hcju/isabelleps huggingface dataset:
progress_e: fixes \<S> :: "s_context"   and i :: "nat"   and vs :: "v list"   and cs_es :: "e list"   and ts' :: "t list"   and s :: "s" assumes "\<S>\<bullet>None \<tturnstile>_ i vs;cs_es : ts'"   and "\<And>k lholed. \<not> Lfilled k lholed [$Return] cs_es"   and "\<And>i k lholed. Lfilled k lholed [$Br i] cs_es \<Longrightarrow> i < k"   and "cs_es \<noteq> [Trap]"   and "\<not> const_list cs_es"   and "store_typing s \<S>" shows "\<exists>a s' vs' es'. \<lparr>s;vs;cs_es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
intro: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" assumes "ContractsWithBilateralSubstitutes Xd Xh Pd Ch"   and "ContractsWithIRC Xd Xh Pd Ch" shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch"
OUT_IN: fixes x :: "'v" assumes "x \<notin> A \<Gamma>"   and "x \<notin> B \<Gamma>" shows "d_OUT f x \<le> d_IN f x"
take_add: fixes i :: "nat"   and j :: "nat"   and xs :: "'c list" shows "take (i + j) xs = take i xs \<cdot> take j (drop i xs)"
unfoldE_1: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "P \<leadsto>\<^sup>^<weakBisim> Q"
assms_3: fixes s :: "'a"   and si :: "'b" shows "<I s si * F> bi si <\<lambda>r. I s si * F * \<up> (r = b s)>\<^sub>t"
stutter_reduced_reduced: fixes \<sigma> :: "nat \<Rightarrow> 'f" shows "\<natural>\<natural>\<sigma> = \<natural>\<sigma>"
adj_sub_assign_fact: fixes i :: "'d"   and e :: "('d, 'e) trm"   and j :: "'f + 'f"   and \<sigma> :: "('d, 'g, 'f, 'h) subst_scheme" assumes "i \<in> SIGT e"   and "j \<in> (case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Inl i \<in> ({Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}) \<inter> {Inl x |x. x \<in> SIGT e}"
provRulesp_provRules_eq: shows "provRulesp = (\<lambda>x xa. (x, xa) \<in> provRules)"
insert_insert_message: fixes M :: "('c, 'd) message set"   and m :: "('c, 'd) insert_message" shows "insert_messages (M \<union> {Insert m}) = insert_messages M \<union> {m}"
hen: shows "local.quadratic_hensel_loop j = (q, S, T, D, H)"
integrable_g: fixes U :: "'b set" assumes "U \<in> sets borel"   and "U \<subseteq> X'" shows "set_integrable lborel U g"
nth_prime_eqI: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "card {q. prime q \<and> q < p} = n" shows "nth_prime n = p"
sup_absorb1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "sup x y = x"
simps: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "run\<^sub>t \<delta> q\<^sub>0 w i = (run \<delta> q\<^sub>0 w i, w i, run \<delta> q\<^sub>0 w (Suc i))"
simps_2: fixes k :: "nat"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "almost_order k (Node ts t) = (length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t)"
simps_1: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and f\<^sub>l :: "real"   and f\<^sub>r :: "real"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" shows "down' d 0 p f\<^sub>l f\<^sub>r \<alpha> = \<alpha>"
ltl\<^sub>P_abs_rep: fixes \<phi> :: "'b ltl_prop_equiv_quotient" shows "Abs (Rep \<phi>) = \<phi>"
Seed_holds_minGraphProps': fixes p :: "nat" shows "minGraphProps' Seed\<^bsub>p\<^esub>"
rcp_specD1: fixes rcp :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "rcp_spec rcp" shows "0 \<notin> fst ` set (fst (rcp gs bs ps sps data))"
exhaust: fixes y :: "'a trm" obtains x1 :: "Preliminaries.var" where "y = Var x1"   | x21 :: "'a" and x22 :: "'a trm list" where "y = Fn x21 x22"
reduce_rulesD_2: fixes r :: "('a, 'b) ta_rule"   and \<delta> :: "('a, 'b) ta_rule set"   and P :: "'a set"   and q :: "'a" assumes "r \<in> reduce_rules \<delta> P"   and "q \<in> rule_states r" shows "q \<in> P"
simps_2: fixes t :: "'b tree" shows "T_merge t \<langle>\<rangle> = 1"
integral_const: fixes a :: "'b"   and b :: "'b"   and c :: "'a" shows "integral (cbox a b) (\<lambda>x. c) = content (cbox a b) *\<^sub>R c"
of_bl_length_less: fixes x :: "bool list"   and k :: "nat" assumes "length x = k"   and "k < LENGTH('b)" shows "of_bl x < 2 ^ k"
one_sum_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" shows "\<one>\<^bsub>sum_group I G\<^esub> = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)"
ag_pOp_add_l: fixes A :: "('d, 'e) aGroup_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "a = b" shows "c \<plusminus>\<^bsub>A\<^esub> a = c \<plusminus>\<^bsub>A\<^esub> b"
marg_dens_def: shows "marg_dens = (\<lambda>(V, V', \<Gamma>, \<delta>) x \<rho> v. \<integral>\<^sup>+ \<sigma>. \<delta> (merge V V' (\<sigma>(x := v), \<rho>)) \<partial>state_measure (V - {x}) \<Gamma>)"
varsPB_def: fixes \<Phi> :: "('a, 'b) lit list set" shows "varsPB \<Phi> = \<Union> {varsC c |c. c \<in> \<Phi>}"
pseudo_proj_preimage': fixes g :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure"   and z :: "bool stream" assumes "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N" shows "pseudo_proj_True n -` g -` {g z} = g -` {g z}"
g: fixes x :: "'d" shows "spmf.mono_body (\<lambda>f. G f x)"
cmp_prod_alt: shows "cmp_prod = (\<lambda>cmp1 cmp2 (a1, a2) (b1, b2). case cmp1 a1 b1 of LESS \<Rightarrow> LESS | EQUAL \<Rightarrow> cmp2 a2 b2 | GREATER \<Rightarrow> GREATER)"
measure_pmf_in_subprob_algebra: fixes x :: "'b pmf" shows "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
transversal_segment_exists: fixes x :: "'a" assumes "x \<in> X"   and "(- f) x \<noteq> (0::'a)" obtains a :: "'a" and b :: "'a" where "x \<in> {a<--<b}" and "rev.transversal_segment a b"
that_2: assumes "\<And>z. z \<in> inside (path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = - 1" shows "thesis"
cr_envT_stateT_eval: fixes X :: "'b \<Rightarrow> int option" shows "((=) ===> cr_envT_stateT X (rel_optionT (cr_id_prob (rel_option (cr_prod1 X (=)))))) (RFI.eval RFI_lookup) (SFP.eval SFP_lookup)"
tag_Star_empty_injI: fixes A :: "'a list set"   and x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "tag_Star A x = tag_Star A y"   and "x @ z \<in> A\<star>"   and "x = []" shows "y @ z \<in> A\<star>"
borel_measurable_partial_derivatives: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'b) vec"   and m :: "'b"   and n :: "'a" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)" shows "(\<lambda>x. matrix (f' x) $ m $ n) \<in> borel_measurable (lebesgue_on S)"
fv_get_and: fixes b :: "nat"   and \<phi> :: "formula" shows "\<Union> (fvi b ` set (get_and_list \<phi>)) = fvi b \<phi>"
ComplD: fixes c :: "'f"   and A :: "'f set" assumes "c \<in> - A" shows "c \<notin> A"
signed_real_of_word_sum_less_eq_neg_numeral_iff: fixes v :: "'a Word.word"   and w :: "'b Word.word"   and n :: "num" shows "(signed_real_of_word v + signed_real_of_word w \<le> - numeral n) = (sint v + sint w \<le> - numeral n)"
no_jmp: shows "\<forall>j. fst s0 = Some (Jump j) \<longrightarrow> j \<in> jmps"
step_Nil: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and ass :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> ([], ass @ css, s) \<rightarrow> (cs', css', t)"   and "ass \<noteq> []" shows "css' = tl ass @ css \<and> (case s of Abrupt s' \<Rightarrow> cs' = snd (hd ass) \<and> t = Normal s' | _ \<Rightarrow> cs' = fst (hd ass) \<and> t = s)"
P_of_Liminf_state_inf: shows "P_of_state (Liminf_state Sts') \<subseteq> P_of_state (Liminf_state Sts)"
strict_border_simps_2: fixes x :: "'a"   and xs :: "'a list" shows "strict_border [] (x # xs) = True"
some_H'h't: fixes M :: "('c \<times> real) set set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real"   and c :: "('c \<times> real) set set"   and H :: "'c set"   and h :: "'c \<Rightarrow> real"   and x :: "'c" assumes "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "graph H h = \<Union> c"   and "x \<in> H" shows "\<exists>H' h'. graph H' h' \<in> c \<and> (x, h x) \<in> graph H' h' \<and> linearform H' h' \<and> Subspace.subspace H' E \<and> Subspace.subspace F H' \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x\<in>H'. h' x \<le> p x)"
xor_int_rec: shows "x BIT b XOR y BIT c = of_bool (odd (x BIT b) \<noteq> odd (y BIT c)) + 2 * (x BIT b div 2 XOR y BIT c div 2)"
i: shows "has_bochner_integral lebesgue (\<lambda>x. indicat_real {a..b} x *\<^sub>R f x) i"
emeasure_subadditive: fixes A :: "'c set"   and M :: "'c measure"   and B :: "'c set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
crdt_ops_rem_last: fixes xs :: "('a \<times> 'b) list"   and x :: "'a \<times> 'b"   and deps :: "'b \<Rightarrow> 'a set" assumes "crdt_ops (xs @ [x]) deps" shows "crdt_ops xs deps"
rcsTr6: fixes H :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<cdot> \<rho> a \<in> H"   and "x \<in> H \<bullet> a" shows "x \<in> H \<bullet> b"
assms_2: fixes a :: "real" shows "continuous (at_right a) F"
ys: shows "ys = map (\<lambda>(n, d). n * dd div d) nds"
get_parent_def: shows "Shadow_DOM.get_parent \<equiv> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent Shadow_DOM.get_child_nodes"
path_component_of_def: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "path_component_of X x y \<equiv> \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
divide_eq_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(divide b c = a) = (if c \<noteq> zero then b = times a c else a = zero)"
map_cong: fixes x :: "'a \<Rightarrow> 'b"   and ya :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. s2p (range ya) z \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> ya"
less_add_same_cancel1: fixes a :: "'a"   and b :: "'a" shows "(a < a + b) = ((0::'a) < b)"
poly_no_roots_leq_leq: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> poly p x \<noteq> 0) = (b < a \<or> p \<noteq> 0 \<and> poly p a \<noteq> 0 \<and> count_roots_between p a b = 0)"
\<mu>'_def: fixes i :: "nat"   and j :: "nat" shows "gs.\<mu>' i j \<equiv> gs.d (Suc j) * gs.\<mu> i j"
Mlc: shows "M (lead_coeff f) \<in> {1..<p ^ n}"
f2: shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> ((r' |\<^bsub>rel\<^esub> s') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'')) = (s' \<otimes> s'' \<otimes> r \<oplus> s \<otimes> (s'' \<otimes> r' \<oplus> s' \<otimes> r'') |\<^bsub>rel\<^esub> s \<otimes> (s' \<otimes> s''))"
extensional_arb: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b" assumes "f \<in> extensional A"   and "x \<notin> A" shows "f x = undefined"
congruent2_def: fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "congruent2 r1 r2 f = (\<forall>(y1, z1)\<in>r1. \<forall>(y2, z2)\<in>r2. f y1 y2 = f z1 z2)"
zip_same: fixes a :: "'a"   and b :: "'a"   and xs :: "'a list" shows "((a, b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
coboundedI: fixes A :: "'g set"   and a :: "'g" assumes "finite A"   and "a \<in> A" shows "a \<le> Max A"
of_nat_add: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m + n) = plus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
fact2: shows "\<bar>spmf (connect \<A> (RES (lazy_channel_insec' \<oplus>\<^sub>O lazy_channel_send' \<oplus>\<^sub>O lazy_channel_recv_f') (False, Void, None, Map.empty))) True - spmf (connect \<A> (RES (lazy_channel_insec' \<oplus>\<^sub>O lazy_channel_send' \<oplus>\<^sub>O lazy_channel_recv') (False, Void, None, Map.empty))) True\<bar> \<le> Sigma_Algebra.measure (measure_spmf (game \<A> lazy_channel_recv_f')) {x. snd x}"
lm018: fixes R :: "('a \<times> 'b) set"   and X1 :: "'a set"   and X2 :: "'a set" shows "Outside' X2 (Outside' X1 R) = Outside' X1 (Outside' X2 R)"
less_add_same_cancel1: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subset># a + b) = ({#} \<subset># b)"
P6_b: fixes x :: "'a"   and y :: "'a" shows "x r\<rightarrow> y l\<rightarrow> x = (1::'a)"
PP_def: shows "PP \<equiv> (B ===> B ===> B) ===> B ===> (B ===> B ===> B) ===> (B ===> B) ===> ((=) ===> B ===> B) ===> (B ===> B ===> B) ===> B ===> (B ===> B ===> B) ===> (B ===> B) ===> ((=) ===> B ===> B) ===> (B ===> B) ===> (=)"
mat_mult_seq_def: fixes A :: "complex mat"   and X :: "nat \<Rightarrow> complex mat" shows "mat_mult_seq A X = (\<lambda>n. A * X n)"
IH: fixes r :: "Core.term"   and s :: "Core.term" assumes "\<forall>x\<in>set u. is_closed x"   and "r \<rightarrow>\<^sub>\<beta> s" shows "subst_bvs1' r k u \<rightarrow>\<^sub>\<beta> subst_bvs1' s k u"
object_ptr_kinds_def: shows "object_ptr_kinds = fmdom \<circ> the_heap"
j0_not_eq_j1: shows "J.j0 \<noteq> J.j1"
fst_netgmap_none: fixes i :: "nat"   and s :: "'a net_state"   and sr :: "'a \<Rightarrow> 'b \<times> 'c" assumes "i \<notin> net_ips s" shows "fst (netgmap sr s) i = None"
list_of_vec_index: fixes v :: "'a Matrix.vec"   and j :: "nat" shows "list_of_vec v ! j = v $ j"
flat_interpretation: fixes b :: "'f" shows "partial_function_definitions (flat_ord b) (flat_lub b)"
fpairsE: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> fpairs r" obtains a :: "V" and b :: "V" where "x = [a, b]\<^sub>\<circ>" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
affine_hull_translation: fixes a :: "'c"   and S :: "'c set" shows "affine hull (+) a ` S = (+) a ` (affine hull S)"
safe_isnFDERIV: fixes x :: "'c"   and ys :: "real list"   and n :: "nat" assumes "x \<in> Csafe" shows "isnFDERIV DIM('c) ode_e [0..<DIM('c)] [DIM('c)..<2 * DIM('c)] (list_of_eucl x @ ys) n"
N1_enabled_at_both_a: shows "\<turnstile> pc2 = #a \<and> PsiInv \<and> pc1 = #a \<longrightarrow> Enabled (<N1>_(x, y, sem, pc1, pc2))"
seq_if_composable: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition_with_homs V H src trg"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "src \<nu> = trg \<mu>"
inner_dirsumD: fixes Ms :: "'b set list" assumes "add_independentS Ms" shows "(\<Oplus>M\<leftarrow>Ms. M) = (\<Sum>M\<leftarrow>Ms. M)"
assms: shows "\<forall>f\<in>funs_term t. \<not> is_Val f"
simps_27: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat" shows "(case Rrep x21 x22 x23 x24 x25 of Rreq x xa xb xc xd xe xf xg xh \<Rightarrow> f1 x xa xb xc xd xe xf xg xh | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f2 x21 x22 x23 x24 x25"
integrableI_bounded: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "f \<in> borel_measurable M"   and "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>" shows "integrable M f"
s: shows "s = s \<inter> h -` {..a} \<union> s \<inter> h -` {a..}"
is_faithful_functor: shows "faithful_functor S.comp C S.map"
simps_7: fixes x31 :: "acom"   and x32 :: "acom" shows "SKIP \<noteq> x31;; x32"
vifunion_vempty_conv_2: fixes I :: "V"   and f :: "V \<Rightarrow> V" shows "(\<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I) = 0) = (\<forall>i\<in>elts I. f i = 0)"
minus_r_inv_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A - A = 0\<^sub>m nr nc"
partial_get_put: fixes x :: "'d \<Longrightarrow> 'e"   and \<rho> :: "'e"   and \<sigma> :: "'e" assumes "pbij_lens x"   and "\<rho> \<in> \<S>\<^bsub>x\<^esub>" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>"
lift_Suc_mono_le: fixes f :: "nat \<Rightarrow> 'b list"   and n :: "nat"   and n' :: "nat" assumes "\<And>n. prefix (f n) (f (Suc n))"   and "n \<le> n'" shows "prefix (f n) (f n')"
Ri_subs_Ri_diff_Rf: fixes \<Gamma> :: "'b inference set"   and N :: "'b literal multiset set" shows "standard_redundancy_criterion.Ri \<Gamma> N \<subseteq> standard_redundancy_criterion.Ri \<Gamma> (N - standard_redundancy_criterion.Rf N)"
AssociationEndCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and role :: "String.literal option"   and "from" :: "String.literal"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E AssociationEndCall src k role from : \<tau>" obtains \<C> :: "'a" and \<D> :: "'a" and "end" :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" where "k = DotCall" and "\<tau> = assoc_end_type end" and "\<Gamma> \<turnstile>\<^sub>E src : \<langle>\<C>\<rangle>\<^sub>\<T>[1]" and "association_end \<C> role from \<D> end"
subset_mset_zmsetE: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" assumes "M \<subset>#\<^sub>z N" obtains A :: "'b multiset" and B :: "'b multiset" and C :: "'b zmultiset" where "M = zmset_of A + C" and "N = zmset_of B + C" and "A \<subset># B"
inputPres: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and M :: "'a"   and N :: "'a" assumes "\<And>Tvec. length xvec = length Tvec \<Longrightarrow> (\<Psi>, P[xvec::=Tvec], Q[xvec::=Tvec]) \<in> Rel" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<leadsto>[Rel] M\<lparr>\<lambda>*xvec N\<rparr>.Q"
Least_def: fixes P :: "'a \<Rightarrow> bool" shows "dual_wellorder.Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> y \<le> x))"
bind_spmf_const: fixes p :: "'m spmf"   and q :: "'l spmf" shows "p \<bind> (\<lambda>x. q) = scale_spmf (weight_spmf p) q"
Spec: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> special_axioms" shows "H \<turnstile> A"
assms_4: shows "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"
leftderives1_implies_derives1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftderives1 \<NN> \<TT> \<RR> u v" shows "CFG.derives1 \<NN> \<TT> \<RR> u v"
list_set_bind: fixes S :: "'e set"   and g :: "'e \<Rightarrow> 'f set"   and xs :: "'g list"   and A :: "('g \<times> 'e) set"   and f :: "'g \<Rightarrow> 'h list"   and B :: "('h \<times> 'f) set" assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; x \<noteq> y\<rbrakk> \<Longrightarrow> g x \<inter> g y = {}"   and "(xs, S) \<in> \<langle>A\<rangle>list_set_rel"   and "(f, g) \<in> A \<rightarrow> \<langle>B\<rangle>list_set_rel" shows "(xs \<bind> f, S \<bind> g) \<in> \<langle>B\<rangle>list_set_rel"
divisor_sum_altdef: fixes n :: "nat" shows "divisor_sum n = divisor_sigma 1 n"
iso_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.iso f = (VxV.iso f \<and> VV.arr f \<and> VV.arr (VxV.inv f))"
uint8_shiftr_def: fixes x :: "uint8"   and n :: "integer" shows "uint8_shiftr x n = (if n < 0 \<or> 8 \<le> n then undefined (>>) x n else x >> nat_of_integer n)"
three_neq_6: shows "\<^bold>3 \<noteq> \<^bold>2"
tb_3: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>b"
norm1_ge_norm: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "norm v \<le> norm1 v"
irreflp_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "irreflp R = (\<forall>a. \<not> R a a)"
conductor_induced: shows "\<Phi>.induced_modulus \<Phi>.conductor"
nodes_list: fixes xs :: "('c, 'd) Node list" assumes "2 \<le> length xs" shows "nlistToList (nodes xs) = nlistToList xs"
dm: shows "rhs * n + m = \<lfloor>r\<rfloor>"
list_set_cons_def: fixes a :: "'b"   and S :: "'b list set" shows "a \<cdot> S = (#) a ` S"
le_RES_nofailI: fixes a :: "'c nres"   and x :: "'c set" assumes "a \<le> RES x" shows "nofail a"
simps_1: fixes uu :: "bool \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and uv :: "bool" shows "pairs uu uv [] = []"
in_spanE: fixes v :: "'a Matrix.vec"   and W :: "'a Matrix.vec set" assumes "v \<in> NC.span W" shows "\<exists>a A. v = NC.lincomb a A \<and> finite A \<and> A \<subseteq> W"
has_limits_of_shape_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "has_limits_of_shape J \<equiv> \<forall>D. diagram J (\<cdot>) D \<longrightarrow> (\<exists>a \<chi>. limit_cone J (\<cdot>) D a \<chi>)"
k: shows "real_of_int k = real_of_float (floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))))"
zero_a_k: shows "is_zero_row_upt_k a k A"