hcju/isabelleps huggingface dataset:
jvm_one_step1: fixes \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma>'' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "P \<turnstile> \<sigma> -jvm\<rightarrow>\<^sub>1 \<sigma>'"   and "P \<turnstile> \<sigma>' -jvm\<rightarrow> \<sigma>''" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>''"
igAbsRenSTR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igAbsRenSTR MOD \<equiv> \<forall>xs y x X. igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
ext_inject: fixes tdir :: "'b"   and xdir :: "'b"   and ydir :: "'b"   and zdir :: "'b"   and more :: "'c"   and tdir' :: "'b"   and xdir' :: "'b"   and ydir' :: "'b"   and zdir' :: "'b"   and more' :: "'c" shows "(\<lparr>tdir = tdir, xdir = xdir, ydir = ydir, zdir = zdir, \<dots> = more\<rparr> = \<lparr>tdir = tdir', xdir = xdir', ydir = ydir', zdir = zdir', \<dots> = more'\<rparr>) = (tdir = tdir' \<and> xdir = xdir' \<and> ydir = ydir' \<and> zdir = zdir' \<and> more = more')"
unit_dvdE: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" obtains c :: "'b" where "a \<noteq> zero" and "b = times a c"
carrier_FactGroup: fixes G :: "('a, 'b) monoid_scheme"   and N :: "'a set" shows "carrier (G Mod N) = (#>\<^bsub>G\<^esub>) N ` carrier G"
l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf ShadowRootClass.known_ptrs Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_disconnected_nodes Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs"
gauss_jordan_integer_inverse: fixes gs :: "int vec list"   and fs :: "int vec list" assumes "set gs \<subseteq> carrier_vec n"   and "length gs = n"   and "set fs \<subseteq> carrier_vec n"   and "length fs = n"   and "gauss_jordan_integer_inverse n (of_int_hom.mat_hom (mat_of_cols n fs)) (of_int_hom.mat_hom (mat_of_cols n gs)) (1\<^sub>m n)" shows "\<exists>U. U \<in> carrier_mat n n \<and> mat_of_rows n gs = U * mat_of_rows n fs"
imply_head: fixes p :: "('c, 'd) fm"   and ps :: "('c, 'd) fm list" shows "p # ps \<turnstile> p"
take_strict: fixes xs :: "['b]" shows "take\<cdot>\<bottom>\<cdot>xs = \<bottom>"
finite_fold_lderiv: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and embed :: "'a \<Rightarrow> 'a list"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'b rexp" assumes "Pi_Derivatives.embed \<Sigma> project embed" shows "finite {fold (\<lambda>a r. \<guillemotleft>embed.lderiv lookup embed a r\<guillemotright>) w \<guillemotleft>s\<guillemotright> |w. True}"
transitive: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs"   and Rel'' :: "(ccs \<times> ccs) set" assumes "P \<leadsto><Rel> Q"   and "Q \<leadsto><Rel'> R"   and "Rel O Rel' \<subseteq> Rel''"   and "\<And>S T. (S, T) \<in> Rel \<Longrightarrow> S \<leadsto>\<^sup>^<Rel> T" shows "P \<leadsto><Rel''> R"
Nom_6: fixes left :: "(('b, 'a) fm list \<times> 'a) list"   and right :: "(('b, 'a) fm list \<times> 'a) list" assumes "(p # ps, a) # branch = left @ right" shows "A, Suc m \<turnstile> filter nonempty left @ right"
from_conjugate_space: fixes x :: "'d conjugate_space" shows "from_conjugate_space x \<in> UNIV"
simps_5: fixes Inv :: "'a"   and b :: "bexp"   and c :: "'a acom"   and P :: "'a"   and Inv' :: "'a"   and b' :: "bexp"   and c' :: "'a acom"   and P' :: "'a" shows "({Inv} WHILE b DO c {P} \<le> {Inv'} WHILE b' DO c' {P'}) = (b = b' \<and> c \<le> c' \<and> Inv \<le> Inv' \<and> P \<le> P')"
makeCArray_def: fixes arraySize :: "nat"   and fillValue :: "'b"   and index :: "nat" shows "makeCArray arraySize fillValue index \<equiv> if index < arraySize then fillValue else undefined"
valid_frac_def: fixes X :: "('a \<times> 'a) set" shows "local.valid_frac X \<equiv> \<exists>r\<in>R. \<exists>s\<in>R\<setminus>I. local.frac r s = X"
resource_of_oracle_run_resource: fixes res :: "('h, 'i) resource" shows "RES run_resource res = res"
sorted_list_of_set_eq_Nil_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A" shows "(linorder.sorted_list_of_set less_eq A = []) = (A = {})"
valid_decomp_def: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "valid_decomp X ps = (\<forall>(h, U)\<in>set ps. h \<in> P[X] \<and> h \<noteq> 0 \<and> U \<subseteq> X)"
comp_simp: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_B.seq (fst g) (fst f)"   and "A.seq (snd g) (snd f)" shows "g \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]\<^sub>x\<^sub>A f = (fst g \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] fst f, snd g \<cdot>\<^sub>A snd f)"
lexer_correct_None: fixes s :: "'b list"   and r :: "'b rexp" shows "(s \<notin> lang r) = (lexer r s = None)"
convex_halfspace_ge: fixes b :: "real"   and a :: "'b" shows "convex {x. b \<le> a \<bullet> x}"
Cod_memb: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "t \<in> f" shows "monoidal_language.Cod C t = monoidal_language.Cod C (free_monoidal_category.rep C f)"
top_closed: shows "\<^bold>\<top> \<in> carrier thy_order"
assms: shows "wf\<^sub>l\<^sub>s\<^sub>t X A"
greedy_path_deadend_v: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a"   and P :: "'a llist" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v P" shows "(P = LNil) = deadend v"
ode_def: fixes x :: "'c" shows "ode x = einterpret ode_e (list_of_eucl x)"
bw_diff_const_pow: fixes k :: "nat"   and c :: "'c" shows "(bw_diff ^^ k) (\<lambda>_. c) = (if k = 0 then \<lambda>_. c else (\<lambda>_. 0::'c))"
assms: shows "subst_domain \<theta> \<inter> range_vars \<theta> = {}"
wf_disc_sel_common_primitive_3: shows "wf_disc_sel (is_Src, src_sel) Src"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a"   and x :: "eint" shows "P (case x of eint x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = ((\<forall>int. x = eint int \<longrightarrow> P (f1 int)) \<and> (x = \<infinity> \<longrightarrow> P f2))"
iso_is_section: fixes f :: "'b" assumes "D.iso f" shows "D.section f"
right_imp_left: assumes "pvalid_left p x" shows "pvalid p x"
unique_2: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and V2 :: "'b mod_ring poly"   and W2 :: "'b mod_ring poly"   and ty_q :: "'c itself" assumes "Knuth_ex_4_6_2_22_prime a b u v w"   and "v = #V2"   and "w = #W2"   and "u = V2 * W2"   and "degree V2 = degree v"   and "degree W2 = degree w"   and "lead_coeff V2 = @lead_coeff v" shows "W2 = #w + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q ty_q)) (#Knuth_ex_4_6_2_22_main.w' ty_q a b u v w)"
cjb: fixes k :: "nat"   and ev :: "'a" assumes "k \<noteq> 0" shows "local.compute_nr_of_jordan_blocks A ev k = length (filter ((=) (k, ev)) n_as)"
pow_A_1_def: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and x :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A" shows "perron_frobenius.pow_A_1 A x = perron_frobenius.A1n A *v x"
proots_count_times: fixes p :: "'a poly"   and q :: "'a poly"   and s :: "'a set" assumes "p * q \<noteq> 0" shows "proots_count (p * q) s = proots_count p s + proots_count q s"
DN_def: fixes \<mu> :: "('a E.term, 'a) arr" shows "DN \<mu> \<equiv> if arr \<mu> then Map \<mu> else B.null"
genideal_zero: shows "Idl {\<zero>} = {\<zero>}"
InvDomCod_3: fixes f :: "'b" assumes "ciso f" shows "Cinv f \<in> mor"
size_list_estimation: fixes x :: "'d"   and xs :: "'d list"   and y :: "nat"   and f :: "'d \<Rightarrow> nat" assumes "x \<in> set xs"   and "y < f x" shows "y < size_list f xs"
that: fixes x :: "Vars.var"   and e :: "exp" assumes "\<Gamma> = [(x, e)]"   and "x \<notin> fv e" shows "thesis"
elims_2: fixes x :: "arctic"   and xa :: "arctic" assumes "x < xa" obtains xb :: "arctic" where "x = MinInfty" and "xa = xb__"   | y :: "int" and xb :: "int" where "x = Num_arc y" and "xa = Num_arc xb" and "y < xb"
p_preserves_inv_implications_nonneg: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t" assumes "next_propagate' c0 c1 loc t"   and "inv_implications_nonneg c0"   and "inv_imp_plus_work_nonneg c0" shows "inv_implications_nonneg c1"
narrow1_acom: fixes y :: "'a acom"   and x :: "'a acom" assumes "y \<sqsubseteq> x" shows "y \<sqsubseteq> x \<triangle>\<^sub>c y"
token_smallest_accepting_rank: fixes i :: "nat" assumes "smallest_accepting_rank = Some i" shows "\<forall>\<^sub>\<infinity>n. \<forall>x. token_succeeds x = (n < x \<or> (\<exists>j\<ge>i. rank x n = Some j) \<or> token_run x n \<in> F)"
zdiv_pos_mono_r: fixes w :: "int"   and z :: "int"   and z' :: "int" assumes "0 < w"   and "w * z \<le> w * z'" shows "z \<le> z'"
diff_zero: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
pochhammer_prod: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a n = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero i)) {0..<n}"
bdia_def: fixes x :: "'a"   and y :: "'a" shows " |x\<rangle> y = ad (ad (x \<cdot> y))"
when_when: fixes zero :: "'a"   and a :: "'a"   and P :: "bool"   and Q :: "bool" shows "zero.when zero (zero.when zero a P) Q = zero.when zero a (P \<and> Q)"
mtx_nonzero_iff: shows "mtx_nonzero c = E"
red1_preserves_len: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "length (lcl s') = length (lcl s)"
fatom_of_hatom_hatom_of_fatom: fixes l :: "char list \<times> fterm list" assumes "ground\<^sub>t\<^sub>s (snd l)" shows "fatom_of_hatom (hatom_of_fatom l) = l"
ide_in_hom_2: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>f : f \<rightarrow>\<^sub>C f\<guillemotright>"
w_simps_5: shows "local.dom \<pi>.chine = \<pi>.chine"
simps: fixes a1 :: "'b sequent list \<times> 'b sequent"   and a2 :: "'b form" shows "rightPrincipal a1 a2 = (\<exists>C F Fs Ps. a1 = (Ps, C) \<and> a2 = Compound F Fs \<and> C = ( \<Empt> \<Rightarrow>* \<LM> Compound F Fs  \<RM>))"
comp_in_homI: fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>C c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>C f : a \<Rightarrow>\<^sub>C c\<guillemotright>"
adj_mat_1: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "adj_mat A \<in> carrier_mat n n"
init_set2: fixes x :: "'node" assumes "x \<in> reach root"   and "x \<notin> path {root} {root}" shows "x = root"
bind_return_pmf': fixes N :: "'c pmf" shows "N \<bind> return_pmf = N"
Busemann_function_inside_approx: fixes e :: "real"   and t :: "nat \<Rightarrow> 'b"   and xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" assumes "0 < e"   and "(\<lambda>n. to_Gromov_completion (t n)) \<longlonglongrightarrow> xi" shows "\<forall>\<^sub>F n in sequentially. Busemann_function_at (to_Gromov_completion (t n)) x y \<le> Busemann_function_at xi x y + e \<and> Busemann_function_at xi x y - 2 * deltaG TYPE('b) - e \<le> Busemann_function_at (to_Gromov_completion (t n)) x y"
R_S: fixes y :: "'a" assumes "y \<in> S" shows "(\<mu> \<rho>'. (\<rho> ++\<^bsub>S\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>S\<^esub> e1\<cdot>\<rho>''))(x := e2\<cdot>\<rho>')) y = (e1\<cdot>(\<mu> \<rho>''. (\<mu> \<rho>'. (\<rho> ++\<^bsub>S\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>S\<^esub> e1\<cdot>\<rho>''))(x := e2\<cdot>\<rho>')) ++\<^bsub>S\<^esub> e1\<cdot>\<rho>'')) y"
transclp_eq_implies_confluent_imp: fixes r1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "r1\<^sup>*\<^sup>* = r2\<^sup>*\<^sup>*"   and "confluentp r1" shows "confluentp r2"
bisim1FAssNull: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and e2 :: "'addr expr1"   and xs :: "'addr val list"   and v :: "'addr val" shows "P,e\<bullet>F{D} := e2,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([v, Null], xs, length (compE2 e) + length (compE2 e2), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
ordinal_rec_0: fixes z :: "ordinal"   and s :: "ordinal \<Rightarrow> ordinal \<Rightarrow> ordinal" shows "ordinal_rec z s 0 = z"
differentiable_Pair: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'e" assumes "f differentiable at x within s"   and "g differentiable at x within s" shows "(\<lambda>x. (f x, g x)) differentiable at x within s"
shortestPath_single_predecessor: fixes g :: "'g"   and n :: "'node"   and m :: "'node" assumes "predecessors g n = [m]"   and "invar g" shows "shortestPath g m < shortestPath g n"
hom_minus: fixes x :: "'b poly"   and y :: "'b poly"   and a :: "'b" shows "poly (x - y) a = poly x a - poly y a"
sigma_finite_measure_count_space_finite: fixes A :: "'d set" assumes "finite A" shows "sigma_finite_measure (count_space A)"
mult_scalar_sum_monomials: fixes q :: "'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "q \<odot> p = (\<Sum>t\<in>keys q. monom_mult (lookup q t) t p)"
Abs_inject: fixes x :: "nat"   and y :: "nat" assumes "x \<in> {0..<n}"   and "y \<in> {0..<n}" shows "(abs x = abs y) = (x = y)"
6: shows "coreflexive (prim_P w v e\<^sup>T * e\<^sup>T)"
sel_1: fixes x1 :: "'d" shows "result (Done x1) = x1"
lm106: fixes Z :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "Z \<subseteq> X \<union> Y" shows "sum (X <| Y) Z = sum (chi X) Z"
leg_simps_1: shows "C.arr \<nu>'.cod.leg0"
cases: fixes a1 :: "('test, 'atom) com"   and a2 :: "'state" assumes "mayDiverge a1 a2" obtains c :: "('test, 'atom) com" and s :: "'state" and c' :: "('test, 'atom) com" and s' :: "'state" where "a1 = c" and "a2 = s" and "(c, s) \<rightarrow>c (c', s') \<and> mayDiverge c' s'"
not_zero_aA: shows "\<not> is_zero_row_upt_k a k A"
rebase_poly_pCons: fixes a :: "'b mod_ring"   and p :: "'b mod_ring poly" shows "#pCons a p = pCons (@a) (#p)"
ran_sorted_alt: fixes a :: "int \<Rightarrow> int"   and l :: "int"   and h :: "int" shows "ran_sorted a l h = (\<forall>i j. l \<le> i \<and> i < j \<and> j < h \<longrightarrow> a i \<le> a j)"
max_y_impl: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and j :: "nat" shows "max_y_impl a b x j = hlde_ops.max_y a b x j"
linord_of_list_def: fixes xs :: "'b list" shows "linord_of_list xs \<equiv> {(x, y). linord_of_listP x y xs}"
subset_Collect_iff: fixes B :: "'c set"   and A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "B \<subseteq> A" shows "(B \<subseteq> {x \<in> A. P x}) = (\<forall>x\<in>B. P x)"
helper: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y * z) = y * x * z"
typeof_addr_sys_xcp: fixes h :: "'heap"   and C :: "String.literal" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "typeof_addr h (addr_of_sys_xcpt C) = \<lfloor>Class_type C\<rfloor>"
A: shows "\<integral>\<^sup>+ x. ennreal (f x * indicat_real {g a..g b} x) \<partial>lborel = \<integral>\<^sup>+ x. ennreal (f (g x) * g' x * indicat_real {a..b} x) \<partial>lborel"
card_order_on: fixes A :: "'a set" shows "\<exists>r. card_order_on A r"
induct: fixes P :: "('c \<Rightarrow> nat \<Rightarrow> 'd) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> nat \<Rightarrow> 'd"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "'c"   and a4 :: "'e \<Rightarrow> bool"   and a5 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a6 :: "'e" assumes "\<And>get sz i l c f \<sigma>. (\<not> (i = 0 \<or> \<not> c \<sigma>) \<Longrightarrow> P get sz (i - 1) l c f (f (get l (sz - i)) \<sigma>)) \<Longrightarrow> P get sz i l c f \<sigma>" shows "P a0 a1 a2 a3 a4 a5 a6"
set: fixes ls :: "'a list"   and h :: "Heap.heap"   and new_ls :: "'a list" shows "Array.set (fst (Array.alloc ls h)) new_ls (snd (Array.alloc ls h)) = snd (Array.alloc new_ls h)"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "F.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. F.satC \<xi> c"
intros_2: fixes g :: "'g"   and ns :: "'node list"   and m :: "'node" assumes "EntryPath g ns"   and "shortestPath g m = Suc (shortestPath g (last ns))" shows "EntryPath g (ns @ [m])"
valid_action_valid_succ: fixes h :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" assumes "h \<in> PROB"   and "s \<in> valid_states PROB" shows "state_succ s h \<in> valid_states PROB"
counit_in_vhom: shows "\<guillemotleft>\<epsilon>' : f \<star> g' \<Rightarrow> src g'\<guillemotright>"
brec_initial_abs: fixes \<delta> :: "('a, 'b) ta_rule dlist"   and Qi :: "('a, unit) HashMap.hashmap" assumes "ls.invar \<delta>"   and "hs.invar Qi" shows "brec_\<alpha> (brec_initial Qi \<delta>) \<in> brw_initial (ls.\<alpha> \<delta>)"
assms_2: shows "\<forall>a\<in>set as. anormal a"
bdt_Some_var0_Zero: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat"   and x :: "bdt" assumes "bdt t var = Some x"   and "var (root t) = 0" shows "x = Zero \<and> t = Node Tip (root t) Tip"
S2: shows "nat_to_sch n = Rec_op (nat_to_sch (c_fst (c_snd n))) (nat_to_sch (c_snd (c_snd n)))"
poly_nate_def: fixes bs :: "'b list"   and p :: "poly" shows "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
assms_1: shows "\<exists>n'''<n' + n''. \<not> Option.is_none (devBC t n nid n''')"
eigenvalue_def: fixes A :: "'b mat"   and k :: "'b" shows "eigenvalue A k = (\<exists>v. eigenvector A v k)"
affected_empty_iff: fixes f :: "'a perm" shows "(affected f = {}) = (f = 1)"
left_neutral: fixes a :: "'a" shows "\<one> \<^bold>\<sqinter> a = a"
convex_hull_list_mono: fixes Ws :: "'a vec list"   and Vs :: "'a vec list" assumes "set Ws \<subseteq> carrier_vec n"   and "set Vs \<subseteq> set Ws" shows "convex_hull_list Vs \<subseteq> convex_hull_list Ws"
SN_onE: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "SN_on r A" obtains "\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)"
mem_affine_3_minus: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "z \<in> S" shows "x + v *\<^sub>R (y - z) \<in> S"
step: fixes v :: "'a"   and w :: "'a"   and P :: "'a llist" assumes "Q (LCons v (LCons w P))" shows "v \<rightarrow> w \<and> (Q (LCons w P) \<or> valid_path (LCons w P))"
prefixes_not_Nil: fixes xs :: "'b list" shows "prefixes xs \<noteq> []"
Times: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x :: "'a language"   and y :: "'a language"   and x' :: "'a language"   and y' :: "'a language" assumes "regular_cong R x y"   and "regular_cong R x' y'" shows "regular_cong R (Times x x') (Times y y')"
awalk_simps_2: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and e :: "'c"   and es :: "'c list"   and w :: "'b" assumes "wf_digraph G" shows "pre_digraph.awalk G u (e # es) w = (e \<in> arcs G \<and> u = tail G e \<and> pre_digraph.awalk G (head G e) es w)"
SkipRule: fixes p :: "'a set"   and q :: "'a set" assumes "p \<subseteq> q" shows "{p}   SKIP   {q}"
k_def_3: shows "\<not> (\<exists>k'>0. k' < k \<and> [[a\<^sub>1 b f k']])"
locally_compactE: fixes S :: "'c set" assumes "locally compact S" obtains u :: "'c \<Rightarrow> 'c set" and v :: "'c \<Rightarrow> 'c set" where "\<And>x. x \<in> S \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> S \<and> openin (top_of_set S) (u x) \<and> compact (v x)"
nonprimitive_dchar_axioms_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" shows "nonprimitive_dchar_axioms n \<chi> \<equiv> \<exists>d<n. dcharacter.induced_modulus n \<chi> d"
set_ext_strong_lift: fixes alts' :: "'a set"   and alts :: "'a set"   and set_ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "liftable_set_extension alts' alts set_ext"   and "total_preorder_on alts' R"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "A \<subseteq> alts'"   and "B \<subseteq> alts'"   and "A \<prec>[set_ext R] B" shows "A \<prec>[set_ext (lift_prefs alts' alts R)] B"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>F' f : F' a \<rightarrow>\<^sub>C F' b\<guillemotright>"
finite_mult_not_eq_zero_rightI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite {a. f a \<noteq> (0::'c)}" shows "finite {a. f a * g a \<noteq> (0::'c)}"
set_ConsD: fixes y :: "'x"   and x :: "'x"   and xs :: "'x list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
execn_to_execn_sequence_flatten: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>sequence Seq (flatten c),s\<rangle> =n\<Rightarrow> t"
min2_def: shows "min2 \<equiv> \<lambda>(k, p) (k', p'). if p \<le> p' then (k, p) else (k', p')"