hcju/isabelleps huggingface dataset:
simps: fixes a :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state" shows "Sstd.final a = (\<exists>F f fd pc H \<Sigma>. a = State F H [Frame f pc \<Sigma>] \<and> Fstd_get F f = Some fd \<and> pc = length (body fd))"
\<theta>\<^sub>f'_def: shows "\<theta>\<^sub>f' = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]"
cross_ratio_product: fixes p :: "proj2"   and q :: "proj2"   and s :: "proj2"   and r :: "proj2"   and t :: "proj2" assumes "proj2_Col p q s"   and "p \<noteq> q"   and "s \<noteq> p"   and "s \<noteq> q" shows "cross_ratio p q r s * cross_ratio p q s t = cross_ratio p q r t"
seq_char: fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "\<F>C.seq g f = (g \<cdot> f \<noteq> \<F>C.null)"
tmap_tllist_of_llist: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and b :: "'d"   and xs :: "'c llist" shows "tmap f g (tllist_of_llist b xs) = tllist_of_llist (g b) (lmap f xs)"
le_halfspace_def: fixes sctn :: "'b sctn"   and x :: "'b" shows "le_halfspace sctn x = (x \<bullet> normal sctn \<le> pstn sctn)"
poincare_distance_formula: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance u v = poincare_distance_formula (to_complex u) (to_complex v)"
elims: fixes x :: "'b list \<times> 'b list"   and y :: "'b list \<times> 'b list" assumes "deq x = y" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)" and "y = (if fs = [] then (fs, rs) else norm (tl fs, rs))"
mbisim_redT_upd: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x1' :: "'x1"   and m1' :: "'m1"   and s1' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x2' :: "'x2"   and m2' :: "'m2"   and s2' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x1 :: "'x1"   and ln :: "'l \<Rightarrow>f nat"   and x2 :: "'x2"   and x1'' :: "'x1"   and x2'' :: "'x2" assumes "redT_upd s1 t ta1 x1' m1' s1'"   and "redT_upd s2 t ta2 x2' m2' s2'"   and "wset s1 = wset s2"   and "locks s1 = locks s2"   and "wset s1' = wset s2'"   and "interrupts s1' = interrupts s2'"   and "finite (dom (thr s1))"   and "wset_thread_ok (wset s1) (thr s1)"   and "thr s1 t = \<lfloor>(x1, ln)\<rfloor>"   and "thr s2 t = \<lfloor>(x2, ln)\<rfloor>"   and "r1.actions_ok s1 t ta1"   and "r2.actions_ok s2 t ta2"   and "ta1 \<sim>m ta2"   and "t \<turnstile> (x1', m1') \<approx> (x2', m2')"   and "wset s1' t = None \<or> x1' \<approx>w x2'"   and "r1.silent_moves t (x1'', shr s1) (x1, shr s1)"   and "r1.r_syntax t x1 (shr s1) ta1 x1' m1'"   and "r2.silent_moves t (x2'', shr s2) (x2, shr s2)"   and "r2.r_syntax t x2 (shr s2) ta2 x2' m2'"   and "t \<turnstile> (x1'', shr s1) \<approx> (x2'', shr s2)"   and "\<not> \<tau>move1 (x1, shr s1) ta1 (x1', m1')"   and "\<not> \<tau>move2 (x2, shr s2) ta2 (x2', m2')"   and "\<And>t'. t \<noteq> t' \<Longrightarrow> tbisim (wset s1 t' = None) t' (thr s1 t') (shr s1) (thr s2 t') (shr s2)" shows "s1' \<approx>m s2'"
Ch_f_range: fixes h :: "'c" shows "f_range (Ch h)"
nat_term_compare: fixes x :: "'a nat_term_order" shows "nat_term_compare x \<in> {cmp. comparator cmp \<and> nat_term_comp cmp}"
distinct_11: shows "T_f32 \<noteq> T_f64"
intersect_cons_rel_family: shows "consequence_relation Bot (\<Turnstile>Q)"
Icc_plus_Icc: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a \<le> b"   and "c \<le> d" shows "{a..b} + {c..d} = {a + c..b + d}"
simps: fixes e :: "'c"   and a :: "'d"   and q :: "('c, 'd) BsSkewBinomialTree list" shows "elem_invar (Element e a q) = ((\<forall>x. x \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q)) \<longrightarrow> a \<le> eprio x \<and> elem_invar x) \<and> SkewBinomialHeapStruc.invar (map bsmapt q))"
assoc_naturality\<^sub>A\<^sub>W\<^sub>C: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<tau> :: "'b"   and \<mu> :: "'b"   and \<nu> :: "'b" assumes "associative_weak_composition V H \<a>"   and "H \<tau> \<mu> \<noteq> partial_magma.null V"   and "H \<mu> \<nu> \<noteq> partial_magma.null V" shows "V (\<a> (partial_magma.cod V \<tau>) (partial_magma.cod V \<mu>) (partial_magma.cod V \<nu>)) (H (H \<tau> \<mu>) \<nu>) = V (H \<tau> (H \<mu> \<nu>)) (\<a> (partial_magma.dom V \<tau>) (partial_magma.dom V \<mu>) (partial_magma.dom V \<nu>))"
im_sheaf_is_presheaf: shows "presheaf_of_rings S is_open local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf"
cod_cod: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.cod (VoV.cod f) = VoV.cod f"
sminus_def: fixes v :: "'a"   and t :: "'a" shows "punit.sminus v t = fst (punit.pp_of_term v - t, punit.component_of_term v)"
inclusion: fixes f :: "'a" assumes "Arr f" shows "C.arr f"
left_unique_def: fixes R :: "'j \<Rightarrow> 'k \<Rightarrow> bool" shows "left_unique R = (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
exec_elim_cases_9: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Cond b c1 c2,s\<rangle> \<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | s :: "'e" where "sb__ = Normal sc__" and "sc__ \<in> b" and "\<Gamma>\<turnstile> \<langle>c1,Normal sc__\<rangle> \<Rightarrow> t"   | s :: "'e" where "sb__ = Normal sd__" and "sd__ \<notin> b" and "\<Gamma>\<turnstile> \<langle>c2,Normal sd__\<rangle> \<Rightarrow> t"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
R1_SOME_2: fixes f :: "nat \<Rightarrow> nat option"   and r :: "recf"   and x :: "nat" assumes "f \<in> \<R>"   and "r = (SOME r'. recfn 1 r' \<and> Partial_Recursive.total r' \<and> f = (\<lambda>x. eval r' [x]))" shows "eval r [x] \<down>"
map_idI: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'c" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = x" shows "map f xs = xs"
InvIsMor: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "Category C"   and "cinv\<^bsub>C\<^esub> f g" shows "f \<in> mor\<^bsub>C\<^esub> \<and> g \<in> mor\<^bsub>C\<^esub>"
param_uncurry: fixes A :: "('c \<times> 'f) set"   and B :: "('d \<times> 'g) set"   and C :: "('e \<times> 'h) set" shows "(uncurry, uncurry) \<in> (A \<rightarrow> B \<rightarrow> C) \<rightarrow> A \<times>\<^sub>r B \<rightarrow> C"
set_subset_Cons: fixes xs :: "'a list"   and x :: "'a" shows "order.greater_eq (set (x # xs)) (set xs)"
rep_eq: fixes x :: "uint" shows "bit x = bit (Rep_uint x)"
assms_3: shows "mix_pmf (local.my_U p) local.some_best local.some_worst \<succeq>[\<R>] mix_pmf (local.my_U q) local.some_best local.some_worst \<and> (mix_pmf (local.my_U q) local.some_best local.some_worst, mix_pmf (local.my_U p) local.some_best local.some_worst) \<notin> \<R>"
open_negations: fixes S :: "'a set" assumes "open S" shows "open (uminus ` S)"
cases: fixes \<Delta> :: "('k, 'l) network_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "flow \<Delta> f" obtains "\<And>e. f e \<le> capacity \<Delta> e" and "\<And>x. \<lbrakk>x \<noteq> source \<Delta>; x \<noteq> sink \<Delta>\<rbrakk> \<Longrightarrow> KIR f x"
le_trans_equiv_rsp_right: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and v :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans v t = le_trans v u"
ChiStar_Self: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto" shows "(S, S) \<in> ChiStar A"
prems_2: shows "reads_consistent pending_write \<O> m (x # xs)"
l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_class_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog" shows "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order first_in_tree_order get_attribute get_element_by_id get_elements_by_class_name get_elements_by_tag_name \<equiv> get_element_by_id = l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_element_by_id first_in_tree_order get_attribute \<and> get_elements_by_class_name = l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_class_name to_tree_order get_attribute \<and> get_elements_by_tag_name = l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_tag_name to_tree_order"
assms_4: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs"
induct: fixes P :: "('a + 'n) list \<Rightarrow> bool"   and a0 :: "('a + 'n) list" assumes "P []"   and "\<And>uu_ uv_. P (Inl uu_ # uv_)"   and "\<And>N \<alpha>. P \<alpha> \<Longrightarrow> P (Inr N # \<alpha>)" shows "P a0"
out2_def: fixes B :: "bool \<times> bool"   and A :: "bool \<times> bool"   and s2 :: "bool" shows "out2 B A s2 = (let (a1, b1) = A; (a2, b2) = B; s1 = s2 \<oplus> (a1 \<oplus> a2 \<and> b1 \<oplus> b2) in return_spmf (s1, s2))"
cdep_slice2: fixes n'' :: "'node SDG_node"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'"   and "n' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 n"
simps_1: fixes n :: "nat"   and x :: "name" shows "lookup [] n x = DBVar x"
af\<^sub>G_sat_core_generalized: fixes \<G> :: "'b ltl set"   and i :: "nat"   and j :: "nat"   and \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set" assumes "Only_G \<G>"   and "i \<le> j"   and "\<G> \<Turnstile>\<^sub>P af\<^sub>G \<phi> (prefix i w)" shows "\<G> \<Turnstile>\<^sub>P af\<^sub>G \<phi> (prefix j w)"
neg_conj_cancel2: fixes Q :: "'a upred"   and P :: "'a upred" shows "(\<not> Q \<and> (P \<or> Q)) = (\<not> Q \<and> P)"
rOutput: fixes M :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and K :: "'a" assumes "cP = M\<langle>N\<rangle>.P"   and "cRs = K\<langle>N\<rangle> \<prec> P"   and "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop"
zero_not_in_residue_units: fixes n :: "nat" assumes "1 \<le> n" shows "0 \<notin> Units (residue_ring (p ^ n))"
csmall_nstep_SucD: fixes \<sigma>' :: "'state"   and coll' :: "'coll"   and P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" assumes "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> (Suc n)" shows "\<exists>\<sigma>1 coll1. (\<sigma>1, coll1) \<in> csmall P \<sigma> \<and> (\<exists>coll. coll' = combine coll1 coll \<and> (\<sigma>', coll) \<in> csmall_nstep P \<sigma>1 n)"
"1_2": shows "v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1 @- w"
strict_runit: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.runit f = f"
second_vertices_new_path: fixes xs :: "'a list" assumes "hd (tl xs) \<notin> G.second_vertices" shows "xs \<notin> paths"
Limsup_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" shows "Limsup F f = (\<Sqinter>P\<in>{P. eventually P F}. \<Squnion> (f ` {x. P x}))"
of_nat_def: fixes n :: "nat" shows "of_nat n = ((+) (1::'a) ^^ n) (0::'a)"
gfp_unfold: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "gfp f = f (gfp f)"
DDH0_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "local.ddh.DDH0 \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>s. sample_uniform (order \<G>) \<bind> (\<lambda>r. let h = \<^bold>g [^] s in \<A> h (\<^bold>g [^] r) (h [^] r)))"
godel_code'_not0: fixes xs :: "nat list"   and n :: "nat" shows "godel_code' xs n \<noteq> 0"
nz_impl_b_least_less: fixes y :: "nat"   and x :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "y < x"   and "f x y \<noteq> 0" shows "b_least f x < x"
saturated_def: fixes N :: "'f set" shows "reduc_calc.saturated N = (Inf_from N \<subseteq> Red_Red_I N)"
\<Sigma>_protocol_def: shows "Schnorr_\<Sigma>.\<Sigma>_protocol \<equiv> Schnorr_\<Sigma>.completeness \<and> Schnorr_\<Sigma>.special_soundness \<and> Schnorr_\<Sigma>.HVZK"
Ls_def: shows "Ls \<equiv> cnj Lq (neg (eql (Var xx) (Var yy)))"
assms: shows "e \<in> set (elements (prefixToLevel level M))"
Nil_eq_unfoldr_iff: fixes g :: "('a, 'b) generator"   and s :: "'b" shows "([] = list.unfoldr g s) = (\<not> list.has_next g s)"
refl_command: fixes c :: "cmd" shows "c \<^sub>C\<longlonglongrightarrow>\<^sup>* c"
fds_exp_def: fixes f :: "'a fds" shows "fds_exp f = (let f' = fds (\<lambda>n. if n = 1 then 0::'a else fds_nth f n) in fds (\<lambda>n. exp (fds_nth f 1) * (\<Sum>k. fds_nth (f' ^ k) n /\<^sub>R fact k)))"
measure_subadditive_countably: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>" shows "Sigma_Algebra.measure M (\<Union> (range A)) \<le> (\<Sum>i. Sigma_Algebra.measure M (A i))"
obj_def: fixes a :: "'a" shows "B.obj a \<equiv> B.arr a \<and> src a = a"
pol_expr_unique2: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d"   and "fst c = fst d" shows "(polyn_expr R X (fst c) c = polyn_expr R X (fst d) d) = (\<forall>j\<le>fst c. snd c j = snd d j)"
simps_2: shows "subcomp sA2 = {sA21, sA22, sA23}"
pell_power_Suc: fixes z :: "'b \<times> 'b"   and n :: "nat" shows "pell_power z (Suc n) = pell_mul z (pell_power z n)"
ordLeq_omax: fixes R :: "('c \<times> 'c) set set"   and r :: "('c \<times> 'c) set"   and p :: "('d \<times> 'd) set" assumes "finite R"   and "\<forall>r\<in>R. Well_order r"   and "r \<in> R"   and "p \<le>o r" shows "p \<le>o omax R"
add: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
DirProds_def: fixes G :: "'c \<Rightarrow> ('d, 'e) monoid_scheme"   and I :: "'c set" shows "DirProds G I = \<lparr>carrier = Pi\<^sub>E I (carrier \<circ> G), monoid.mult = \<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i, one = \<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>\<rparr>"
walk_in_V: fixes xs :: "'a list" assumes "Q.walk xs" shows "set xs \<subseteq> V\<^bsub>H_x\<^esub>"
integral_unique: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and k :: "'b set" assumes "(f has_integral y) k" shows "integral k f = y"
list_all2_rev: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" shows "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
apply_outputs_3: fixes p :: "'a aexp list"   and s :: "'a \<Rightarrow> value option" shows "apply_outputs p s = map (\<lambda>p. aval p s) p"
lm006: fixes X :: "'c set"   and Y :: "'d set" shows "injections X Y \<subseteq> injectionsUniverse"
inf_have_prems: fixes \<iota>F :: "'f inference" assumes "\<iota>F \<in> Inf_F" shows "prems_of \<iota>F \<noteq> []"
AppI: fixes A :: "type_scheme list"   and e1 :: "expr"   and t2 :: "typ"   and t1 :: "typ"   and e2 :: "expr" assumes "A |- e1 :: t2 -> t1"   and "A |- e2 :: t2" shows "A |- App e1 e2 :: t1"
non_neg_mat_st_mat: shows "non_neg_mat (st_mat x)"
exec_bind_SE_success2: fixes A :: "'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e"   and b :: "'d"   and \<sigma>' :: "'e"   and M :: "'e \<Rightarrow> (bool \<times> 'e) option" assumes "A \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> A ;- M) = (\<sigma>' \<Turnstile> M)"
WT_gpv_Fail: fixes \<Gamma> :: "('a, 'b) \<I>" shows "\<Gamma> \<turnstile>g Fail \<surd>"
hom_deg_set_closed_plus: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and z :: "nat"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>a1 a2. \<lbrakk>a1 \<in> A; a2 \<in> A\<rbrakk> \<Longrightarrow> a1 + a2 \<in> A"   and "p \<in> hom_deg_set z A"   and "q \<in> hom_deg_set z A" shows "p + q \<in> hom_deg_set z A"
igSubstIGVar2STR_imp_igSubstIGVar2: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstIGVar2STR Delta MOD" shows "FixSyn.igSubstIGVar2 Delta MOD"
sndTSconfig': fixes initH :: "nat list"   and init :: "nat list"   and qs :: "nat list" shows "snd (Partial_Cost_Model.config' (rTS initH) (init, []) qs) = rev qs @ []"
f8: shows "s'' \<otimes> s \<otimes> s' \<otimes> (s'' \<otimes> s' \<otimes> (r'' \<otimes> r) \<oplus> s'' \<otimes> s \<otimes> (r'' \<otimes> r')) = s'' \<otimes> s \<otimes> s' \<otimes> (s'' \<otimes> s' \<otimes> (r'' \<otimes> r)) \<oplus> s'' \<otimes> s \<otimes> s' \<otimes> (s'' \<otimes> s \<otimes> (r'' \<otimes> r'))"
ordLeq_Well_order_simp: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r \<le>o r'" shows "Well_order r \<and> Well_order r'"
exhaust: fixes y :: "Mode" obtains "y = AsmNoReadOrWrite"   | "y = AsmNoWrite"   | "y = GuarNoReadOrWrite"   | "y = GuarNoWrite"
simps: fixes a1 :: "'c"   and a2 :: "'c"   and U :: "'c set"   and r :: "('c \<times> 'c) set" shows "((a1, a2) \<in> on U r\<^sup>+) = ((\<exists>a b. a1 = a \<and> a2 = b \<and> a \<in> U \<and> b \<in> U \<and> (a, b) \<in> r) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> a \<in> U \<and> b \<in> U \<and> c \<in> U \<and> (a, b) \<in> on U r\<^sup>+ \<and> (b, c) \<in> r))"
is_lasso_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and m :: "'a"   and L :: "('Q \<times> nat) lasso" shows "gb_graph.is_lasso (b_graph.to_gbg_ext (degeneralize_ext ma) m) L \<equiv> graph.is_lasso_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) L \<and> (\<forall>A\<in>gbg_F (b_graph.to_gbg_ext (degeneralize_ext ma) m). set (lasso_cycle L) \<inter> A \<noteq> {})"
order_iff_strict: fixes b :: "'a list"   and a :: "'a list" shows "suffix b a = (strict_suffix b a \<or> a = b)"
isUb_UNIV_I: fixes S :: "'a set"   and u :: "'a" assumes "\<And>y. y \<in> S \<Longrightarrow> y \<le> u" shows "isUb UNIV S u"
preserves_arr: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and f :: "'a" assumes "functor A B F"   and "partial_magma.arr A f" shows "partial_magma.arr B (F f)"
product_run_Some: fixes \<iota>\<^sub>m :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and q\<^sub>0 :: "'a"   and \<delta>\<^sub>m :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'a"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "\<iota>\<^sub>m k = Some q\<^sub>0" shows "run (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w i k = Some (run (\<delta>\<^sub>m k) q\<^sub>0 w i)"
assms_3: shows "nsqn rt1 dip = nsqn rt2 dip"
gen_wf_predecessors_def: shows "gen_wf_predecessors \<equiv> graph_path_base.predecessors gen_wf_inEdges'"
generated_ringI_Basic: fixes A :: "'a set" assumes "A \<in> sigma_sets \<Omega> M" shows "A \<in> M.generated_ring"
arity\<^sub>h_conv: fixes s :: "('s, 'v) tm" shows "arity\<^sub>h s = arity_hd\<^sub>h (head s) - of_nat (num_args s)"
mem_simps_5: fixes c :: "'a"   and A :: "'a set" shows "(c \<in> - A) = (c \<notin> A)"
length_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "length (list.map f xs) = length xs"
proj_O_dim: shows "proj_O \<in> carrier_mat N N"
find_SomeD: fixes f :: "'b \<Rightarrow> bool"   and x :: "'b list"   and y :: "'b" assumes "find f x = Some y" shows "f y \<and> y \<in> set x"
add_3: fixes X :: "'a multiset" assumes "X' = add_mset x X" obtains z :: "'a" and Z :: "'a multiset" where "Y' = add_mset z Z" and "(x, z) \<in> ns" and "(X, Z) \<in> multpw ns"
aff_dim_subspace: fixes S :: "'a set" assumes "subspace S" shows "aff_dim S = int (dim S)"
simps_87: fixes x121 :: "expr"   and x122 :: "expr"   and x31 :: "char list"   and x32 :: "expr" shows "x121;; x122 \<noteq> \<lparr>x31\<rparr>x32"
mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> o[F](g1)"   and "f2 \<in> o[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> o[F](\<lambda>x. g1 x * g2 x)"
assms_2: shows "theta\<inverse> \<subseteq> Sretr (theta\<inverse> \<union> Sbis)"
elims: fixes x :: "real"   and xa :: "nat"   and xb :: "'b"   and y :: "real" assumes "disc_rfr_proc x xa xb = y" obtains r :: "real" and w :: "'b" where "x = ra__" and "xa = 0" and "xb = w__" and "y = 1"   | r :: "real" and n :: "nat" and w :: "'b" where "x = r" and "xa = Suc n" and "xb = w" and "y = (1 + r) * disc_rfr_proc r n w"
S_intersections_incident_1: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec"   and l :: "proj2_line" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "proj2_abs p \<noteq> proj2_abs q"   and "proj2_incident (proj2_abs p) l"   and "proj2_incident (proj2_abs q) l" shows "proj2_incident (S_intersection1 p q) l"
Union: fixes X :: "V set" assumes "small X" shows "small (\<Union> (elts ` X))"
Int_stable_prod_algebra: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "Int_stable (prod_algebra I M)"
GSMP_def: fixes P :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" shows "GSMP P \<equiv> {t \<in> SMP P. fv t = {}}"
distinct_113: fixes x8 :: "shift_type"   and x9 :: "ctrl_type" shows "shift_type x8 \<noteq> ctrl_type x9"