hcju/isabelleps huggingface dataset:
conv_abscissa_add_le: fixes f :: "'a fds"   and g :: "'a fds" shows "conv_abscissa (f + g) \<le> max (conv_abscissa f) (conv_abscissa g)"
that: fixes u :: "real" assumes "0 \<le> u"   and "u \<le> 1"   and "g u \<notin> interior S"   and "u = 0 \<or> (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S" shows "thesis"
scale_zero_left: fixes x :: "'a" shows "0 *R x = (0::'a)"
M: shows "sets M = sets (subprob_algebra (subprob_algebra N))"
Pi_pmf_if_set': fixes A :: "'b set"   and dflt :: "'c"   and b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c pmf" assumes "finite A" shows "Pi_pmf A dflt (\<lambda>x. if b x then return_pmf dflt else f x) = Pi_pmf {x \<in> A. \<not> b x} dflt f"
delete_type: fixes t :: "'a bro"   and h :: "nat"   and x :: "'a" assumes "t \<in> B h" shows "delete x t \<in> B h \<union> B (h - 1)"
preserves_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src\<^sub>C (F \<mu>) = F.map\<^sub>0 (src\<^sub>B \<mu>)"
right_cancel: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "group f z inverse" shows "(f b a = f c a) = (b = c)"
dbl_inc_simps_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one one = numeral.numeral one plus (num.Bit1 num.One)"
bubble_sort: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> y" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
c_hd_aux1: fixes u :: "nat" assumes "c_len u = 1" shows "c_hd u = c_snd (u - 1)"
assms_2: shows "\<exists>k. \<forall>x\<in>MPLS PROB. x < k"
inject_5: fixes x7 :: "('a, 'b) form"   and y7 :: "('a, 'b) form" shows "(Neg x7 = Neg y7) = (x7 = y7)"
insert_compr: fixes a :: "'b::type"   and B :: "'b::type set" shows "insert (a::'b::type) (B::'b::type set) = {x::'b::type. x = a \<or> x \<in> B}"
spr_abs_tLastD: fixes v :: "'s"   and t :: "'s Trace"   and a :: "'a" assumes "v \<in> spr_abs t" shows "envObs a v = envObs a (tLast t)"
comm_group_abelian_groupI: fixes G :: "('c, 'd) ring_scheme" assumes "comm_group (add_monoid G)" shows "abelian_group G"
open_map_imp_quotient_map: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "T \<subseteq> f ` S"   and "\<And>T. openin (top_of_set S) T \<Longrightarrow> openin (top_of_set (f ` S)) (f ` T)" shows "openin (top_of_set S) (S \<inter> f -` T) = openin (top_of_set (f ` S)) T"
simps_4: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "is_heap (T v (T va vb vc) E) = (val (T va vb vc) \<le> v \<and> is_heap (T va vb vc))"
that: fixes b :: "nat" assumes "local.prefixes t j \<down>= b" shows "thesis"
Map_in_Hom: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f" shows "concrete_category.Map f \<in> Hom (concrete_category.Dom f) (concrete_category.Cod f)"
continuous_at: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "isCont f x = f \<midarrow>x\<rightarrow> f x"
partizan_r: shows "\<forall>e. Elem e r \<longrightarrow> e \<in> games_lfp"
word_pred_rbl: fixes w :: "'b word"   and bl :: "bool list" assumes "to_bl w = bl" shows "to_bl (word_pred w) = rev (rbl_pred (rev bl))"
entt_star_true_simp_1: fixes A :: "assn"   and B :: "assn" shows "(A \<Longrightarrow>\<^sub>t B * true) = (A \<Longrightarrow>\<^sub>t B)"
infnorm_triangle: fixes x :: "'a"   and y :: "'a" shows "infnorm (x + y) \<le> infnorm x + infnorm y"
ide_in_hom: fixes a :: "'a" shows "Adj.C.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>C a\<guillemotright>"
edge_succ_w_2: shows "edge_succ HM (rev_H vw) = rev_H uw"
simps: fixes M :: "('e, 'f, 'g) FSM"   and q :: "'g"   and xs :: "'e list" shows "language_state_for_input M q xs = {xs || ys |ys. length xs = length ys \<and> xs || ys \<in> LS M q}"
linear_order_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
Class_in_Partition: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c" assumes "equivalence S E"   and "a \<in> S" shows "equivalence.Class S E a \<in> S / E"
externals_def: fixes asig :: "'e signature" shows "externals asig \<equiv> inputs asig \<union> outputs asig"
integral_divide_zero: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'b"   and c :: "'b" shows "LINT x|M. f x / c = integral\<^sup>L M f / c"
eq: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y = Eq) = (x = y)"
inject: fixes x1 :: "'d"   and x2 :: "'e"   and x3 :: "'d list"   and y1 :: "'d"   and y2 :: "'e"   and y3 :: "'d list" shows "(x1 \<rightarrow> x2 x3 = y1 \<rightarrow> y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
bex_triv_one_point1: fixes A :: "'a set"   and a :: "'a" shows "(\<exists>x\<in>A. x = a) = (a \<in> A)"
assert_nsc_def: fixes assert_all_s :: "('a \<times> 'b) list \<Rightarrow> 'd \<Rightarrow> 'a list + 'e"   and i :: "'a" shows "assert_nsc assert_all_s i = (\<lambda>((asi, tv, ui), s). if i \<in> set ui then Inl [i] else case assert_all_s (list_map_to_fun asi i) s of Inl I \<Rightarrow> Inl I | Inr s' \<Rightarrow> Inr ((asi, tv, ui), s'))"
Cauchy_convergent_iff: fixes X :: "nat \<Rightarrow> 'e" shows "Cauchy X = convergent X"
Fun_mkArr'_1: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "partial_magma.in_hom S (set_category_given_img.mkArr S img \<AA> A B F) (set_category_given_img.mkIde S img \<AA> A) (set_category_given_img.mkIde S img \<AA> B)"
Derives1_rule: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "r \<in> \<RR>"
dom_eq_empty_conv: fixes f :: "'b \<Rightarrow> 'c option" shows "(dom f = {}) = (f = \<emptyset>)"
nat_filtration_space: fixes n :: "nat" shows "space (nat_filtration n) = UNIV"
simps_1: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and x :: "'a"   and i :: "nat" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.coef_hd coef_sym (Var x) i = PVar (PCoef x i)"
tg_def: shows "e' 0 \<in>\<^sub>i range_tm e (tm_sub tg (tm_const ga))"
Kripke_axioms: shows "Kripke S s0 \<delta> AP L"
lt2p_lem: fixes n :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n" shows "uint w < 2 ^ n"
avl_dist: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat"   and x :: "'b" assumes "avl (MKT n l r h)"   and "is_ord (MKT n l r h)"   and "x \<in> set_of l" shows "x \<notin> set_of r"
assms_1: shows "optim_instr n instr \<Sigma>1 = Ok (instr', \<Sigma>2)"
or_and_s: fixes w1 :: "word32"   and w2 :: "word32" assumes "w1 AND 128 = 0 \<and> w2 AND 128 = 0" shows "(w1 OR w2) AND 128 = 0"
strict_implies_order: fixes b :: "('d, 'e) Term.term"   and a :: "('d, 'e) Term.term" assumes "b \<rhd> a" shows "b \<unrhd> a"
da_elim_cases_15: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Throw e\<rangle>\<guillemotright> A" obtains E :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "nrm A = UNIV" and "brk A = (\<lambda>l. UNIV)"
compress_normalize_output_interfaces_def: fixes m :: "'c common_primitive match_expr" shows "compress_normalize_output_interfaces m \<equiv> compress_normalize_primitive (is_Oiface, oiface_sel) OIface compress_interfaces m"
maxR_absorb1: fixes r :: "('a \<times> 'a) set"   and y :: "'a"   and x :: "'a" assumes "MaxR r"   and "(y, x) \<in> r" shows "MaxR.maxR r x y = x"
isOK_I: fixes m :: "'a + 'b"   and x :: "'b" assumes "m = Inr x" shows "isOK m"
IH1: fixes T :: "secLevel" assumes "\<Gamma> \<turnstile> e1 : T"   and "x \<in> fv e1"   and "\<Gamma> x = Some High" shows "\<Gamma> \<turnstile> e1 : High"
wf_envs_update: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and P' :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and db :: "(String.literal, event_data list set list) mapping"   and m :: "nat"   and \<phi> :: "Formula.formula"   and xs :: "event_data option list set list"   and p :: "String.literal" assumes "wf_envs \<sigma> j P P' V db"   and "m = Formula.nfv \<phi>"   and "{0..<m} \<subseteq> fv \<phi>"   and "list_all2 (\<lambda>i. qtable m (fv \<phi>) (mem_restr UNIV) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) [Monitor.progress \<sigma> P \<phi> j..<Monitor.progress \<sigma> P' \<phi> (Suc j)] xs" shows "wf_envs \<sigma> j (P(p \<mapsto> Monitor.progress \<sigma> P \<phi> j)) (P'(p \<mapsto> Monitor.progress \<sigma> P' \<phi> (Suc j))) (V(p \<mapsto> \<lambda>i. {v. length v = m \<and> Formula.sat \<sigma> V v i \<phi>})) (Mapping.update p (map ((`) (map the)) xs) db)"
l4port_logic: fixes xc :: "16 prefix_match"   and a :: "16 word"   and b :: "16 word"   and xaa :: "(16 word \<times> 16 word) option"   and pp :: "16 word" assumes "xc \<in> set (wordinterval_CIDR_split_prefixmatch (WordInterval a b))"   and "xaa = Some (pfxm_prefix xc, ~~ pfxm_mask xc)"   and "pp && ~~ pfxm_mask xc = pfxm_prefix xc" shows "a \<le> pp \<and> pp \<le> b"
hyps: assumes "fold (\<or>?) l true = invalid" shows "fold (\<or>?) (rev l) false = invalid"
list_update_append1: fixes i :: "nat"   and xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "i < length xs" shows "(xs @ ys)[i := x] = xs[i := x] @ ys"
1: shows "(\<rightarrow>) (freeword_funlift signed_lconjpermutation r) = signed_list_lconjaction (pair_relator_list s t)"
distinct_29: fixes x5 :: "String.literal"   and x6 :: "ty" shows "Class x5 \<noteq> x6\<lfloor>\<rceil>"
subtopology_Times: fixes X :: "'c topology"   and Y :: "'d topology"   and S :: "'c set"   and T :: "'d set" shows "subtopology (prod_topology X Y) (S \<times> T) = prod_topology (subtopology X S) (subtopology Y T)"
integrable_discrete_difference: fixes X :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "integrable M f = integrable M g"
tr_rev_step_3: shows "c' \<turnstile> ev \<mapsto> c''"
lincomb_index: fixes i :: "nat"   and X :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "i < n"   and "X \<subseteq> carrier_vec n" shows "lincomb a X $ i = (\<Sum>x\<in>X. a x * x $ i)"
in_gamma_join_UpI: fixes s :: "char list \<Rightarrow> int"   and S1 :: "'av Abs_State.st option"   and S2 :: "'av Abs_State.st option" assumes "s \<in> \<gamma>\<^sub>o S1 \<or> s \<in> \<gamma>\<^sub>o S2" shows "s \<in> \<gamma>\<^sub>o (S1 \<squnion> S2)"
outstanding_non_volatile_write_disj: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<inter> (outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>j \<union> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb\<^sub>j \<union> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) \<union> (outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) - read_only_reads \<O>\<^sub>j (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j)) \<union> (\<O>\<^sub>j \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j))) = {}"
prec_split1_aux: fixes A :: "assn"   and B :: "assn" shows "A * B * true \<Longrightarrow>\<^sub>A A * true"
bind_refine': fixes M :: "'a nres"   and R' :: "('a \<times> 'b) set"   and M' :: "'b nres"   and f :: "'a \<Rightarrow> 'c nres"   and R :: "('c \<times> 'd) set"   and f' :: "'b \<Rightarrow> 'd nres" assumes "M \<le> \<Down> R' M'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; inres M x; inres M' x'; nofail M; nofail M'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "M \<bind> f \<le> \<Down> R (M' \<bind> f')"
nba_g_E_rtrancl: fixes A :: "('d, 'c) nba" shows "(g_E (nba_g A))\<^sup>* = {(p, q). q \<in> reachable A p}"
condCatch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and b :: "'e set"   and R :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 Q,(b \<inter> R \<union> - b \<inter> A)"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P condCatch c\<^sub>1 b c\<^sub>2 Q,A"
pi': shows "set pi \<subseteq> (supp p \<union> supp p') \<times> (supp p \<union> supp p')"
assms_4: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
reduced_def: fixes n :: "nat"   and x :: "nat \<Rightarrow> nat" shows "reduced n x = (LEAST k. k = n \<or> x k \<noteq> 0)"
Bno: fixes x :: "'a" assumes "x \<in> S" shows "norm x \<le> B"
closure_subset_eq: fixes S :: "'a set" shows "(closure S \<subseteq> S) = closed S"
circuit_in_min_dep_in: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set"   and x :: "'b" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.circuit_in indep \<E> X"   and "x \<in> X" shows "indep_system.indep_in indep \<E> (X - {x})"
A_def: shows "A \<equiv> \<lambda>N. {(m, n). (m, n) \<in> {1..N} \<times> {1..N} \<and> coprime m n}"
reach_obs_upd: fixes x :: "'e \<Rightarrow> 'f"   and S :: "('e, 'f, 'g) spec_scheme" shows "reach (S\<lparr>obs := x\<rparr>) = reach S"
exchange_injective: fixes e :: "'a"   and v :: "'a"   and w :: "'a" assumes "(e \<sqinter> top \<sqinter> id (e \<sqinter> top) \<le> top \<and> top \<sqinter> (e \<sqinter> top) = top) \<and> id e \<sqinter> top \<sqinter> id (id e \<sqinter> top) \<le> top \<and> top \<sqinter> (id e \<sqinter> top) = top"   and "e \<le> v \<sqinter> - id v"   and "w \<sqinter> id w \<le> top \<and> w \<sqinter> top \<le> - top"   and "v \<sqinter> top = v" shows "(w \<sqinter> - (w \<sqinter> - id v \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> id (w \<sqinter> (- v \<sqinter> - id v) \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> e) \<sqinter> id (w \<sqinter> - (w \<sqinter> - id v \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> id (w \<sqinter> (- v \<sqinter> - id v) \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> e) \<le> top"
zero_floatarith_def: shows "0 = floatarith.Num 0"
lcos_m_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "g <# (h <# M) = g \<otimes> h <# M"
of_nat_0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 0 = zero"
oc1: shows "fs ! i - (fs ! i - v) \<in> orthogonal_complement (local.span (gso ` {0..<i}))"
uninitAborts_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and s :: "('e, 'f, 'g) SLin_state" assumes "SLin \<delta> \<gamma> bot" shows "SLin.uninitAborts \<delta> s \<equiv> {d. \<exists>ivs\<in>initSets s. \<exists>rs\<in>SLin.pendingSeqs s. d = pre_RDR.exec \<delta> (semilattice_set.F (pre_RDR.inf \<delta>) ivs) rs}"
simps_9: fixes v :: "Term.var" shows "assignsE (Acc v) = assignsV v"
rep_eq: fixes x :: "'a reg_lan"   and xa :: "'a reg_lan" shows "Rep_reg_lan (x \<cdot> xa) = Rep_reg_lan x \<cdot> Rep_reg_lan xa"
subst_poly_lec_add: fixes \<sigma> :: "nat \<Rightarrow> linear_poly"   and c1 :: "'a le_constraint"   and c2 :: "'a le_constraint" shows "subst_poly_lec \<sigma> (c1 + c2) = subst_poly_lec \<sigma> c1 + subst_poly_lec \<sigma> c2"
join_shunt: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "(sup x y = top) = less_eq (uminus x) y"
serial_sas_plus_equivalent_to_serial_strips: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi>" shows "STRIPS_Semantics.is_serial_solution_for_problem (\<phi> \<Psi> ) (map (sasp_op_to_strips \<Psi>) \<psi>)"
trm_rep_preserves_lit_semantics: fixes I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"   and t1 :: "'a trm"   and \<sigma> :: "('a \<times> 'a trm) list"   and t2 :: "'a trm"   and L :: "'a literal"   and s :: "'a trm"   and polarity :: "sign"   and \<sigma>' :: "('a \<times> 'a trm) list" assumes "fo_interpretation I"   and "I (t1 \<lhd> \<sigma>) (t2 \<lhd> \<sigma>)"   and "orient_lit_inst L t1 s polarity \<sigma>'"   and "\<not> validate_ground_lit I (subst_lit L \<sigma>)" shows "\<not> validate_ground_lit I (subst_lit (mk_lit polarity (Eq t2 s)) \<sigma>)"
seq_char: fixes g :: "'a"   and f :: "'a" shows "S.seq g f = (S.arr f \<and> S.arr g \<and> B.seq g f)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>trg f : trg a \<Rightarrow> trg b\<guillemotright>"
weakly_sequential_def: fixes P :: "'c option process" shows "weakly_sequential P \<equiv> \<forall>xs\<in>traces P. None \<notin> set (butlast xs)"
inference_computation: fixes N :: "'a literal multiset set"   and Q :: "'a literal multiset set"   and C :: "'a literal multiset"   and P :: "'a literal multiset set" assumes "N = concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) Q C)" shows "({}, P \<union> {C}, Q) \<leadsto> (N, P, Q \<union> {C})"
ladder_early_stage_4: fixes index :: "nat" assumes "index + 1 < length L" shows "ladder_j L index < length (ladder_\<gamma> \<alpha> E L' index)"
digraphD_5: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vcard \<CC> = 4\<^sub>\<nat>"
intro: fixes agent :: "'b" assumes "\<And>a. a = agent" shows "FiniteSingleAgentEnvironment_axioms agent"
intros_10: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs\<^sub>1 :: "'c rule list"   and m' :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2)"   and "matches \<gamma> m' p"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> Undecided"
dual_dual_ord: fixes dual :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ord_with_dual dual less_eq" shows "less_eq (dual x) (dual y) = less_eq y x"
LPDi_safe: fixes g :: "safety"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex"   and n :: "nat" assumes "safe_regex Futu g (from_mregex mr \<phi>s)"   and "ms \<in> LPDi n mr" shows "safe_regex Futu g (from_mregex ms \<phi>s)"
case_rews_1: fixes f :: "'e \<rightarrow> 'f \<rightarrow> 'd" shows "Tuple2_case\<cdot>f\<cdot>\<bottom> = \<bottom>"
fresh_at_Nil: fixes t :: "('form, 'rule, 'c, 'd) itnode rose_tree" shows "fresh_at t [] = {}"
lunit'_in_vhom: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and b :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "trg f = b" shows "partial_magma.in_hom V (category.inv V (bicategory.lunit V H \<a> \<i> src trg f)) f (H b f)"
Abs_preservation: fixes \<Gamma> :: "binding list"   and T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and t\<^sub>1\<^sub>2 :: "trm"   and t\<^sub>2 :: "trm"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> (\<lambda>:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet> t\<^sub>2 : T" shows "\<Gamma> \<turnstile> t\<^sub>1\<^sub>2[0 \<mapsto> t\<^sub>2] : T"
disc_compl_I: fixes H :: "ocircline"   and z :: "complex_homo" assumes "out_ocircline H z" shows "z \<in> disc_compl H"
done_only_from_recv_marker: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "channel cid = Some (p, q)"   and "snd (cs c cid) \<noteq> Done"   and "snd (cs c' cid) = Done" shows "ev = RecvMarker cid q p"
insert_first_part: fixes ref :: "'a option"   and r :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and oid :: "'a" assumes "ref = None \<or> ref = Some r \<and> r \<in> set xs" shows "insert_spec (xs @ ys) (oid, ref) = insert_spec xs (oid, ref) @ ys"
lex_prod_def: fixes ra :: "('g \<times> 'g) set"   and rb :: "('h \<times> 'h) set" shows "ra <*lex*> rb = {((a, b), a', b'). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
rep_eq: fixes x :: "uint16" shows "nat_of_uint16 x = unat (Rep_uint16 x)"