hcju/isabelleps huggingface dataset:
get_deterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and l :: "'b"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Read (VE (Loc l))])" shows "substitution.revision_step subst r s s' = (l \<in> dom (\<sigma>;;\<tau>) \<and> s' = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (the ((\<sigma>;;\<tau>) l))])))"
simps: fixes a :: "lam" shows "SN a = (\<exists>t. a = t \<and> (\<forall>x.  t \<longrightarrow>\<^sub>\<beta> x \<longrightarrow> SN x))"
cases: fixes x :: "tm \<times> tm" obtains xa :: "name" and r :: "tm" and d :: "tm" and y :: "name" and z :: "name" where "atom xa \<sharp> (r, d, y, z)" and "atom y \<sharp> (r, d, z)" and "atom z \<sharp> (r, d)" and "x = (r, d)"
ord_isom_restricted_by_Tw: fixes E :: "('e, 'f) Order_scheme"   and D1 :: "'a set" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))"   and "D1 \<subseteq> carrier D" shows "ord_isom (Iod D D1) (Iod E ((Tw\<^bsub>D,E\<^esub>) ` D1)) (restrict (Tw\<^bsub>D,E\<^esub>) D1)"
nn_integral_set_ennreal: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and A :: "'b set" shows "\<integral>\<^sup>+x\<in>A. ennreal (f x)\<partial>M = \<integral>\<^sup>+ x. ennreal (f x * indicat_real A x) \<partial>M"
right_adjoint_simps_2: fixes f :: "'a" assumes "is_left_adjoint f" shows "src f\<^sup>* = trg f"
canonical_retraction_def: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and C :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>" shows "ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A C = (\<lambda>a. restrict1 (ChamberComplexWithApartmentSystem.the_apartment_iso (SOME A. A \<in> \<A> \<and> (SOME C. SimplicialComplex.maxsimp X C \<and> a \<in> C) \<in> A \<and> C \<in> A) A) (\<Union> X) a)"
mulex_on_multisetsD_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset" assumes "mulex_on P F M N" shows "M \<in> multisets F"
inst_ok_code1: fixes \<Theta> :: "exetheory"   and insts :: "('b \<times> typ) list" assumes "exe_wf_theory' \<Theta>" shows "list_all (exetyp_ok \<Theta>) (map snd insts) = list_all (typ_ok (translate_theory \<Theta>)) (map snd insts)"
circline_line: fixes A :: "real"   and BC :: "complex"   and cl :: "complex set"   and D :: "real"   and z1 :: "complex"   and z2 :: "complex" assumes "A = 0"   and "BC \<noteq> 0"   and "cl = circline A BC D"   and "z1 = - cor D * BC / (2 * BC * cnj BC)"   and "z2 = z1 + \<i> * sgn (if 0 < arg BC then - BC else BC)" shows "cl = line z1 z2"
multiplicative_dirichlet_prod_def: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "multiplicative_dirichlet_prod f g \<equiv> multiplicative_function f \<and> multiplicative_function g"
finite_measure_pair_measure: fixes M :: "'a measure"   and N :: "'b measure" assumes "finite_measure M"   and "finite_measure N" shows "finite_measure (N \<Otimes>\<^sub>M M)"
Entry_target: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "targetnode a = Entry" shows "False"
combine_r_distr: fixes Ks :: "'a list"   and Us :: "'a list"   and k :: "'a" assumes "set Ks \<subseteq> carrier R"   and "set Us \<subseteq> carrier R"   and "k \<in> carrier R" shows "k \<otimes> combine Ks Us = combine (map ((\<otimes>) k) Ks) Us"
type_definition_lifted_pair: shows "type_definition Rep_lifted_pair Abs_lifted_pair {pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)}"
lselect_llength: fixes w :: "'a llist"   and s :: "nat set" shows "llength (lselect s w) = esize {i \<in> s. enat i < llength w}"
neq2: shows "uint w \<noteq> 2 ^ LENGTH(31)"
Cantor_dec_Cons_iff: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<beta>s :: "V list" shows "sorted_wrt (\<lambda>x y. y < x) (\<alpha> # \<beta> # \<beta>s) = (\<beta> < \<alpha> \<and> sorted_wrt (\<lambda>x y. y < x) (\<beta> # \<beta>s))"
sel_1: fixes x1 :: "('b \<times> 'c \<times> 'c) set"   and x2 :: "'c set" shows "edges (LG x1 x2) = x1"
INF_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd \<times> 'e"   and D :: "'c \<Rightarrow> 'd \<times> 'e" assumes "A = B"   and "\<And>x. s2p B x \<Longrightarrow> C x = D x" shows "Inf_prod (\<P> C A) = Inf_prod (\<P> D B)"
Resolution_insert: fixes S :: "'b literal set set"   and R :: "'b literal set"   and \<A> :: "'b \<Rightarrow> bool" assumes "S \<turnstile> R"   and "cnf_semantics \<A> S" shows "cnf_semantics \<A> {R}"
Smith_1xn_aux_S'_AQ': fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S' :: "'a mat"   and Q' :: "'a mat"   and i :: "nat"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and n :: "nat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "(S', Q') = Smith_Impl.Smith_1xn_aux Smith_1x2 i A (S, Q)"   and "A \<in> carrier_mat 1 n"   and "S \<in> carrier_mat 1 n"   and "Q \<in> carrier_mat n n"   and "S = A * Q"   and "i < n" shows "S' = A * Q'"
d: fixes prod :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'c"   and x :: "'f"   and S :: "'f set"   and g :: "'f \<Rightarrow> 'd" assumes "bounded_bilinear prod"   and "f differentiable at x within S"   and "g differentiable at x within S" shows "(\<lambda>x. prod (f x) (g x)) differentiable at x within S"
sp_spec_subj_obj_def: fixes configuration_subj_obj :: "'a \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and a :: "'a"   and x :: "obj_id_t"   and m :: "mode_t" shows "abstract_policy_derivation.sp_spec_subj_obj configuration_subj_obj a x m \<equiv> configuration_subj_obj a x m \<or> (\<exists>y. x = PAGE y \<and> m = READ \<and> configuration_subj_obj a x WRITE)"
j: shows "atom j \<sharp> (k1, k2, k, x)"
ring_hom_cringI: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring_hom_ring R S h"   and "cring R"   and "cring S" shows "ring_hom_cring R S h"
poly_of_pm_one: fixes x :: "'b" shows "poly_of_pm x 1 = 1"
inner_lv_rel_def: fixes a :: "'b list"   and b :: "'b list" shows "inner_lv_rel a b = sum_list (map2 (*) a b)"
dom_max_mapping: fixes P1 :: "'a \<Rightarrow> 'b option"   and P2 :: "'a \<Rightarrow> 'b option" shows "dom (max_mapping P1 P2) = dom P1 \<inter> dom P2"
common_sudiv_exists_refl: fixes C1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and C2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "common_sudiv_exists C1 C2" shows "common_sudiv_exists C2 C1"
powr_mult_2: fixes x :: "real"   and y :: "real" assumes "0 < x" shows "x powr (y * 2) = x\<^sup>2 powr y"
simps_6: fixes k :: "nat"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "pre ({k} Ab C) Q = (\<lambda>s. ediv (pre C (\<lambda>s. enat k * Q s) s) k)"
r_null: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
assms: shows "(\<lambda>n. fds_nth f (Suc n) / nat_power (Suc n) s) sums L"
imod_enat_enat: fixes a :: "nat"   and b :: "nat" shows "enat a mod enat b = enat (a mod b)"
Nmlize_Nml: fixes t :: "'c local.term" assumes "Nml t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = t"
F_tprog_def: shows "F_tprog = tm_of F_aprog"
shift: fixes I :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat" assumes "I \<subseteq> {..<brn (cl ! n)}" shows "local.shift cl n I \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}"
hoare_conj_right: fixes P :: "'a set"   and R :: "('a \<times> 'a) set"   and Q1 :: "'a set"   and Q2 :: "'a set" assumes "{P} R {> Q1}"   and "{P} R {> Q2}" shows "{P} R {> Q1 \<inter> Q2}"
OO_Grp_alt: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e" shows "(BNF_Def.Grp A f)\<inverse>\<inverse> OO BNF_Def.Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
trace_parallel_plan_step_effect_is: fixes k :: "nat"   and I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" shows "trace_parallel_plan_strips I \<pi> ! Suc k = execute_parallel_operator (trace_parallel_plan_strips I \<pi> ! k) (\<pi> ! k)"
length_removeAll_less: fixes x :: "'g"   and xs :: "'g list" assumes "x \<in> set xs" shows "length (removeAll x xs) < length xs"
fun_mult_assoc: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a"   and z :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "z \<in> carrier F" shows "x \<otimes>\<^bsub>F\<^esub> y \<otimes>\<^bsub>F\<^esub> z = x \<otimes>\<^bsub>F\<^esub> (y \<otimes>\<^bsub>F\<^esub> z)"
pre_zeta_aux_cnj: fixes a :: "real"   and n :: "nat"   and z :: "complex" assumes "0 < a" shows "pre_zeta_aux n a (cnj z) = cnj (pre_zeta_aux n a z)"
lderiv_preserves: fixes r :: "'b rexp"   and s :: "'b rexp"   and as :: "'a" assumes "\<guillemotleft>r\<guillemotright> = \<guillemotleft>s\<guillemotright>" shows "\<guillemotleft>lderiv as r\<guillemotright> = \<guillemotleft>lderiv as s\<guillemotright>"
arr_dom: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.arr (B.VV.dom f)"
encode1_alter: fixes dims :: "nat list"   and A :: "nat set"   and vars :: "nat set"   and i :: "nat" assumes "{0..<length dims} \<subseteq> A" shows "partial_state.encode1 dims vars i = digit_decode (partial_state.dims1 dims vars) (nths (digit_encode dims i) (A \<inter> vars))"
borel_measurable_vimage_closed: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>T. closed T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
s_int: shows "expectation (\<lambda>x. \<bar>u s x / real s\<bar>) < rho"
corollary5: fixes 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"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "\<forall>a b. \<exists>a1 b1 d. a = times a1 d \<and> b = times b1 d \<and> ring.ideal_generated plus zero uminus times {a1, b1} = ring.ideal_generated plus zero uminus times {one}"   and "ring.ideal_generated plus zero uminus times {a, b, c} = ring.ideal_generated plus zero uminus times {d}" shows "\<exists>a1 b1 c1. a = times a1 d \<and> b = times b1 d \<and> c = times c1 d \<and> ring.ideal_generated plus zero uminus times {a1, b1, c1} = ring.ideal_generated plus zero uminus times {one}"
ALength_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list"   and a' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gt uf P t h (a, xs) (a', xs')" shows "\<tau>red1gt uf P t h (a\<bullet>length, xs) (a'\<bullet>length, xs')"
induct: fixes P :: "Core.term \<Rightarrow> bool"   and a0 :: "Core.term" assumes "\<And>all S T t. (all = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT \<Longrightarrow> P t) \<Longrightarrow> P (Ct all S $ Abs T t)"   and "\<And>v va. P (Ct v va)"   and "\<And>v va. P (Fv v va)"   and "\<And>v. P (Bv v)"   and "\<And>v va. P (Abs v va)"   and "\<And>vb vc va. P (Fv vb vc $ va)"   and "\<And>vb va. P (Bv vb $ va)"   and "\<And>vb vc va. P (Abs vb vc $ va)"   and "\<And>vb vc va. P (vb $ vc $ va)"   and "\<And>v vb vc. P (v $ Ct vb vc)"   and "\<And>v vb vc. P (v $ Fv vb vc)"   and "\<And>v vb. P (v $ Bv vb)"   and "\<And>v vb vc. P (v $ (vb $ vc))" shows "P a0"
simps_5: fixes ns :: "nat list" shows "bn_count (Suc (Suc 0) # ns) = Suc (Suc 0) + bn_count ns"
append_set_rem_last: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and zs :: "'a list" assumes "set (xs @ [x]) = set (ys @ [x] @ zs)"   and "distinct (xs @ [x])"   and "distinct (ys @ [x] @ zs)" shows "set xs = set (ys @ zs)"
fp_cop_F_range_inv': fixes x :: "'x"   and ds :: "'d set" assumes "x \<in> fp_cop_F ds" shows "x \<in> Field (Pd (Xd x)) \<and> Xd x \<in> ds"
mult_minus1_right: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "'d" assumes "class.ring_1 one times plus zero minus uminus" shows "times z (uminus one) = uminus z"
DBM_val_bounded_len_3'_aux: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and c :: "'c"   and d :: "'c"   and vs :: "nat list" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "v c \<le> n"   and "v d \<le> n"   and "\<forall>k\<in>set vs. 0 < k \<and> k \<le> n \<and> (\<exists>c. v c = k)" shows "dbm_entry_val u (Some c) (Some d) (len m (v c) (v d) vs)"
integrable_distr: fixes T :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and M' :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "T \<in> M \<rightarrow>\<^sub>M M'"   and "integrable (distr M M' T) f" shows "integrable M (\<lambda>x. f (T x))"
p'_def: shows "p' = G.awalk_to_apath p"
conj_comm: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(P \<and> Q) = (Q \<and> P)"
get_add_eq: fixes e :: "'fenv_inca"   and x :: "'fun"   and v :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef" shows "Finca_get (Finca_add e x v) x = Some v"
assms_1: shows "P \<turnstile> C sees M, Static :  []\<rightarrow>Void = m in D"
a: shows "\<forall>x y. T (i x (i y x))"
not_lfinite_llength: fixes xs :: "'b llist" assumes "\<not> lfinite xs" shows "llength xs = \<infinity>"
partition_between_vars_def: shows "partition_between_vars = partition_between_ref (\<le>) id"
same_orbit_rel_def: shows "conj.same_orbit_rel = {pa \<in> subgroups_of_size (p ^ a) \<times> subgroups_of_size (p ^ a). \<exists>g\<in>carrier G. conjugation_action (p ^ a) g (snd pa) = fst pa}"
ordLeq_def: shows "ordLeq = {(r, r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
exhaust: fixes y :: "diagram" obtains x1 :: "node fset" and x2 :: "node \<Rightarrow> assertion_gadget" and x3 :: "(node fset \<times> command_gadget \<times> node fset) list" where "y = Graph x1 x2 x3"
a: fixes x :: "'a" assumes "x \<in> deps (Insert z)" shows "\<not> (a pm < a \<lbrakk>x\<rbrakk> \<and> a \<lbrakk>x\<rbrakk> < a sm)"
seq_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(c\<^sub>1 ;; c\<^sub>2, mem) \<leadsto>\<^sub>s (c', mem')" obtains "c\<^sub>1 = Stop" and "c' = c\<^sub>2" and "mem' = mem"
nb_1: shows "\<forall>op. op \<in> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (add_effects_of op)} \<longrightarrow> \<not> lit_semantics \<A> op"
CAND_AndL1_elim: fixes x :: "name"   and y :: "name"   and M :: "trm"   and B :: "ty" assumes "x:AndL1 y.M x \<in> \<parallel>(B)\<parallel>"   and "x:AndL1 y.M x \<notin> BINDINGn B (\<parallel><B>\<parallel>)" shows "\<exists>B1 B2. B = B1 AND B2 \<and> y:M \<in> \<parallel>(B1)\<parallel>"
validE: fixes X :: "msg" assumes "X \<in> valid" obtains x :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "X = implem (Chan x A B M)" and "M \<in> payload"
enabled_mono: fixes F :: "state \<times> state \<Rightarrow> bool"   and s :: "state"   and G :: "state \<times> state \<Rightarrow> bool" assumes "s \<Turnstile> Enabled F"   and "\<turnstile> F \<longrightarrow> G" shows "s \<Turnstile> Enabled G"
preserves_cod: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "cod (R f) = R (cod f)"
inducts_1: fixes x1 :: "('exp, 'id) MWLfCom"   and P1 :: "('exp, 'id) MWLfCom \<Rightarrow> bool"   and P2 :: "('exp, 'id) MWLfCom list \<Rightarrow> bool" assumes "\<turnstile>\<^bsub>\<C>\<^esub> x1"   and "P1 skip"   and "\<And>x e. AssignSideCondition x e \<Longrightarrow> P1 (x:=e)"   and "\<And>c V. \<lbrakk>\<turnstile>\<^bsub>\<C>\<^esub> c; P1 c; \<turnstile>\<^bsub>\<V>\<^esub> V; P2 V\<rbrakk> \<Longrightarrow> P1 (fork c V)"   and "\<And>c1 c2. \<lbrakk>\<turnstile>\<^bsub>\<C>\<^esub> c1; P1 c1; \<turnstile>\<^bsub>\<C>\<^esub> c2; P1 c2\<rbrakk> \<Longrightarrow> P1 (c1;c2)"   and "\<And>c b. \<lbrakk>\<turnstile>\<^bsub>\<C>\<^esub> c; P1 c; WhileSideCondition b\<rbrakk> \<Longrightarrow> P1 (while b do c od)"   and "\<And>c1 c2 b. \<lbrakk>\<turnstile>\<^bsub>\<C>\<^esub> c1; P1 c1; \<turnstile>\<^bsub>\<C>\<^esub> c2; P1 c2; IfSideCondition b c1 c2\<rbrakk> \<Longrightarrow> P1 (if b then c1 else c2 fi)"   and "\<And>V. \<forall>i<length V. \<turnstile>\<^bsub>\<C>\<^esub> V ! i \<and> P1 (V ! i) \<Longrightarrow> P2 V" shows "P1 x1"
FinDimVectorSpace: shows "FinDimVectorSpace (\<sharp>\<cdot>) V"
limit_frequently_eq: fixes "open" :: "'b set \<Rightarrow> bool"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b"   and c :: "'b"   and d :: "'b" assumes "class.t1_space open"   and "F \<noteq> bot"   and "\<exists>\<^sub>F x in F. f x = c"   and "filterlim f (topological_space.nhds open d) F" shows "d = c"
add_top: fixes x :: "'a" assumes "(0::'a) \<le> x" shows "x + \<top> = \<top>"
l9_19: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and Y :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet X Y P"   and "Tarski_neutral_dimensionless.Col Bet A B P" shows "Tarski_neutral_dimensionless.OS Bet X Y A B = (Tarski_neutral_dimensionless.Out Bet P A B \<and> \<not> Tarski_neutral_dimensionless.Col Bet X Y A)"
coboundedI2: fixes b :: "'c DBMEntry"   and c :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<preceq> c" shows "linordered_monoid.min a b \<preceq> c"
cases: fixes x :: "'b heap list" obtains h1 :: "'b heap" and h2 :: "'b heap" and hs :: "'b heap list" where "x = h1 # h2 # hs"   | "x = []"   | v :: "'b heap" where "x = [v]"
commutes_with_composition: fixes \<eta> :: "'b \<Rightarrow> 'c"   and M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and M' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c"   and x :: "'b"   and y :: "'b" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'"   and "x \<in> M"   and "y \<in> M" shows "\<eta> (composition x y) = composition' (\<eta> x) (\<eta> y)"
hyps_1: shows "rtrancl3p (redT_updW t) ws' was ws''"
chop_subset2: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "N_Chop(i,j,k) \<longrightarrow> k \<le> i"
Equiv_def: fixes P :: "('d, 'e, 'f) formula"   and Q :: "('d, 'e, 'f) formula" shows "(P \<leftrightarrow> Q) = (P && Q || ! P && ! Q)"
hyps_4: fixes b :: "(name \<times> ty) list" assumes "valid b"   and "(x, T\<^sub>1) # \<Gamma>\<^sub>1 \<subseteq> b" shows "b \<turnstile> t : T\<^sub>2"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "braun \<langle>l, x, r\<rangle> = ((size l = size r \<or> size l = size r + 1) \<and> braun l \<and> braun r)"
wls_psubstAbs_compose: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsAbs (us, s) A"   and "wlsEnv rho"   and "wlsEnv rho'" shows "A $[rho] $[rho'] = A $[rho &[rho']]"
add_le_imp_le_left: fixes c :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "c + a \<subseteq># c + b" shows "a \<subseteq># b"
loop_invar: shows "loop \<in> dwa_invar (brc_det_algo (build_rqrm \<delta>) \<delta>)"
simps_10: fixes va :: "pol"   and vb :: "nat"   and vd :: "nat"   and ve :: "pol"   and v :: "nat"   and Q :: "pol" shows "isnorm (PX (PX va vb (Pinj vd ve)) (Suc v) Q) = (isnorm (PX va vb (Pinj vd ve)) \<and> isnorm Q)"
defined_split: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<delta> x) = (\<tau> |\<noteq> x \<triangleq> invalid \<and> \<tau> |\<noteq> x \<triangleq> null)"
border_def: fixes xs :: "'a list"   and ys :: "'a list" shows "border xs ys = (prefix xs ys \<and> suffix xs ys)"
urel_integer_normalize: fixes pp :: "integer"   and x :: "integer"   and y :: "int" assumes "p = int_of_integer pp"   and "urel_integer x y" shows "urel_integer (if x = 0 then 0 else 1) (if y = 0 then 0 else 1)"
block_original_count_le: fixes n :: "nat"   and b :: "'a set" assumes "0 < n" shows "multiplicity b \<le> count (multiple_blocks n) b"
finite_sorted_distinct_unique: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less"   and "finite A" shows "\<exists>!xs. set xs = A \<and> linorder.sorted less_eq xs \<and> distinct xs"
LLL_4: fixes f :: "int poly" assumes "degree u < Suc j'"   and "f \<noteq> 0"   and "pll.dvdm u f"   and "degree f < Suc j'" shows "\<parallel>local.g\<parallel>\<^sup>2 \<le> 2 ^ (Suc j' - 1) * \<parallel>f\<parallel>\<^sup>2"
assms_4: shows "f \<star> g \<cong> trg f"
cod_trg_eq_dom_trg: shows "HHfgh.cod.trg = HHfgh.dtrg"
one_closed: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "h \<one>\<^bsub>G\<^esub> \<in> carrier H"
sum_lessThan_telescope': fixes f :: "nat \<Rightarrow> 'a"   and m :: "nat" shows "(\<Sum>n<m. f n - f (Suc n)) = f 0 - f m"
select_convs_2: fixes sprFst :: "('b, 'c, 'd) BEState"   and sprLst :: "('b, 'c, 'd) BEState"   and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set"   and more :: "'e" shows "sprLst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprLst"
Act_eq_iff_perm_renaming: fixes f1 :: "'e"   and \<alpha>1 :: "'d"   and x1 :: "('b, 'c, 'd, 'e) formula"   and f2 :: "'e"   and \<alpha>2 :: "'d"   and x2 :: "('b, 'c, 'd, 'e) formula" shows "(Act f1 \<alpha>1 x1 = Act f2 \<alpha>2 x2) = (f1 = f2 \<and> (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1))"