sig :: "sig"   and m :: "mhead"   and G :: "prog"   and P :: "pname" assumes "length (parTs sig) = length (pars m)"   and "\<forall>T\<in>set (parTs sig). is_acc_type G P T"   and "is_acc_type G P (resTy m)"   and "distinct (pars m)" shows "wf_mhead G P sig m"
Borsuk_map_into_sphere: fixes a :: "'a"   and s :: "'a set" shows "((\<lambda>x. (x - a) /\<^sub>R norm (x - a)) ` s \<subseteq> sphere (0::'a) 1) = (a \<notin> s)"
islist_def: fixes h :: "'c \<Rightarrow> 'c ref"   and p :: "'c ref" shows "islist h p = (\<exists>as. List h p as)"
d2_4: fixes K :: "'a set" assumes "K \<in> p2" shows "\<exists>a b. K = cbox a b"
injD: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "inj f"   and "f x = f y" shows "x = y"
hom_mult: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<phi> (x \<otimes>\<^bsub>R\<^esub> y) = \<phi> x \<otimes>\<^bsub>S\<^esub> \<phi> y"
pure_split: fixes P :: "'a \<Rightarrow> bool" shows "pure P = (P = (\<lambda>s. True) \<or> P = (\<lambda>s. False))"
control_dependence_def: fixes n :: "'node"   and n' :: "'node" shows "n controls n' \<equiv> \<exists>a a' as. n -a # as\<rightarrow>\<^sub>\<iota>* n' \<and> n' \<notin> set (sourcenodes (a # as)) \<and> intra_kind (kind a) \<and> n' postdominates targetnode a \<and> valid_edge a' \<and> intra_kind (kind a') \<and> sourcenode a' = n \<and> \<not> n' postdominates targetnode a'"
corec_prod_apply: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and s :: "'e" shows "corec_prod f g s = (f s, g s)"
ih2: assumes "\<Gamma> \<turnstile> e\<^sub>2 : T'" shows "\<Gamma> \<turnstile> e\<^sub>2' : T'"
kde_def: shows "kde \<equiv> min k (min d e) / 2"
rtrancl_into_rtrancl: fixes a :: "'x"   and b :: "'x"   and r :: "('x \<times> 'x) set"   and c :: "'x" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
simps_5: fixes e\<^sub>1 :: "expr"   and bop :: "bop"   and e\<^sub>2 :: "expr"   and A :: "char list set option" shows "\<D> (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
tickFree_Cons: fixes a :: "'b event"   and t :: "'b event list" shows "tickFree (a # t) = (a \<noteq> tick \<and> tickFree t)"
induced_map_meas: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.induced_map T A \<in> M \<rightarrow>\<^sub>M M"
divisors_zero: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_no_zero_divisors plus zero times"   and "times a b = zero" shows "a = zero \<or> b = zero"
M_Rel_def: fixes x :: "int"   and x' :: "'a mod_ring" shows "M_Rel x x' \<equiv> M x = to_int_mod_ring x'"
Y: fixes U :: "'b set" assumes "U \<subseteq> topspace Y" shows "closedin X {x \<in> topspace X. f x \<in> U} = closedin Y U"