hcju/isabelleps huggingface dataset:
xfer_run_aux2: fixes cs :: "nat \<Rightarrow> 'c" assumes "C.is_run cs" obtains as :: "nat \<Rightarrow> 'a" where "A.is_run as" and "\<forall>i. R (as i) (cs i)"
assms_2: shows "s' \<in> set_pmf (action cfg)"
expand: fixes prod :: "'c \<times> 'd"   and prod' :: "'c \<times> 'd" assumes "fst prod = fst prod' \<and> snd prod = snd prod'" shows "prod = prod'"
Entry_no_phis: fixes g :: "'a"   and v :: "'e" shows "Mapping.lookup (phis g) (Entry g, v) = None"
swapWatches_def: fixes clause :: "nat"   and state :: "State" shows "swapWatches clause state \<equiv> state\<lparr>getWatch1 := (getWatch1 state)(clause := getWatch2 state clause), getWatch2 := (getWatch2 state)(clause := getWatch1 state clause)\<rparr>"
derives_trans: fixes a :: "symbol list"   and b :: "symbol list"   and c :: "symbol list" assumes "derives a b"   and "derives b c" shows "derives a c"
sub2: shows "BVO ODE2 \<subseteq> BVO (OProd ODE1 ODE2)"
monoid_def: fixes G :: "('d, 'e) monoid_scheme" shows "Group.monoid G \<equiv> ((\<forall>x y. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G) \<and> (\<forall>x y z. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> z \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z))) \<and> \<one>\<^bsub>G\<^esub> \<in> carrier G \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x) \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x)"
fmadd_empty_1: fixes m :: "('c, 'd) fmap" shows "fmempty ++\<^sub>f m = m"
LcmI: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and b :: "'a" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times a b"   and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times a c) \<Longrightarrow> dvd.dvd times b c"   and "normalize b = b" shows "b = Lcm A"
fundamental_theorem_network_capacity_v2: shows "AE wd in lborel_f weight_space_dim. \<nexists>ws Z. Z < r ^ N_half \<and> (\<forall>inputs. input_sizes (deep_model_l rs) = map dim_vec inputs \<longrightarrow> evaluate_net (insert_weights shared_weights (deep_model_l rs) wd) inputs = evaluate_net (insert_weights shared_weights (shallow_model (rs ! 0) Z (last rs) (2 * N_half - 1)) ws) inputs)"
rev_in_conv: fixes u :: "'b list"   and A :: "'b list set" shows "(rev u \<in> A) = (u \<in> rev ` A)"
Pi_UNIV: fixes A :: "'a set" shows "A \<rightarrow> UNIV = UNIV"
ex231d: fixes x :: "'a" shows "x \<le> x \<sqinter> top \<sqinter> x"
elims_2: fixes x :: "'a \<times> nat"   and xa :: "('a, 'b) event" assumes "event_at x xa" obtains i :: "'a \<times> nat" and m :: "('a, 'b) event" where "x = i" and "xa = m" and "event_pred i ((=) m)"
MaxR_f_code: fixes d :: "'d"   and x :: "'x"   and acc :: "'x option" shows "MaxR.MaxR_f (Pd d) x acc = (if x \<in> Field (Pd d) then Some (case acc of None \<Rightarrow> x | Some y \<Rightarrow> MaxR.maxR (Pd d) x y) else acc)"
uint16_mod_def: fixes x :: "uint16"   and y :: "uint16" shows "uint16_mod x y = (if y = 0 then undefined (mod) x 0 else x mod y)"
equalityI: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
wiringD_2: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and f :: "'i \<Rightarrow> 'k"   and g :: "'l \<Rightarrow> 'j" assumes "wiring \<I> \<I>' cnv (f, g)" shows "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>"
simps_3: fixes \<A> :: "'a ltln set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<A> \<Turnstile>\<^sub>P \<phi> and\<^sub>n \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<and> \<A> \<Turnstile>\<^sub>P \<psi>)"
rd_int_is_rd_ass: fixes P :: "interface" shows "rd_ass (asn P) = rd_int P"
simps_2: fixes u :: "'a tree"   and t :: "'a tree"   and ts :: "'a tree list" shows "insortTree u (t # ts) = (if cachedWeight u \<le> cachedWeight t then u # t # ts else t # insortTree u ts)"
minimal_in_set: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> nat" assumes "xs \<noteq> []" shows "minimal f xs \<in> set xs"
iso_is_retraction: fixes f :: "'a" assumes "B.iso f" shows "B.retraction f"
simps_1: fixes x2 :: "bool"   and y2 :: "bool" shows "(DBool x2 = DBool y2) = (x2 = y2)"
relcompEpair: fixes a :: "'e"   and c :: "'f"   and r :: "('e \<times> 'g) set"   and s :: "('g \<times> 'f) set" assumes "(a, c) \<in> r O s" obtains b :: "'g" where "(a, b) \<in> r" and "(b, c) \<in> s"
prime_multiplicity_other: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "p \<noteq> q" shows "factorial_semiring.multiplicity times one p q = 0"
in_homI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "C.arr f"   and "C.dom f = a"   and "C.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"
FOREACHci_refine_rcg: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi> it \<sigma>; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> \<Phi> it \<sigma>"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; c \<sigma>; c' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> R (f' x' \<sigma>')" shows "FOREACH\<^sub>C\<^bsup>\<Phi>\<^esup> S c f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>C\<^bsup>\<Phi>'\<^esup> S' c' f' \<sigma>0')"
vars_subst_var: fixes subst_var :: "nat \<Rightarrow> linear_poly \<Rightarrow> linear_poly \<Rightarrow> linear_poly"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly"   and lp :: "linear_poly" assumes "SubstVar TYPE('a) subst_var" shows "vars (subst_var x\<^sub>j lp' lp) \<subseteq> vars lp - {x\<^sub>j} \<union> vars lp'"
constrains_UN: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i co A' i" shows "F \<in> \<Union> (A ` I) co \<Union> (A' ` I)"
subset_mset_def: fixes A :: "'k multiset"   and B :: "'k multiset" shows "(A \<subset># B) = (A \<subseteq># B \<and> A \<noteq> B)"
holomorphic_pochhammer: fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and n :: "nat" assumes "f holomorphic_on A" shows "(\<lambda>s. pochhammer (f s) n) holomorphic_on A"
af_letter_lang_congruent: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" assumes "\<phi> \<sim>\<^sub>L \<psi>" shows "af_letter \<phi> \<nu> \<sim>\<^sub>L af_letter \<psi> \<nu>"
leftderives1_implies_LeftDerives1: fixes u :: "symbol list"   and v :: "symbol list" assumes "leftderives1 u v" shows "\<exists>i r. LeftDerives1 u i r v"
sep_conj_true_P: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) (sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) P) = sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) P"
wf_digraph_axioms: shows "wf_digraph (with_proj slim)"
orderE: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" obtains "a = a + b"
normalize_quot_div_unit_right: fixes u :: "'b"   and x :: "'b"   and y :: "'b" assumes "is_unit u" shows "normalize_quot (x, y div u) = (fst (normalize_quot (x, y)) * u, snd (normalize_quot (x, y)))"
V0_def: shows "V\<^sub>0 = {f \<in> \<R>. \<phi> (the (f 0)) = f}"
add_closed_converse_right: fixes G :: "'a set"   and g :: "'a"   and x :: "'a" assumes "Group G"   and "g + x \<in> G"   and "g \<in> G" shows "x \<in> G"
r_pdec22_prim: shows "prim_recfn 1 r_pdec22"
ivl_cong: fixes a :: "'d"   and c :: "'d"   and b :: "'d"   and d :: "'d"   and g :: "'d \<Rightarrow> 'e"   and h :: "'d \<Rightarrow> 'e" assumes "a = c"   and "b = d"   and "\<And>x. \<lbrakk>c \<le> x; x < d\<rbrakk> \<Longrightarrow> g x = h x" shows "sum g {a..<b} = sum h {c..<d}"
fst_apsnd: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'d \<times> 'f" shows "fst (apsnd f x) = fst x"
const_res_subseq_def: fixes k :: "nat"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "const_res_subseq k s = (SOME s'. \<exists>n. is_subseq_of s s' \<and> s' = filtered_seq s (kth_res_equals k n) \<and> (\<forall>m. s' m k = n))"
fmrel_tranclp_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'e \<rightharpoonup>\<^sub>f 'd"   and b :: "'e \<rightharpoonup>\<^sub>f 'd"   and P :: "'e \<rightharpoonup>\<^sub>f 'd \<Rightarrow> bool" assumes "fmrel r\<^sup>+\<^sup>+ a b"   and "reflp r"   and "\<And>y. fmrel r a y \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(fmrel r)\<^sup>+\<^sup>+ a y; fmrel r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
evalc_evaln: fixes c :: "com"   and s :: "state"   and t :: "state" assumes "<c,s> -c-> t" shows "\<exists>n. <c,s> -n-> t"
jm: shows "jordan_matrix n_as \<in> carrier_mat d d"
add_leD2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "order.greater_eq n (m + k)" shows "order.greater_eq n k"
finite_cone_def: fixes Vs :: "'a vec set" shows "n.finite_cone Vs = {b. \<exists>c. n.nonneg_lincomb c (if finite Vs then Vs else {}) b}"
simps_5: fixes x :: "octo"   and y :: "octo" shows "Im4 (x - y) = Im4 x - Im4 y"
induct: fixes P :: "'a atom formula \<Rightarrow> bool"   and a0 :: "'a atom formula" assumes "\<And>uu_ uv_. P (Atom (predAtm uu_ uv_))"   and "P \<bottom>"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (\<phi>\<^sub>1 \<^bold>\<and> \<phi>\<^sub>2)"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (\<phi>\<^sub>1 \<^bold>\<or> \<phi>\<^sub>2)"   and "P (\<^bold>\<not> \<bottom>)"   and "\<And>va vb. P (Atom (Eq va vb))"   and "\<And>va. P (\<^bold>\<not> (Atom va))"   and "\<And>va. P (\<^bold>\<not> (\<^bold>\<not> va))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<and> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<or> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<rightarrow> vb))"   and "\<And>v va. P (v \<^bold>\<rightarrow> va)" shows "P a0"
fst_echelon_form_of_det: fixes A :: "(('c, 'd) vec, 'd) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" shows "fst (echelon_form_of_det A bezout) = fst (echelon_form_of_det_iarrays (matrix_to_iarray A) bezout)"
exp_le_one_iff: fixes x :: "real" shows "(exp x \<le> 1) = (x \<le> 0)"
Red_F_of_subset: fixes N :: "'a literal multiset set"   and N' :: "'a literal multiset set" assumes "N \<subseteq> N'" shows "G.Red_F N \<subseteq> G.Red_F N'"
lang_nTimes: fixes r :: "'a rexp"   and s :: "'a rexp" shows "lang (nTimes r s) = lang (Times r s)"
compatible_iff_seq: fixes g :: "'b"   and f :: "'b" shows "(D.domains g \<inter> D.codomains f \<noteq> {}) = D.seq g f"
apply_operations_added_ids: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and f :: "'b \<Rightarrow> 'a set \<times> 'a set"   and x :: "'b" assumes "es prefix of j"   and "apply_operations es = Some f" shows "fst (f x) \<subseteq> set (added_ids es x)"
strong_induct: fixes P :: "'a \<Rightarrow> tyS \<Rightarrow> bool"   and z :: "'a"   and tyS :: "tyS" assumes "\<And>ty z. P z (Ty ty)"   and "\<And>tvar tyS z. \<lbrakk>tvar \<sharp> z; \<And>z. P z tyS\<rbrakk> \<Longrightarrow> P z (\<forall>[tvar].tyS)" shows "P z tyS"
E_validD_2: fixes v :: "'v"   and e :: "'w"   and v' :: "'v" assumes "(v, e, v') \<in> E" shows "v' \<in> V"
positive_Real: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X" shows "Real.positive (Real.Real X) = (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
conf_lemma: fixes m :: "nat"   and r :: "nat"   and t :: "nat" shows "rec_exec rec_conf [m, r, t] = conf m r t"
starlike_negligible_strong: fixes S :: "'c set"   and a :: "'c" assumes "closed S"   and "\<And>c x. \<lbrakk>0 \<le> c; c < 1; a + x \<in> S\<rbrakk> \<Longrightarrow> a + c *\<^sub>R x \<notin> S" shows "negligible S"
coin_def: fixes dc :: "nat option \<times> bool list"   and c :: "nat" shows "coin dc c = snd dc ! (c mod n)"
simps_3: fixes uu :: "('e set \<times> 'e set) list"   and v :: "('f, 'g, 'e) prot_fun"   and va :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and vb :: "'h" shows "equal_mod_timpls uu (Fun v va) (Var vb) = False"
vend_def: shows "vend \<equiv> \<lparr>Label = STR ''vend'', Arity = 0, Guards = [Ge (V (R 2)) (L (Num 100))], Outputs = [V (R 1)], Updates = [(1, V (R 1)), (2, V (R 2))]\<rparr>"
lm131: fixes Y :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "Y \<subseteq> f -` {0::'d}" shows "sum f Y = (0::'d)"
con_comp_divergences_def: fixes P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e" shows "con_comp_divergences P Q p q \<equiv> {xs @ ys |xs ys. set xs \<subseteq> range p \<union> range q \<and> map (inv p) (filter (\<lambda>x. x \<in> range p) xs) \<in> divergences P \<and> map (inv q) (filter (\<lambda>x. x \<in> range q) xs) \<in> divergences Q}"
simps_2: fixes x :: "complex"   and y :: "complex" shows "Im (x - y) = Im x - Im y"
funcset_carrier': fixes f :: "'a \<Rightarrow> 'a"   and A :: "('a, 'b) partial_object_scheme"   and x :: "'a" assumes "f \<in> carrier A \<rightarrow> carrier A"   and "x \<in> carrier A" shows "f x \<in> carrier A"
closure_contains_Inf: fixes S :: "real set" assumes "S \<noteq> {}"   and "bdd_below S" shows "Inf S \<in> closure S"
bt_augment_v_in_2: fixes v :: "'c"   and q :: "('c, 'd) pq"   and ts :: "('d, 'c) bintree list" assumes "v \<in> set |q|" shows "v \<in> set |bts_augment ts q|"
eval_fds_remainder: fixes m :: "nat"   and f :: "'a fds"   and s :: "'a" shows "eval_fds (fds_remainder m f) s = (\<Sum>n. fds_nth f (n + Suc m) / nat_power (n + Suc m) s)"
assms_4: shows "\<forall>\<^sub>F x in at_top. u x \<le> u' x"
open_chart_image: fixes X :: "'b set"   and c :: "('b, 'c) chart" assumes "open X"   and "X \<subseteq> domain c" shows "open (apply_chart c ` X)"
assms: fixes x :: "'a"   and y :: "'a" shows "\<exists>m. dist x m = dist x y / 2 \<and> dist m y = dist x y / 2"
leg0_commutes: shows "\<mu>.cod.leg0 \<cdot> \<mu>.chine = \<mu>.leg0"
minimal: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.pcpo below" shows "below (pcpo.bottom below) x"
assms_2: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>" shows "a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b = b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a"
T_on_embed: fixes s0 :: "'a list"   and qs :: "'a list"   and A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)" shows "real (Move_to_Front.T_on A s0 qs) = Move_to_Front.T_on_rand' (Partial_Cost_Model.embed A) (fst (Partial_Cost_Model.embed A) s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs"
cmod_def: fixes z :: "complex" shows "cmod z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
eq2: shows "punit.lt (rep_list g') = punit.lt (rep_list g)"
N_of_norm_1: fixes f :: "'b" shows "eNorm N_of_norm f = ennreal (norm f)"
F_def: fixes k :: "nat" shows "F k = recurrent_subset_infty (E k)"
diff_cancel: fixes a :: "'r" shows "a - a = (0::'r)"
simps_2: fixes uu :: "'v"   and uv :: "nat" shows "min_passign (PCoef uu uv) = 1"
nn_integral_spmf: fixes A :: "'a set"   and p :: "'a spmf" shows "(\<Sum>\<^sup>+ x\<in>A. ennreal (spmf p x)) = emeasure (measure_spmf p) A"
gcd_left_idem: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd a (gcd a b) = gcd a b"
pow_free_Abelian_group: fixes S :: "'b set"   and x :: "'b \<Rightarrow>\<^sub>0 int"   and n :: "nat" shows "x [^]\<^bsub>free_Abelian_group S\<^esub> n = frag_cmul (int n) x"
unfoldr_rbt_keys_generator: fixes t :: "'c set_rbt" shows "list.unfoldr rbt_keys_generator (RBT_Set2.init t) = RBT_Set2.keys t"
preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>D (G \<mu>) = G.map\<^sub>0 (src\<^sub>C \<mu>)"
permutes_image: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "p ` S = S"
sig_red_rtrancl_mono: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and F' :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q"   and "F \<subseteq> F'" shows "(sig_red sing_reg top_tail F')\<^sup>*\<^sup>* p q"
residue_holomorphic_over_power: fixes A :: "complex set"   and z0 :: "complex"   and f :: "complex \<Rightarrow> complex"   and n :: "nat" assumes "open A"   and "z0 \<in> A"   and "f holomorphic_on A" shows "residue (\<lambda>z. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
reflpD: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "reflp r" shows "r x x"
Vinv2_inductive_1: shows "init v_TS \<subseteq> Vinv2"
r2f_ad_rel_hom: fixes R :: "('a \<times> 'b) set" shows "\<F> (ad_rel R) = kad (\<F> R)"
add_0_l: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "'c point" assumes "ell_field R" shows "ell_field.add R a Infinity p = p"
bal_stackD: fixes step :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and stack :: "'a \<Rightarrow> 'b list"   and c :: "'a"   and T :: "'a list"   and c' :: "'a" assumes "balance_trace step stack"   and "balance_trace.bal step stack c T c'" shows "stack c' = stack c"
a3: shows "bdd_above (L2_set (cmod \<circ> y) ` Collect finite)"
gcd_factorial_dvd1: fixes a :: "'a"   and b :: "'a" shows "gcd_factorial a b dvd a"
fct_gen_subalgebra_def: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c" shows "fct_gen_subalgebra M N X = gen_subalgebra M (sigma_sets (space M) {X -` B \<inter> space M |B. B \<in> sets N})"
less_imp_le: fixes D :: "('d, 'e) Order_scheme"   and a :: "'d"   and b :: "'d" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b" shows "a \<preceq>\<^bsub>D\<^esub> b"
\<C>_vars_consistent: fixes x\<^sub>A :: "'Var\<^sub>A" shows "var\<^sub>C_of ` \<C>_vars\<^sub>A x\<^sub>A = \<C>_vars\<^sub>C (var\<^sub>C_of x\<^sub>A)"
simps: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and a :: "'a" shows "BD_Security.reachNT istate step T a = (a = istate \<or> (\<exists>trn. a = tgtOf trn \<and> BD_Security.reachNT istate step T (srcOf trn) \<and> step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn) \<and> \<not> T trn))"
iT_Mult_iWeakUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<W> t2 I \<otimes> k. Q t2) = (P (t1 * k). t1 \<W> t2 I. Q (t2 * k))"
intros_4: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n\<^sub>0 :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n\<^sub>1 :: "'b SDG_node"   and V :: "'c"   and p :: "'f"   and n\<^sub>2 :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n\<^sub>3 :: "'b SDG_node"   and V' :: "'c"   and n\<^sub>4 :: "'b SDG_node"   and a :: "'a"   and a' :: "'a" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>0 ns n\<^sub>1"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>1 (Some V) (Some (p, True)) n\<^sub>2"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>2 ns' n\<^sub>3"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>3 (Some V') (Some (p, False)) n\<^sub>4"   and "valid_edge a"   and "a' \<in> get_return_edges a"   and "sourcenode a = parent_node n\<^sub>1"   and "targetnode a = parent_node n\<^sub>2"   and "sourcenode a' = parent_node n\<^sub>3"   and "targetnode a' = parent_node n\<^sub>4" shows "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>0 (ns @ n\<^sub>1 # ns' @ [n\<^sub>3]) n\<^sub>4"
Ide_Dom: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Ide V src trg (bicategorical_language.Dom V src trg t)"
s: shows "supp dbfm \<subseteq> atom ` Vs"
refl_onD2: fixes A :: "'g set"   and r :: "('g \<times> 'g) set"   and x :: "'g"   and y :: "'g" assumes "refl_on A r"   and "(x, y) \<in> r" shows "y \<in> A"