hcju/isabelleps huggingface dataset:
snapshot_state_unchanged_trace_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and j :: "nat"   and p :: "'d"   and u :: "'e" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "i \<le> j"   and "j \<le> length t"   and "ps (distributed_system.s channel trans send recv init t i) p = Some u" shows "ps (distributed_system.s channel trans send recv init t j) p = Some u"
smult_prod: fixes a :: "'a"   and b :: "'a poly" shows "Polynomial.smult a b = Polynomial.monom a 0 * b"
prems: shows "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel (a # S))"
Sup_fold_sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A" shows "Sup A = Finite_Set.fold sup bot A"
ffUnion_subset_elem: fixes x :: "'a fset"   and X :: "'a fset fset" assumes "x |\<in>| X" shows "x |\<subseteq>| ffUnion X"
kbo_code: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "kbo s t = (let wt = weight t; ws = weight s in if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> wt \<le> ws then if wt < ws then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> True | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> let ff = (f, length ss); gg = (g, length ts) in if pr_strict ff gg then (True, True) else if pr_weak ff gg then lex_ext_unbounded kbo ss ts else (False, False) else (False, False))"
assms_2: shows "I.mpt_factor proj1 M2 T2"
p_3: shows "insertion (\<lambda>i. exp (\<alpha> i)) p = 0"
admits_SNF_JNF_def: fixes A :: "'b mat" shows "admits_SNF_JNF A = (square_mat A \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> Q \<in> carrier_mat (dim_row A) (dim_row A) \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q)))"
f_dim: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "dim_vec (map of_int_hom.vec_hom fs ! i) = n"
peek_and_subst_res: fixes P :: "'b assn"   and p :: "state \<Rightarrow> bool"   and w :: "vals" shows "(P \<and>. p)\<leftarrow>w = (P\<leftarrow>w \<and>. p)"
diff_intersect_right_idem: fixes M :: "'a multiset"   and N :: "'a multiset" shows "M - N \<inter># M = M - N"
isomorphic_def: fixes a :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" shows "VoV.isomorphic a a' = (\<exists>f. VoV.in_hom f a a' \<and> VoV.iso f)"
Rep_monom_inject: fixes x :: "'c monom"   and y :: "'c monom" shows "(Rep_monom x = Rep_monom y) = (x = y)"
eq3: shows "spair g1 h = monom_mult ((1::'b) / punit.lc (rep_list g1)) a g1 - monom_mult ((1::'b) / punit.lc (rep_list h)) b h"
z_lemma: fixes sign :: "rat list"   and p :: "real poly"   and qs :: "real poly list"   and I :: "nat list" assumes "sign \<in> set (characterize_consistent_signs_at_roots_copr p qs)"   and "list_constr I (length qs)" shows "z I sign = 1 \<or> z I sign = - 1"
cQuiet: fixes P :: "('a, 'b, 'c) psi" assumes "quiet P" shows "(\<Psi>, \<zero>, P) \<in> Rel"
simps_9: fixes n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" shows "lang n (rexp.Inter r s) = lang n r \<inter> lang n s"
\<Delta>_in_hom_1: shows "\<guillemotleft>(\<mu> \<star> r\<^sub>0) \<cdot> \<rho> : src \<rho> \<rightarrow> trg \<sigma>\<guillemotright>"
zin_def: fixes x :: "'b"   and A :: "'b zet" shows "zin x A \<equiv> x \<in> Rep_zet A"
asian_option_def: fixes T :: "nat" shows "asian_option T = (\<lambda>w. (\<Sum>i = 1..T. prices Mkt stk i w) / real T)"
GmultD': fixes g :: "'g"   and v :: "'w" shows "g *\<star> v = ((1::'f) \<delta>\<delta> g) \<star> v"
fcard_fminus1_less: fixes x :: "'c"   and A :: "'c fset" assumes "x |\<in>| A" shows "fcard (A |-| {|x|}) < fcard A"
fact_prod: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero n = semiring_1.of_nat one plus zero (\<Prod>{1..n})"
Per_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Per A B C \<or> Per C B A" shows "Per A B C"
wordinterval_list_to_set_compressed: fixes xs :: "'a wordinterval list" shows "wordinterval_to_set (wordinterval_compress (foldr wordinterval_union xs Empty_WordInterval)) = wordinterval_list_to_set xs"
coprime_power_left_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one (power.power one times a n) b = (algebraic_semidom.coprime times one a b \<or> n = 0)"
sff': shows "p = Polynomial.smult c (\<Prod>(a, i)\<leftarrow>qis. a ^ Suc i)"
mon': shows "monic (\<Prod>i\<leftarrow>[0..<n]. [:- cis (real i * 2 * pi / real n), 1:])"
simps_1: fixes x :: "'b"   and r :: "'b tree" shows "del_left \<langle>\<langle>\<rangle>, x, r\<rangle> = (x, r)"
add_leaves_bdd_all_eq: fixes P :: "'a \<Rightarrow> bool"   and tr :: "'a bdd"   and xs :: "'a list" shows "list_all P (add_leaves tr xs) = (bdd_all P tr \<and> list_all P xs)"
dom_map_add: fixes m :: "'a \<Rightarrow> 'b option"   and n :: "'a \<Rightarrow> 'b option" shows "dom (m ++ n) = dom n \<union> dom m"
WTrt_iffs_1: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'b J_prog"   and h :: "'a"   and E :: "String.literal \<Rightarrow> ty option"   and v :: "'b val"   and T :: "ty" shows "J_heap_base.WTrt typeof_addr P h E (Val v) T = (heap_base.typeof_h typeof_addr h v = \<lfloor>T\<rfloor>)"
N: fixes ij :: "'n \<times> 'n" shows "\<exists>n\<le>N. ij \<in> G ^^ n"
sets_lim_stream: fixes x :: "real \<times> 'a" shows "sets (K.lim_stream x) = sets (stream_space S)"
u_4: fixes i :: "nat"   and x :: "'a" shows "u i x < top"
bind_tell: fixes w :: "'w"   and m :: "'m"   and f :: "'a \<Rightarrow> 'm" shows "bind (tell w m) f = tell w (bind m f)"
mult_right_sub_dist_sup_left: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<odot> z \<preceq> (x \<oplus> y) \<odot> z"
simps: fixes X :: "'b set" shows "the_RES (RES X) = X"
loop_sem: fixes I :: "('b, 'c, 'a) interp"   and \<alpha> :: "('b, 'c, 'a) hp" shows "prog_sem I \<alpha>** = (prog_sem I \<alpha>)\<^sup>*"
map_\<Gamma>_replace_prefix: fixes f :: "'c set \<Rightarrow> 'c set"   and \<pi> :: "'c prefix"   and \<sigma> :: "'c trace" assumes "\<forall>x. f (f x) = f x"   and "prefix_of (pmap_\<Gamma> f \<pi>) \<sigma>" shows "map_\<Gamma> f (replace_prefix \<pi> \<sigma>) = map_\<Gamma> f \<sigma>"
ch2ch_fst: fixes Y :: "nat \<Rightarrow> 'a \<times> 'b" assumes "chain Y" shows "chain (\<lambda>i. fst (Y i))"
h2: shows "all_list_constr_R [([], []), ([0], []), ([], [0])] (Suc 0)"
IF1set_simps: fixes x :: "('d, 'd IF1, 'd IF2) F1" shows "IF1set (ctor1 x) = F1set1 x \<union> (\<Union> (IF1set ` F1set2 x) \<union> \<Union> (IF2set ` F1set3 x))"
assms_2: shows "(\<box>(\<lambda>s. Q s \<longrightarrow> Q' s)) \<sigma>"
ordermap_mono_le: fixes w :: "'a"   and x :: "'a"   and r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "(w, x) \<in> r \<or> w = x"   and "wf r"   and "trans r"   and "w \<in> A"   and "x \<in> A"   and "small A" shows "ordermap A r w \<le> ordermap A r x"
drop_yields_results_implies_nbound: fixes n :: "nat"   and x :: "'a list" shows "drop n x \<noteq> [] \<longrightarrow> n < length x"
primepow_odd_def: fixes q :: "nat" shows "primepow_odd q = (\<exists>p k. 1 \<le> k \<and> prime p \<and> q = p ^ (2 * k + 1))"
abs_summable_on_subset: fixes f :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and A :: "'d set" assumes "f abs_summable_on B"   and "A \<subseteq> B" shows "f abs_summable_on A"
L_2: shows "{i \<in> I. f i \<noteq> 0} \<union> {i \<in> I. g i \<noteq> 0} \<subseteq> I"
eventually_const_iff: fixes P :: "bool"   and F :: "'a filter" shows "(\<forall>\<^sub>F x in F. P) = (P \<or> F = bot)"
mojmir_accept_iff_rabin_accept_rank: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set"   and i :: "nat" assumes "mojmir_to_rabin \<Sigma> \<delta> q\<^sub>0 w F" shows "(finite (mojmir_def.fail \<Sigma> \<delta> q\<^sub>0 w F) \<and> finite (mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i) \<and> infinite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i)) = accepting_pair\<^sub>R (semi_mojmir_def.step \<Sigma> \<delta> q\<^sub>0) (semi_mojmir_def.initial q\<^sub>0) (mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<delta> q\<^sub>0 F \<union> mojmir_to_rabin_def.merge\<^sub>R \<delta> q\<^sub>0 F i, mojmir_to_rabin_def.succeed\<^sub>R \<delta> q\<^sub>0 F i) w"
rwof_rel_def: fixes init :: "'b nres"   and cond :: "'b \<Rightarrow> bool"   and step :: "'b \<Rightarrow> 'b nres" shows "rwof_rel init cond step \<equiv> {(s, s'). rwof init cond step s \<and> cond s \<and> inres (step s) s'}"
domintros_2: fixes uv :: "'a list"   and p :: "(real, 'a) vec" shows "build_dom (uv, [p])"
iterates_above_successor: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and y :: "'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "y \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"   and "a \<in> Field leq" shows "y = a \<or> y \<in> bourbaki_witt_fixpoint.iterates_above lub leq f (f a)"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : ts _> ts'"
finite_Rs: fixes init :: "'a set" assumes "finite init" shows "finite (R\<^sup>* `` init)"
d_weakly_future_consistent_def: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> ('c list \<times> 'c list) set" shows "d_weakly_future_consistent P I D R \<equiv> \<forall>u\<in>range D \<inter> (- I) `` range D. \<forall>xs ys. (xs, ys) \<in> R u \<longrightarrow> (xs \<in> traces P) = (ys \<in> traces P) \<and> next_dom_events P D u xs = next_dom_events P D u ys"
finite_set_mset_mset_set: fixes A :: "'b set" assumes "finite A" shows "set_mset (mset_set A) = A"
sel_17: fixes x71 :: "'a expr"   and x72 :: "call_kind"   and x73 :: "'a call_expr" shows "un_Call3 (Call x71 x72 x73) = x73"
ins_mset: fixes t :: "('c, 'd) BinomialTree"   and q :: "('c, 'd) BinomialTree list" assumes "tree_invar t"   and "queue_invar q" shows "queue_to_multiset (ins t q) = tree_to_multiset t + queue_to_multiset q"
simps_2: fixes f :: "('c \<Rightarrow> 'b) tree"   and x :: "'c tree" shows "left (f \<diamondop> x) = left f \<diamondop> left x"
power_BIT: fixes n :: "nat" shows "2 ^ Suc n - 1 = (2 ^ n - 1) BIT True"
fresh_1: fixes a :: "name"   and x1 :: "nat" shows "a \<sharp> TVar x1 = a \<sharp> x1"
HD: fixes h :: "'a"   and t :: "'a list" shows "hd (h # t) = h"
fls_lr_inverse_const_2: fixes b :: "'b"   and y :: "'b" shows "fls_right_inverse (fls_const b) y = fls_const y"
step_7: shows "\<alpha> = \<omega>_sum \<alpha>s"
coboundedI1: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a \<le> c" shows "inf a b \<le> c"
trg_simps_2: fixes F :: "('a E.term, 'a) arr" assumes "arr F" shows "Cod (trg F) = E.Trg (Dom F)"
da_good_approxE: fixes Env :: "env"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and T :: "tys"   and A :: "assigned" assumes "prg Env\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "Env\<turnstile>t\<Colon>T"   and "Env\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"   and "wf_prog (prg Env)" obtains "normal s1 \<Longrightarrow> nrm A \<subseteq> dom (locals (snd s1))" and "\<And>l. \<lbrakk>fst s1 = Some (Jump (Break l)); normal s0\<rbrakk> \<Longrightarrow> brk A l \<subseteq> dom (locals (snd s1))" and "\<lbrakk>fst s1 = Some (Jump Ret); normal s0\<rbrakk> \<Longrightarrow> Result \<in> dom (locals (snd s1))"
ground_fmla_aux_def: fixes A :: "fmla"   and S :: "atom set" shows "ground_fmla_aux A S \<equiv> supp A \<subseteq> S"
star_inductl_var_eq2: fixes y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" assumes "y = x O y" shows "y = x\<^sup>* O y"
intro: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "\<And>s i. i < PL.brn c \<longrightarrow> (s, PL.eff aval tval cval c s i) \<in> indis \<and> PL_Indis.discr aval tval cval indis (PL.cont aval tval cval c s i)" shows "PL_Indis.discr aval tval cval indis c"
verts_in_translation_finite_1: fixes X :: "'a allegorical_term" shows "finite (vertices (translation X))"
f: fixes s :: "'s" assumes "I s" shows "f (x, s) = z"
strict_partial_order_acyclic: fixes r :: "('a \<times> 'a) set" assumes "strict_partial_order r" shows "acyclic r"
pw_bind_nofail: fixes M :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres" shows "nofail (M \<bind> f) = (nofail M \<and> (\<forall>x. inres M x \<longrightarrow> nofail (f x)))"
double_dual_iso_one: shows "G''.double_dual_iso \<one>\<^bsub>Characters (Characters G)\<^esub> = principal_char (Characters (Characters (Characters G)))"
mahler_landau_graeffe_approximation: fixes g :: "int poly"   and f :: "int poly"   and k :: "nat"   and dd :: "nat"   and d :: "nat"   and kk :: "nat" assumes "g = graeffe_poly_impl f k"   and "dd = d ^ 2 ^ Suc k"   and "kk = 2 ^ Suc k" shows "\<lfloor>real d * mahler_measure f\<rfloor> \<le> mahler_landau_graeffe_approximation kk dd g"
gap_T: shows "\<forall>T\<ge>LEAST t. \<forall>T. gap i T \<le> gap i t. gap i T = gap i (LEAST t. \<forall>T. gap i T \<le> gap i t)"
ringE_2: fixes R :: "('e, 'f) ring_scheme" assumes "ring R" shows "Group.monoid R"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat \<Rightarrow> nat option"   and a3 :: "exp list"   and a4 :: "nat list" assumes "a1;a2 \<turnstile>+ a3 : a4" obtains CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" where "a1 = CTa__" and "a2 = \<Gamma>__" and "a3 = []" and "a4 = []"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and C0 :: "nat" and es :: "exp list" and Cs :: "nat list" where "a1 = CT" and "a2 = \<Gamma>" and "a3 = e0 # es" and "a4 = C0 # Cs" and "CT;\<Gamma> \<turnstile> e0 : C0" and "CT;\<Gamma> \<turnstile>+ es : Cs"
card_gt_0_iff: fixes A :: "'c set" shows "(0 < card A) = (A \<noteq> {} \<and> finite A)"
Lcm_in_lcm_closed_set_nat: fixes M :: "nat set" assumes "finite M"   and "M \<noteq> {}"   and "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> lcm m n \<in> M" shows "Lcm M \<in> M"
stat_override: shows "G \<turnstile> new overrides\<^sub>S old"
Min_singleton: fixes x :: "'b word" shows "signed.Min {x} = x"
netglobal_def: fixes P :: "(nat \<Rightarrow> state) \<Rightarrow> bool" shows "netglobal P \<equiv> \<lambda>s. P (fst (initmissing (netgmap (\<lambda>(p, q). (fst (Fun.id p), snd (Fun.id p), q)) s)))"
fps_2: shows "fps_of_ratfps fps = Abs_fps f"
weak_supplementation: fixes y :: "i"   and x :: "i" assumes "MM2"   and "PP y x" shows "\<exists>z. P z x \<and> \<not> O z y"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m t'"
S5Basic_8: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<phi> in v]"
subset_imp_lepoll: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A \<lesssim> B"
F_mndet: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<noteq> {}" shows "F (mndet A P) = (\<Union>x\<in>A. F (x \<rightarrow> P x))"
primroot_prim: fixes x :: "'b list" assumes "x \<noteq> \<epsilon>" shows "primitive (\<rho> x)"
ennreal_cong_mult: fixes x :: "real"   and y :: "ennreal"   and z :: "ennreal" assumes "x \<noteq> 0 \<Longrightarrow> y = z" shows "ennreal x * y = ennreal x * z"
adjacentI: fixes z :: "'b set"   and x :: "'b set"   and y :: "'b set" assumes "z \<lhd> x"   and "z \<lhd> y" shows "x \<sim> y"
pell_square_solution_nat_aux: fixes D :: "nat"   and x :: "nat"   and y :: "nat" assumes "is_square D"   and "0 < D"   and "x\<^sup>2 = 1 + D * y\<^sup>2" shows "(x, y) = (1, 0)"
natural_isomorphism_axioms: shows "natural_isomorphism A B G F \<sigma>'"
mod_factor_imp_mod_0: fixes x :: "nat"   and m :: "nat"   and k :: "nat"   and y :: "nat" assumes "x mod (m * k) = y * k mod (m * k)" shows "x mod k = 0"
trss_call: fixes u :: "'a"   and p :: "'b"   and v :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and c :: "'a list multiset"   and r :: "'a list" assumes "(u, Call p, v) \<in> edges fg"   and "mon fg p \<inter> mon_c fg c = {}" shows "((u # r, c), LCall p, entry fg p # v # r, c) \<in> trss fg"
getEnv_preserves_wls: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wlsEnv rho"   and "rho xs x = Some X" shows "wls (asSort xs) X"
offset_poly_eq_0_lemma: fixes c :: "'a"   and p :: "'a poly"   and a :: "'a" assumes "Polynomial.smult c p + pCons a p = 0" shows "p = 0"
distinct_32: fixes x17 :: "hash" shows "Hash x17 \<noteq> Unit"
extended_Gromov_product_at_def: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_product_at e x y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e (u n) (v n))) |u v. abs_Gromov_completion u = x \<and> abs_Gromov_completion v = y \<and> Gromov_completion_rel u u \<and> Gromov_completion_rel v v}"
horiz_verti_disjoint: shows "two_chain_vertical_boundary two_chain \<inter> two_chain_horizontal_boundary two_chain = {}"
exec_elim_cases_5: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow> t" obtains f :: "'c" where "s = Fault f" and "t = Fault f"   | sa :: "'b" and s' :: "('b, 'c) xstate" where "s = Normal sa__" and "\<Gamma>\<turnstile> \<langle>c1,Normal sa__\<rangle> \<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow> t"   | "s = Stuck" and "t = Stuck"   | sa :: "'b" where "s = Abrupt sa" and "t = Abrupt sa"
rA': shows "local.sub3 (dim_row A) (dim_row A) (1::'a, A) = (r', A')"
ideal_inv1_closed: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e" assumes "Ring R"   and "ideal R I"   and "x \<in> I" shows "-\<^sub>a\<^bsub>R\<^esub> x \<in> I"
enumFromToS_defined: fixes x :: "int\<^sub>\<bottom>"   and y :: "int\<^sub>\<bottom>" shows "enumFromToS\<cdot>x\<cdot>y \<noteq> \<bottom>"
prems: shows "\<forall>res. \<not> evaluate_prog False env s (top0 # tops) res"
order_iff_strict: fixes b :: "'t"   and a :: "'t" shows "(b \<preceq>\<^sub>t a) = (b \<prec>\<^sub>t a \<or> a = b)"
phull_subset_module: fixes B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "phull B \<subseteq> local.punit.pmdl B"
split_rflow_outgoing: fixes u :: "nat" shows "(\<Sum>v\<in>cf.E `` {u}. f' (u, v)) = (\<Sum>v\<in>E `` {u}. f' (u, v)) + (\<Sum>v\<in>E\<inverse> `` {u}. f' (u, v))"
le: shows "(x' \<le> x) = (int_of_integer x' \<le> int_of_integer x)"
induct: fixes P :: "('a, 'c) Term.term literal list \<Rightarrow> ('a, 'b) Term.term literal list \<Rightarrow> ('c \<Rightarrow> ('a, 'b) Term.term option) \<Rightarrow> bool"   and a0 :: "('a, 'c) Term.term literal list"   and a1 :: "('a, 'b) Term.term literal list"   and a2 :: "'c \<Rightarrow> ('a, 'b) Term.term option" assumes "\<And>Ks \<sigma>. P [] Ks \<sigma>"   and "\<And>L Ls Ks \<sigma>. (\<And>x x2. \<lbrakk>x \<in> set Ks; match_term_list [(atm_of L, atm_of x)] \<sigma> = Some x2\<rbrakk> \<Longrightarrow> P Ls (remove1 x Ks) x2) \<Longrightarrow> P (L # Ls) Ks \<sigma>" shows "P a0 a1 a2"
lincomb_conv_take_right: fixes rs :: "'a list"   and ms :: "'b list" shows "rs \<bullet>\<cdot> ms = rs \<bullet>\<cdot> take (length rs) ms"
A_def: shows "A = {n. n \<le> m \<and> 0 < cls_val_process Mkt pf n y}"
fold_pls'_closed: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "g ` A \<subseteq> S" shows "Finite_Set.fold (pls' \<circ> g) z A \<in> S"
elims_2: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "inv_begin1 x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "(l, r) = ([], Oc \<up> n)"
that: fixes ll :: "'a list"   and e :: "'a" assumes "l = ll @ [e]" shows "thesis"
im_sum_list_map: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'a"   and as :: "'c list" assumes "GroupHom G T"   and "order.greater_eq G (set (map f as))" shows "T (sum_list (map f as)) = (\<Sum>a\<leftarrow>as. T (f a))"
simps_2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "binrelchain P [x] = True"
methd_rec: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "class G C = Some c"   and "ws_prog G" shows "methd G C = (if C = Object then Map.empty else filter_tab (\<lambda>sig m. G \<turnstile> C inherits method sig m) (methd G (super c))) ++ table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
simplex_def: fixes n :: "int"   and S :: "'b set" shows "n simplex S \<equiv> \<exists>C. \<not> affine_dependent C \<and> int (card C) = n + 1 \<and> S = convex hull C"
finite_tree_edge_tree: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and vertices :: "'e fset"   and nodeOf :: "'e \<Rightarrow> ('b, 'd) graph_node"   and edges :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) set"   and vidx :: "'e \<Rightarrow> nat"   and inst :: "'e \<Rightarrow> 'c"   and v :: "'e"   and p :: "('b, 'a) antecedent"   and pth :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) list" assumes "Tasked_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions vertices nodeOf edges vidx inst" shows "tfinite (Tasked_Proof_Graph.tree freshenLC renameLCs subst anyP antecedent vertices nodeOf edges vidx inst v p pth) = tfinite (Tasked_Proof_Graph.edge_tree anyP antecedent nodeOf edges v p)"