hcju/isabelleps huggingface dataset:
Im_interval_plus: fixes A :: "complex_interval"   and B :: "complex_interval" shows "Im_interval (A + B) = Im_interval A + Im_interval B"
fimage_Suc_eq_singleton: fixes Z :: "nat fset"   and y :: "nat" shows "(Suc |`| Z = {|y|}) = (\<exists>x. Z = {|x|} \<and> Suc x = y)"
br_invar_step: fixes \<delta> :: "('d, 'e) ta_rule set"   and \<Sigma> :: "'d set \<times> 'd set"   and \<Sigma>' :: "'d set \<times> 'd set" assumes "finite \<delta>"   and "\<Sigma> \<in> br_cond"   and "\<Sigma> \<in> br_invar \<delta>"   and "(\<Sigma>, \<Sigma>') \<in> br_step \<delta>" shows "\<Sigma>' \<in> br_invar \<delta>"
oseq_step_invariant_ctermI: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "((nat \<Rightarrow> 'f) \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and i :: "nat"   and S :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> 'g seq_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> bool"   and P :: "((nat \<Rightarrow> 'f) \<times> 'h) \<times> 'g seq_action \<times> (nat \<Rightarrow> 'f) \<times> 'h \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "simple_labels \<Gamma>"   and "automaton.trans A = oseqp_sos \<Gamma> i"   and "\<And>p l \<sigma> a q l' \<sigma>' pp. \<lbrakk>p \<in> cterms \<Gamma>; l \<in> labels \<Gamma> p; ((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i; ((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A; l' \<in> labels \<Gamma> q; (\<sigma>, pp) \<in> oreachable A S U; p \<in> sterms \<Gamma> pp; (\<sigma>', q) \<in> oreachable A S U; S \<sigma> \<sigma>' a\<rbrakk> \<Longrightarrow> P ((\<sigma>, l), a, \<sigma>', l')" shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
lepoll_antisym: fixes A :: "'a set"   and B :: "'b set" assumes "A \<lesssim> B"   and "B \<lesssim> A" shows "A \<approx> B"
eval\<Pi>\<^sub>1_inverse: fixes x :: "\<Pi>\<^sub>1" shows "make\<Pi>\<^sub>1 (eval\<Pi>\<^sub>1 x) = x"
abs_zmult_eq_1: fixes m :: "int"   and n :: "int" assumes "\<bar>m * n\<bar> = 1" shows "\<bar>m\<bar> = 1"
backtrack_nsc_code_def: shows "backtrack_nsc_code = backtrack_nsc backtrack_s"
simps: fixes a1 :: "'a list"   and a2 :: "'b sequent" shows "((a1, a2) \<in> Ax) = ((\<exists>i. a1 = [] \<and> a2 = ( \<LM> At i  \<RM> \<Rightarrow>* At i)) \<or> a1 = [] \<and> a2 = ( \<LM> ff  \<RM> \<Rightarrow>* Em))"
Gromov_product_tendsto_PInf_a_b: fixes a :: "'a"   and u :: "nat \<Rightarrow> 'a"   and v :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "(\<lambda>x. ereal (Gromov_product_at a (u x) (v x))) \<longlonglongrightarrow> \<infinity>" shows "(\<lambda>x. ereal (Gromov_product_at b (u x) (v x))) \<longlonglongrightarrow> \<infinity>"
cm_all'_def: fixes c0 :: "('c, 't, 'd) Propagate.configuration_scheme"   and \<Delta> :: "('c \<times> 't) zmultiset" shows "cm_all' c0 \<Delta> = Finite_Set.fold (\<lambda>(loc, t) c. apply_cm c loc t (zcount \<Delta> (loc, t))) c0 (set_zmset \<Delta>)"
num_params_tm_neg_eq: fixes t1 :: "taylor_model" shows "num_params (tm_poly (tm_neg t1)) = num_params (tm_poly t1)"
emeasure_sup_measure': fixes B :: "'b measure"   and A :: "'b measure"   and X :: "'b set" assumes "sets B = sets A"   and "X \<in> sets A" shows "emeasure (sup_measure' A B) X = (SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
SetToIndex: fixes a :: "'b"   and liste :: "'b list" assumes "a \<in> set liste" shows "\<exists>index<length liste. a = liste ! index"
qPsubstAbs_preserves_qGoodAbs: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodEnv rho"   and "qGoodAbs A" shows "qGoodAbs A $[[rho]]"
cyc_permutationss_def: shows "cyc_permutationss = product_lists \<circ> map cyc_permutations"
inverse_rat: fixes a :: "int"   and b :: "int" shows "inverse (Fract a b) = Fract b a"
simps: fixes fs :: "face list"   and n :: "nat"   and f :: "face list list"   and h :: "nat list" shows "faceListAt (Graph fs n f h) = f"
separatedin_mono: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set"   and S' :: "'c set"   and T' :: "'c set" assumes "separatedin X S T"   and "S' \<subseteq> S"   and "T' \<subseteq> T" shows "separatedin X S' T'"
to_nat_on_finite: fixes S :: "'c set" assumes "finite S" shows "bij_betw (to_nat_on S) S {..<card S}"
assms: shows "of_int (fls_subdegree f) \<noteq> (0::'a)"
array_rules_2: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a"   and z :: "'b" shows "(f(x := y)) z = (if z = x then y else f z)"
member_filter: fixes P :: "'a \<Rightarrow> bool"   and L :: "'a list"   and x :: "'a" assumes "List.member (filter P L) x" shows "List.member L x"
snd_sym_preproc_addnew_superset: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and vs :: "'t list"   and v :: "'t" shows "set fs \<subseteq> set (snd (sym_preproc_addnew gs vs fs v))"
intros_2: fixes T :: "('fun, 'var) Term.term list"   and f :: "'fun"   and M :: "('fun, 'var) Term.term set" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t" shows "M \<turnstile>\<^sub>c Fun f T"
exhaust: fixes y :: "'b option" obtains "y = \<bottom>"   | x2 :: "'b" where "y = \<lfloor>x2\<rfloor>"
simps_2: fixes uv :: "('c, 'd) hd"   and uw :: "dir"   and ux :: "dir list" shows "position_of (Hd uv) (uw # ux) = False"
assms: fixes n :: "nat" assumes "0 < n" shows "0 \<le> f n"
inv: fixes h :: "'e"   and H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "h \<in> H" shows "inv\<^bsub>G\<^esub> h \<in> generate G H"
b_empty: fixes T :: "('e, 'f) TS_scheme" shows "[] \<in> beh T"
b: shows "mem\<^sub>1 x \<noteq> mem\<^sub>2 x"
assms_2: shows "d i \<cdot> ad t \<le> d q"
vsubset_vdiff: fixes A :: "V"   and B :: "V"   and C :: "V" assumes "A \<subseteq>\<^sub>\<circ> B -\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> B"
img_fact: fixes f :: "'s" assumes "arr f" shows "incl_of (img f) (cod f) \<cdot> corestr f = f"
abs_summable_on_cmult_right: fixes c :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "c \<noteq> (0::'d) \<Longrightarrow> f abs_summable_on A" shows "(\<lambda>x. c * f x) abs_summable_on A"
vsubset_VPowI: fixes a :: "V"   and b :: "V" assumes "a \<subseteq>\<^sub>\<circ> b" shows "VPow a \<subseteq>\<^sub>\<circ> VPow b"
simps_13: fixes vg :: "nat"   and f :: "'fun"   and \<Sigma> :: "Unboxed.type option list" shows "optim_instr vg (Ubx.instr.ICall f) \<Sigma> = Result.of_option () (get_arity f) \<bind> (\<lambda>ar. gen_pop_push (Ubx.instr.ICall f) (replicate ar None, None) \<Sigma>)"
encode12_inv1: fixes i :: "nat"   and dims :: "nat list"   and vars :: "nat set"   and j :: "nat" assumes "i < partial_state.d1 dims vars"   and "j < partial_state.d2 dims vars" shows "partial_state.encode1 dims vars (partial_state.encode12 dims vars (i, j)) = i"
midpoint_eq_endpoint_1: fixes a :: "'b"   and b :: "'b" shows "(midpoint a b = a) = (a = b)"
igConsIPresIGWlsSTR_imp_igConsIPresIGWls: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igConsIPresIGWlsSTR MOD" shows "igConsIPresIGWls MOD"
countable_insert_eq: fixes x :: "'a"   and A :: "'a set" shows "countable (insert x A) = countable A"
typeBinOp3: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and e1 :: "expr"   and lev :: "secLevel"   and e2 :: "expr"   and bop :: "bop" assumes "\<Gamma> \<turnstile> e1 : lev"   and "\<Gamma> \<turnstile> e2 : High" shows "\<Gamma> \<turnstile> e1 \<guillemotleft>bop\<guillemotright> e2 : High"
unique_resl: fixes x :: "'a"   and z :: "'a"   and z' :: "'a"   and y :: "'a" assumes "D x z"   and "D x z'"   and "x \<oplus> z = y"   and "x \<oplus> z' = y" shows "z = z'"
span_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c poly"   and S :: "'c poly set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> poly_vs.span S" shows "sum f A \<in> poly_vs.span S"
has_derivative_Pair: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set"   and g :: "'d \<Rightarrow> 'f"   and g' :: "'d \<Rightarrow> 'f" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. (f x, g x)) has_derivative (\<lambda>h. (f' h, g' h))) (at x within s)"
simps_16: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and x1 :: "'b" shows "(case Assumption x1 of Assumption x \<Rightarrow> f1 x | Conclusion x \<Rightarrow> f2 x | Rule x \<Rightarrow> f3 x | Helper \<Rightarrow> f4) = f1 x1"
some_lub_conv: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and u :: "'b" assumes "acyclic r"   and "is_lub (r\<^sup>*) x y u" shows "some_lub (r\<^sup>*) x y = u"
rep_eq: fixes x :: "'b pmf" shows "mapping.rep (mapping_of_pmf x) = (\<lambda>xa. if pmf x xa = 0 then None else Some (pmf x xa))"
intros_2: fixes d :: "nat"   and u :: "'d \<Rightarrow> real"   and x :: "'d" assumes "real d < u x"   and "u x < real (d + 1)" shows "intv_elem x u (Intv d)"
m_assoc: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)"   and "z \<in> carrier (poly_ring R)" shows "x \<oplus>\<^bsub>poly_ring R\<^esub> y \<oplus>\<^bsub>poly_ring R\<^esub> z = x \<oplus>\<^bsub>poly_ring R\<^esub> (y \<oplus>\<^bsub>poly_ring R\<^esub> z)"
decode_abs_plan_def: shows "decode_abs_plan \<equiv> map (fst \<circ> the \<circ> lookup_action) \<circ> rem_condless_ops I"
iso_imp_group: fixes H :: "('e, 'f) monoid_scheme" assumes "G \<cong> H"   and "Group.monoid H" shows "Group.group H"
lift_Def_High: fixes V :: "'g"   and H :: "'g set"   and E :: "'h"   and Def :: "'h \<Rightarrow> 'g set"   and X :: "'h"   and L :: "'g set" assumes "V \<in> H" shows "(Node E, V) \<in> lift_Def_set Def E X H L"
split_minD: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "x # inorder t' = inorder t"
points_index_simple_def: fixes ps :: "'a set" shows "\<B> index ps = card {b \<in> design_support. ps \<subseteq> b}"
SUP_bot_conv_1: 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 B :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup (B ` A) = bot) = (\<forall>x\<in>A. B x = bot)"
WT_recv: fixes \<eta> :: "nat"   and s :: "'a list cstate" assumes "pred_cstate (\<lambda>x. length x = \<eta>) s" shows "\<I>_uniform UNIV (insert None (Some ` nlists UNIV \<eta>)) \<turnstile>c S_CHAN.recv_oracle s \<surd>"
fields_noneI: fixes P :: "cld list"   and ty :: "ty" assumes "find_path_ty P ty None" shows "fields P ty None"
data_in'_def: shows "data_in' \<equiv> fst (takedropWhile_queue (\<lambda>(t, X). right I < enat (nt - t)) data_in)"
inferences_between_eq_UNION: fixes Q :: "'a literal multiset set"   and C :: "'a literal multiset" shows "inference_system.inferences_between (ord_FO_\<Gamma> S) Q C = inference_system.inferences_between (ord_FO_\<Gamma> S) {C} C \<union> (\<Union>D\<in>Q. inference_system.inferences_between (ord_FO_\<Gamma> S) {D} C)"
kmax_ge: fixes f :: "nat \<Rightarrow> real" shows "\<forall>i\<in>kmax f PR. \<forall>r\<in>reduce f PR. r \<le> f i"
commutative_squareI: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.cospan f g"   and "C.span h k"   and "C.dom f = C.cod h"   and "f \<cdot> h = g \<cdot> k" shows "C.commutative_square f g h k"
rep_eq: fixes x :: "'a fset"   and xa :: "'b fset" shows "fset (x |\<times>| xa) = fset x \<times> fset xa"
fv_open_Var: fixes xT :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and i :: "nat" assumes "fresh xT t" shows "fv (open_Var i xT t) |\<subseteq>| finsert xT (fv t)"
connectedin_eq_not_separated_subset: fixes X :: "'b topology"   and S :: "'b set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>C1 C2. S \<subseteq> C1 \<union> C2 \<and> S \<inter> C1 \<noteq> {} \<and> S \<inter> C2 \<noteq> {} \<and> separatedin X C1 C2))"
compat_different: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and x :: "'Var" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i \<sqsubset> length cms\<^sub>1"   and "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "mem\<^sub>1 x \<noteq> mem\<^sub>2 x \<and> dma x = Low"
sorted_wrt_insort_wrt_imp_sorted_wrt: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "sorted_wrt r (insort_wrt s x xs)" shows "sorted_wrt r xs"
splitAt_is_prefix: fixes ram :: "'a"   and vs :: "'a list" assumes "ram \<in> set vs" shows "is_prefix (fst (splitAt ram vs) @ [ram]) vs"
chain_def: fixes S :: "nat \<Rightarrow> 'a set" shows "chain S = (\<forall>i. S i \<subseteq> S (Suc i))"
simps_2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'b"   and x :: "'c"   and xs :: "'c list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
remove_child_preserves_type_wf: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.remove_child ptr child \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h'"
FG_advice_b1_helper: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> subfrmlsn \<phi>"   and "suffix i w \<Turnstile>\<^sub>n \<psi>" shows "suffix i w \<Turnstile>\<^sub>n \<psi>[\<F>\<G> \<phi> w]\<^sub>\<mu>"
dense_ge_bounded: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.dense_linorder less_eq less"   and "less z x"   and "\<And>w. \<lbrakk>less z w; less w x\<rbrakk> \<Longrightarrow> less_eq y w" shows "less_eq y z"
map_closed: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set"   and a :: "'c" assumes "Set_Theory.map \<alpha> S T"   and "a \<in> S" shows "\<alpha> a \<in> T"
finite_sgnx_at_left_at_right_1: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and s :: "real set"   and x :: "real" assumes "finite {t. f t = 0 \<and> a < t \<and> t < b}"   and "continuous_on ({a<..<b} - s) f"   and "finite s"   and "x \<in> {a<..<b}" shows "f sgnx_able at_left x"
assms: fixes xs :: "'a list" shows "distinct (fst (run_state (label_list xs) s))"
B2_def: fixes m1 :: "'b option"   and m2 :: "'b option"   and \<alpha> :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "B2 m1 m2 \<alpha> \<equiv> m1 \<rightharpoonup> (\<lambda>a. m2 \<rightharpoonup> \<alpha> a)"
closed_def: fixes e :: "db" shows "closed e \<equiv> \<forall>i. \<not> freedb e i"
DERIV_unique: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c"   and E :: "'c" assumes "(f has_field_derivative D) (at x)"   and "(f has_field_derivative E) (at x)" shows "D = E"
intro: fixes G :: "('c, 'd) pre_digraph" assumes "finite (verts G)"   and "finite (arcs G)" shows "fin_digraph_axioms G"
weaken_A_Int_B_eq_empty: fixes A :: "'a set"   and B :: "'a set" assumes "\<forall>x. x \<in> A \<longrightarrow> x \<notin> B" shows "A \<inter> B = {}"
select_convs_2: fixes nrm :: "lname set"   and brk :: "label \<Rightarrow> lname set"   and more :: "'a" shows "brk \<lparr>nrm = nrm, brk = brk, \<dots> = more\<rparr> = brk"
assms: shows "f \<bullet> ns l \<noteq> \<zero>"
CBF: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) in v]"
nn_integral_indicator: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> sets M" shows "integral\<^sup>N M (indicator A) = emeasure M A"
d_1: shows "distinct (filter P (list_of (ltake (enat (Suc (max i j))) lst)))"
intros_2: fixes get :: "'a \<Rightarrow> ('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef option"   and st1 :: "('a, 'dyn) frame list"   and st2 :: "('a, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and f :: "'a"   and \<Sigma>1 :: "'dyn list"   and \<Sigma>2 :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and fd2 :: "('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef"   and n :: "nat"   and opt :: "'a option" assumes "rel_stacktraces get st1 st2 (Some f)"   and "\<Sigma>1 = norm_stack \<Sigma>2"   and "get f = Some fd2"   and "sp_fundef get fd2 (take n (body fd2)) = Ok (map typeof \<Sigma>2)"   and "pred_option (is_valid_fun_call get fd2 n \<Sigma>2) opt" shows "rel_stacktraces get (Frame f n \<Sigma>1 # st1) (Frame f n \<Sigma>2 # st2) opt"
lookup_empty: fixes k :: "'d" shows "Mapping.lookup Mapping.empty k = None"
supp_eqvt_at: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "eqvt_at f x"   and "finite (supp x)" shows "supp (f x) \<subseteq> supp x"
sortedP_Cons: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "linorder_list0.sortedP le (x # xs)" obtains "\<forall>xa\<in>set xs. le x xa" and "linorder_list0.sortedP le xs"
lset_lconcat_subset: fixes xss :: "'c llist llist" shows "lset (lconcat xss) \<subseteq> \<Union> (lset ` lset xss)"
snull_TT_conv: fixes xs :: "[:'a:]" shows "(snull\<cdot>xs = TT) = (xs = [::])"
ineqs_4: shows "\<forall>n\<le>n\<^sub>0. f n \<le> C\<^sub>4"
RID\<^sub>S_def: fixes \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option" shows "RID\<^sub>S \<sigma> \<equiv> \<Union> (RID\<^sub>V ` ran \<sigma>)"
assms_4: shows "prv (neg (subst \<chi> t x))"
small_big_antisym: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> G * H"   and "h2 \<in> G * H"   and "h1 \<in> o[F](h2)"   and "h2 \<in> O[F](h1)" shows "False"
Le_le_LeI: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "Le a \<le> Le b"
simps_8: fixes f :: "'d \<Rightarrow> 'b" shows "map_option f None = None"
and_extract_pure_right_ctx_iff: fixes P :: "assn"   and Q :: "assn"   and b :: "bool" shows "P \<and>\<^sub>A Q * \<up> b = (P \<and>\<^sub>A Q) * \<up> b"
H: shows "ptrm_infer_type \<Gamma> (PFst P) = Some \<tau>"
unapp_rel: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c"   and u :: "'d" assumes "term_struct_rel_strong P"   and "P t u" shows "rel_option (rel_prod P P) (unapp t) (unapp u)"
that: fixes s :: "nat \<Rightarrow> 'a" assumes "\<And>i. P i (s i)"   and "filterlim s at_top sequentially" shows "thesis"
simps_11: fixes r :: "'a rexp"   and s :: "'a rexp"   and v :: "'a rexp" shows "pnPlus (Plus r s) (Star v) = pnPlus r (pnPlus s (Star v))"
ipurge_tr_rev_t_input_1: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" shows "ipurge_tr_rev_t_inv_1 I D u xs \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = []\<rparr>"
test_def1: fixes x1a :: "'a option"   and x2a :: "'b list" shows "test x1a x2a = (case x1a of None \<Rightarrow> case x2a of [] \<Rightarrow> 1::'a | x2da # x1da \<Rightarrow> 2::'a | Some x1ea \<Rightarrow> x1ea)"
response_def: fixes r :: "nat"   and w :: "nat"   and c :: "nat" shows "response r w c = return_spmf ((w * c + r) mod order \<G>)"
ExN: fixes xs :: "real list" shows "(\<exists>a\<in>set (dnf_modified \<phi>). case a of (al, fl, n) \<Rightarrow> \<exists>L. length L = n \<and> (\<forall>a\<in>set al. aEval a (L @ xs)) \<and> (\<forall>f\<in>set fl. eval f (L @ xs))) = eval \<phi> xs"
wf_base_case: fixes V :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and R :: "'a option list set" assumes "card V = 1"   and "rwf_query n V Q Qn"   and "R = genericJoin V Q Qn" shows "table n V R"
assms_1: shows "S \<equiv> trace_rest_closure rest1 rest2 IA IU p q"
abs_complex_real: fixes x :: "complex" shows "\<bar>x\<bar> \<in> \<real>"
star_of_0_le: fixes y :: "'a" shows "(0 \<le> star_of y) = ((0::'a) \<le> y)"
dec_I1_Extend_FO: fixes \<AA> :: "interp"   and P :: "nat fset"   and x :: "WS1S_Nameful.fo"   and bs1 :: "WS1S_Nameful.fo list" shows "dec_I1 (Extend FO 0 \<AA> P) (x # bs1) = (dec_I1 \<AA> bs1)(x := fMin P)"
map_map: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "map f (map g xs) = map (f \<circ> g) xs"
finite_imp_ex_se_succ: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label" assumes "finite (pred c)"   and "\<forall>e\<in>pred c. finite (Bexp.vars e)" shows "\<exists>c'. se c l c'"
lex_fun_trans: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c"   and u :: "'a \<Rightarrow> 'c" assumes "lex_fun s t"   and "lex_fun t u" shows "lex_fun s u"
eq_elapsed_def: fixes u :: "'c \<Rightarrow> 'b"   and u' :: "'c \<Rightarrow> 'b" shows "eq_elapsed u u' \<equiv> (0::'b) < elapsed u u' \<longrightarrow> (\<forall>c\<in>\<X>. u' c - u c = elapsed u u')"
IH: fixes M :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) multiset" shows "\<exists>\<sigma>. (F + M, E' + subst_mset \<sigma> M) \<in> unif\<^sup>*"
rbt_insert_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.rbt_insert less = ord.rbt_insert_with_key less (\<lambda>_ _ nv. nv)"
digit_less_base: fixes x :: "nat"   and i :: "nat" shows "digit x i < base"
assms_3: fixes A :: "'a trm"   and B :: "'a trm" assumes "P A"   and "P B" shows "P (App A B)"
isometry_on_compose: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "isometry_on X f"   and "isometry_on (f ` X) g" shows "isometry_on X (\<lambda>x. g (f x))"
statEqvt'': fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and p :: "(name \<times> name) list"   and M :: "'f"   and N :: "'f" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "p \<bullet> SChanEq M N = SChanEq (p \<bullet> M) (p \<bullet> N)"
right_idem: fixes a :: "'a"   and b :: "'a" shows "a + b + b = a + b"
rangeE: fixes b :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type" assumes "(b::'b::type) \<in> range (f::'c::type \<Rightarrow> 'b::type)" obtains x :: "'c::type" where "(b::'b::type) = (f::'c::type \<Rightarrow> 'b::type) (x::'c::type)"
quasi_inversesI: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f \<cong> g \<star> f"   and "f \<star> g \<cong> trg f" shows "quasi_inverses f g"
qfree_lift_dnf_qe: fixes qe :: "'a list \<Rightarrow> 'a fm"   and \<phi> :: "'a fm" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)" shows "qfree (lift_dnf_qe qe \<phi>)"
CMD_def: shows "CMD \<equiv> CM \<and> CMD_axioms"
edges_setFinal: fixes f :: "face" shows "\<E> (setFinal f) = \<E> f"
simps_3: fixes states :: "hf set"   and init :: "hf set"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps :: "(hf \<times> hf) set"   and more :: "'c" shows "nfa.final \<lparr>nfa.states = states, init = init, final = final, nxt = nxt, eps = eps, \<dots> = more\<rparr> = final"
hom_0: fixes x :: "'a poly poly" assumes "poly_y_x x = 0" shows "x = 0"
R_def: shows "R = {row i A |i. row i A \<noteq> 0}"
t4: shows "typ_of s = typ_of (Abs_fv x \<tau>' t $ Fv x \<tau>')"
emeasure_pmf: fixes M :: "'b pmf" shows "emeasure (measure_pmf M) (set_pmf M) = 1"