hcju/isabelleps huggingface dataset:
converse_rtrancl_induct: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P b"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>*; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
swap_swap_id: fixes n :: "nat"   and m :: "nat"   and t :: "poly" shows "swap n m (swap m n t) = t"
Abs_sat'_eq_of_nat: fixes n :: "nat" shows "Abs_sat' n = Sat n"
h2: shows "|~ F \<and> Unchanged v \<longrightarrow> \<circle>F"
eligible_literal_coincide: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<sigma>' :: "('a \<times> 'a trm) list"   and C :: "'a eclause"   and L :: "'a literal" assumes "coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C))"   and "eligible_literal L C \<sigma>"   and "L \<in> cl_ecl C" shows "eligible_literal L C \<sigma>'"
simps_1: fixes x :: "nat" shows "digit x 0 = x mod base"
fresh_distinct: fixes S :: "'a set"   and \<Gamma> :: "('a \<times> 'b) list" assumes "atom ` S \<sharp>* \<Gamma>" shows "S \<inter> domA \<Gamma> = {}"
sign_changes'_def: shows "sign_changes' \<equiv> \<lambda>x\<^sub>0 ps x. \<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x"
rel_distinct_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y21 :: "'c"   and y22 :: "'c llist" shows "\<not> lrel R (LCons y21 y22) LNil"
equiv_F_grounding: fixes q :: "'q"   and C1 :: "'f"   and C2 :: "'f" assumes "q \<in> Q"   and "C1 \<doteq> C2" shows "\<G>_F_q q C1 \<subseteq> \<G>_F_q q C2"
diffconst_result_correct: shows "proof_result DiffConstProof = ([], [], [Equals (Differential (Const 0)) (Const 0)])"
range_to_nat: shows "range to_nat = {0..<CARD('a)}"
Cons_eq_appendI: fixes x :: "'d"   and xs1 :: "'d list"   and ys :: "'d list"   and xs :: "'d list"   and zs :: "'d list" assumes "x # xs1 = ys"   and "xs = xs1 @ zs" shows "x # xs = ys @ zs"
star_unfoldl_eq: fixes x :: "'a" shows "1' + x\<^sup>+ = x\<^sup>\<star>"
WT: shows "\<I> \<turnstile>g gpv \<surd>"
acc_modi_Package_le: fixes a :: "acc_modi" assumes "Package \<le> a" shows "a = Package \<or> a = Protected \<or> a = Public"
nat_mult_dvd_cancel_disj: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k * m dvd k * n) = (k = 0 \<or> m dvd n)"
DAs_axioms: shows "DAs alphabet init delta accept wellformed Language wf Lang alphabet init delta accept wellformed Language wf Lang (=)"
Dmonoid_axioms_def: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" shows "Dmonoid_axioms prod one \<equiv> (\<forall>x. prod one x = x) \<and> (\<forall>x. prod x one = x)"
base_case_satisfy_equation_alt_R: fixes qs :: "real poly list"   and p :: "real poly" assumes "length qs = 1"   and "p \<noteq> 0" shows "satisfy_equation_R p qs [([], []), ([0], []), ([], [0])] [[1], [0], [- 1]]"
assign\<^sub>1: fixes x :: "'Var"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'AExp"   and t :: "'BExp set"   and P :: "'BExp set"   and P' :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set" assumes "x \<notin> dom \<Gamma>"   and "x \<notin> \<C>"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t"   and "t \<le>:\<^sub>P dma_type x"   and "P' = assign_post P x e |` {v. stable \<S> v}" shows "\<turnstile> \<Gamma>,\<S>,P {x \<leftarrow> e} \<Gamma>,\<S>,P'"
second_summand_in_sum: fixes y :: "i"   and x :: "i" shows "P y (x \<oplus> y)"
hf_to_ide_mapsto: shows "hf_to_ide \<in> UNIV \<rightarrow> Collect ide"
subdegreeI: fixes f :: "'a fps"   and d :: "nat" assumes "f $ d \<noteq> (0::'a)"   and "\<And>i. i < d \<Longrightarrow> f $ i = (0::'a)" shows "subdegree f = d"
unfold: shows "nub_body = (\<Lambda> f v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>lnil) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(x :@ f\<cdot>(lfilter\<cdot>(neg oo Abs_cfun ((=\<^sub>B) x))\<cdot>xs)))))"
simps_1: fixes p :: "'b" shows "last_prog p [] = p"
balanced_net_Input: fixes M :: "nat" shows "balanced_net (Input M)"
times_int_raw_snd: fixes x :: "nat \<times> nat"   and z :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "x \<approx> z" shows "times_int_raw y x \<approx> times_int_raw y z"
intersection_calculus_axioms_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a inference set"   and Red_F_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set" shows "intersection_calculus_axioms Bot Inf Q entails_q Red_I_q Red_F_q \<equiv> Q \<noteq> {} \<and> (\<forall>q\<in>Q. calculus Bot Inf (entails_q q) (Red_I_q q) (Red_F_q q))"
iter_induct_eq: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> ; (d::'a::refinement_lattice) = lfp (\<lambda>x::'a::refinement_lattice. d \<sqinter> c ; x)"
sel: fixes x :: "((_) object_ptr, (_) Object) fmap" shows "the_heap (Heap x) = x"
rewrite_negated_dst_ports_not_has_disc_negated: fixes m :: "'b common_primitive match_expr" assumes "normalized_nnf_match m" shows "\<not> has_disc_negated is_Dst_Ports False (rewrite_negated_dst_ports m)"
simps_1: fixes n :: "nat"   and n' :: "nat" shows "observe (ENat n) (ONat n') = (n = n')"
A_3: fixes F :: "nat \<Rightarrow> 'a" assumes "\<forall>n. F n \<in> A n" shows "F \<longlonglongrightarrow> x"
category_def: fixes CC :: "('d, 'e, 'f) category_scheme" shows "category CC \<equiv> ((\<forall>f. f \<in> Ar\<^bsub>CC\<^esub> \<longrightarrow> Dom\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>) \<and> (\<forall>f. f \<in> Ar\<^bsub>CC\<^esub> \<longrightarrow> Cod\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>) \<and> (\<forall>f. f \<in> Ar\<^bsub>CC\<^esub> \<longrightarrow> Id\<^bsub>CC\<^esub> (Cod\<^bsub>CC\<^esub> f) \<bullet>\<^bsub>CC\<^esub> f = f)) \<and> ((\<forall>f. f \<in> Ar\<^bsub>CC\<^esub> \<longrightarrow> f \<bullet>\<^bsub>CC\<^esub> Id\<^bsub>CC\<^esub> (Dom\<^bsub>CC\<^esub> f) = f) \<and> (\<forall>A. A \<in> Ob\<^bsub>CC\<^esub> \<longrightarrow> Id\<^bsub>CC\<^esub> A \<in> Hom\<^bsub>CC\<^esub> A A)) \<and> (\<forall>A B C. (\<bullet>\<^bsub>CC\<^esub>) \<in> Hom\<^bsub>CC\<^esub> B C \<rightarrow> Hom\<^bsub>CC\<^esub> A B \<rightarrow> Hom\<^bsub>CC\<^esub> A C) \<and> (\<forall>f g h. f \<in> Ar\<^bsub>CC\<^esub> \<longrightarrow> g \<in> Ar\<^bsub>CC\<^esub> \<longrightarrow> h \<in> Ar\<^bsub>CC\<^esub> \<longrightarrow> Cod\<^bsub>CC\<^esub> h = Dom\<^bsub>CC\<^esub> g \<longrightarrow> Cod\<^bsub>CC\<^esub> g = Dom\<^bsub>CC\<^esub> f \<longrightarrow> f \<bullet>\<^bsub>CC\<^esub> (g \<bullet>\<^bsub>CC\<^esub> h) = f \<bullet>\<^bsub>CC\<^esub> g \<bullet>\<^bsub>CC\<^esub> h)"
intro: fixes set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_attribute_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')" shows "l_set_attribute_get_child_nodes_axioms set_attribute_locs get_child_nodes_locs"
flowsto_subset: fixes X0 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and T :: "real set"   and CX :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and Y :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and Z :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and S :: "real set"   and CZ :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and W :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" assumes "flowsto X0 T CX Y"   and "Z \<subseteq> X0"   and "T \<subseteq> S"   and "CX \<subseteq> CZ"   and "Y \<subseteq> W" shows "flowsto Z S CZ W"
ZFFunToLambda: fixes f :: "ZF" assumes "isFun f"   and "f |\<subseteq>| HOLZF.Domain f |\<times>| HOLZF.Range f" shows "f = Lambda (HOLZF.Domain f) ((\<acute>) f)"
thetaParZO_ZOretr: shows "thetaParZO \<subseteq> ZOretr (thetaParZO \<union> ZObis)"
IDirProds_Un: fixes A :: "'d set"   and S :: "'d \<Rightarrow> 'a set"   and B :: "'d set" assumes "\<forall>i\<in>A. subgroup (S i) G"   and "\<forall>j\<in>B. subgroup (S j) G" shows "IDirProds G S (A \<union> B) = IDirProds G S A <#> IDirProds G S B"
p_max_eq: shows "other.p_max = s\<^sub>o - v\<^sub>o\<^sup>2 / a\<^sub>o / 2"
simps_2: fixes c :: "color"   and l :: "('b \<times> color) tree"   and a :: "'b"   and uu :: "color"   and r :: "('b \<times> color) tree" shows "paint c \<langle>l, (a, uu), r\<rangle> = \<langle>l, (a, c), r\<rangle>"
evaldjf_ex: fixes bs :: "real list"   and f :: "'a \<Rightarrow> fm"   and ps :: "'a list" shows "Ifm bs (evaldjf f ps) = (\<exists>p\<in>set ps. Ifm bs (f p))"
weight_outside: fixes x :: "'v" assumes "x \<notin> \<^bold>V\<^bsub>\<Gamma>'\<^esub>" shows "weight \<Gamma>' x = 0"
covering: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int"   and ps :: "'b set" assumes "t_covering_design point_set block_collection u_block_size grouping min_index"   and "ps \<subseteq> point_set"   and "int (card ps) = grouping" shows "min_index \<le> int (block_collection index ps)"
simps_2: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cMassF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cMassF"
simb: fixes as :: "'a"   and cs :: "'c"   and l :: "'l" assumes "A.reachable as"   and "C.reachable cs"   and "R as cs"   and "A.can_step l as" shows "C.can_step l cs"
intro: fixes state_list :: "'c list"   and \<S> :: "'c set" assumes "set state_list = \<S>" shows "HMM3_axioms \<S> state_list"
perp_in_distinct: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "A \<noteq> B \<and> C \<noteq> D"
wf_filterQuery_2: fixes I :: "nat set"   and V :: "nat set"   and n :: "nat"   and Qp :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and QQp :: "(nat set \<times> 'a option list set) set"   and QQn :: "(nat set \<times> 'a option list set) set" assumes "I \<subseteq> V"   and "1 \<le> card I"   and "rwf_query n V Qp Qn"   and "QQp = filterQuery I Qp"   and "QQn = filterQueryNeg I Qn" shows "non_empty_query QQp"
subcls1: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and J :: "qtname" assumes "G\<turnstile>C\<prec>\<^sub>C1D"   and "G\<turnstile>D\<leadsto>J" shows "G\<turnstile>C\<leadsto>J"
mapping_of_inject: fixes x :: "('a, 'b) pp"   and y :: "('a, 'b) pp" shows "(pp.mapping_of x = pp.mapping_of y) = (x = y)"
abs_abs: fixes abs :: "'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (abs a) = abs a"
wtE_not_glitOfC: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" assumes "Ik.wtE \<xi>" shows "\<not> GE.satC \<xi> (list (glitOfC c))"
simps_4: fixes x2 :: "('e, 'f) atm"   and x1 :: "('e, 'f) atm" shows "Neg x2 \<noteq> Pos x1"
cons: shows "\<forall>l s. P' l s \<longrightarrow> e s \<le> k2 * e' s \<and> (\<forall>t. \<exists>l'. P l' s \<and> (Q l' t \<longrightarrow> Q' l t))"
winding_number_compose_exp: fixes p :: "real \<Rightarrow> complex" assumes "path p" shows "winding_number (exp \<circ> p) 0 = (pathfinish p - pathstart p) / (2 * complex_of_real pi * \<i>)"
simps_2: fixes r :: "('a \<times> 'a) set"   and \<alpha> :: "'a"   and \<sigma> :: "'a list" shows "r|\<alpha> # \<sigma>| = {#\<alpha>#} + r|\<sigma>| -s r \<down>s {\<alpha>}"
psimps_6: fixes c :: "'b"   and d :: "'b" assumes "unify_dom (Const c, Const d)" shows "unify (Const c) (Const d) = (if c = d then Some [] else None)"
prv_cnj_imp_monoR2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> (imp \<chi> \<psi>))" shows "prv (imp (cnj \<phi> \<chi>) \<psi>)"
action_closed: fixes s :: "'s \<times> ('c \<Rightarrow> real) set"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and t :: "'s \<times> ('c \<Rightarrow> real) set" assumes "s \<in> \<S>"   and "cfg \<in> R_G.cfg_on s"   and "t \<in> set_pmf (action cfg)" shows "t \<in> \<S>"
ncols_not_0: fixes A :: "(('d, 'e) vec, 'f) vec" shows "0 \<noteq> ncols A"
is_basis: shows "on U with \<tau> : \<guillemotleft>topological_basis\<guillemotright> B"
Mapping_RBT_inverse: fixes y :: "('c, 'd) rbt" assumes "y \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}" shows "impl_of (Mapping_RBT y) = y"
2: fixes V_of :: "'a \<Rightarrow> V"   and A :: "V" assumes "range V_of \<subseteq> elts A" shows "range (\<lambda>x. ZFC_in_HOL.set (V_of ` x)) \<subseteq> elts (VPow A)"
overlap_symmetry: fixes x :: "i"   and y :: "i" assumes "PM"   and "O x y" shows "O y x"
of_nat_take_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.of_nat one plus zero (take_bit n m) = take_bit n (semiring_1.of_nat one plus zero m)"
simps_1: shows "Missing_Lemmas.max_list [] = (THE x. False)"
the_\<nu>_props_2: shows "local.iso TTfgh_TfTgh.the_\<nu>"
inM: shows "0 < zcount M (results_in t' (sum_weights (map (\<lambda>(s, l, t). l) xs)))"
b_least2_aux3: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "b_least2 f x y = y"   and "f x y \<noteq> 0" shows "b_least2 f x (Suc y) = y"
wt_App_plus_\<delta>\<^sub>h_ge: 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 A :: "'a pvar \<Rightarrow> zhmultiset"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s)) + eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.coef coef_sym s 0)) * eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t)) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym (App s t))) + zhmset_of (of_nat \<delta>)"
resChainFreshSet_2: fixes yvec :: "name list"   and xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "yvec \<sharp>* (\<lparr>\<nu>*xvec\<rparr>P) = (\<forall>x\<in>set yvec. x mem xvec \<or> x \<sharp> P)"
dom_eqI: fixes a :: "'s"   and f :: "'s" assumes "ide a"   and "seq f a" shows "local.dom f = a"
prv_imp_lcnj: fixes \<chi> :: "'fmla"   and \<phi>s :: "'fmla list" assumes "\<chi> \<in> fmla"   and "set \<phi>s \<subseteq> fmla"   and "\<And>\<phi>. \<phi> \<in> set \<phi>s \<Longrightarrow> prv (imp \<chi> \<phi>)" shows "prv (imp \<chi> (lcnj \<phi>s))"
open_halfspace_Re_gt: fixes b :: "real" shows "open {z. b < Re z}"
ih: assumes "rcvd (rho n) q p = Some m" shows "p \<in> HOs (round (rho n) q) q"
None_bot: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ox :: "'b option" shows "None \<sqsubseteq>\<^bsub>Opt.le r\<^esub> ox"
sup_commute: fixes x :: "'a upred"   and y :: "'a upred" shows "(x \<or> y) = (y \<or> x)"
sublens_pres_wb: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e" assumes "wb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "wb_lens X"
set_sortQ: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "set (qsort xs l) = set xs"
assms: shows "A \<in>\<^sub>\<circ> Vset \<alpha>"
map_transform_def: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" shows "map_transform t ae = map_ran (\<lambda>x. lift_transform t (ae x))"
g_def: shows "g \<equiv> Abs_fps (\<lambda>n. sum (($) a) {0..n})"
synthetic_div_eq_0_iff: fixes p :: "'a poly"   and c :: "'a" shows "(synthetic_div p c = 0) = (degree p = 0)"
LeftDerivationIntrosAt_last: fixes index :: "nat" assumes "1 \<le> index \<and> index + 1 = length L" shows "LeftDerivationIntrosAt \<alpha> E L' index"
measure_UNION': fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> fmeasurable M"   and "pairwise (\<lambda>i j. disjnt (F i) (F j)) I" shows "Sigma_Algebra.measure M (\<Union> (F ` I)) = (\<Sum>i\<in>I. Sigma_Algebra.measure M (F i))"
llength_lnull: fixes xs :: "'c llist" assumes "lnull xs" shows "llength xs = 0"
prime_factorization_prod_mset_primes: fixes A :: "'a multiset" assumes "\<And>p. p \<in># A \<Longrightarrow> prime p" shows "prime_factorization (\<Prod>\<^sub># A) = A"
pdom_comp: fixes g :: "('c, 'b) pfun"   and f :: "('a, 'c) pfun" shows "pdom (g \<circ>\<^sub>p f) = pdom (f \<rhd>\<^sub>p pdom g)"
Lambert_W_nonneg_iff: fixes x :: "real" shows "(0 \<le> Lambert_W x) = (0 \<le> x)"
word_less_power_trans: fixes n :: "'b word"   and m :: "nat"   and k :: "nat" assumes "n < 2 ^ (m - k)"   and "k \<le> m"   and "m < LENGTH('b)" shows "2 ^ k * n < 2 ^ m"
lex_ext_SN: fixes g :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and m :: "nat" assumes "\<And>x y z. \<lbrakk>snd (g x y); fst (g y z)\<rbrakk> \<Longrightarrow> fst (g x z)" shows "SN {(ys, xs). (\<forall>y\<in>set ys. SN_on {(s, t). fst (g s t)} {y}) \<and> fst (lex_ext g m ys xs)}"
homomorphism: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "pure f \<diamondop> pure x = pure (f x)"
blank_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list" shows "blank P C \<equiv> (C, init_fields (fields P C))"
bounded_coeffs_imp_fds_abs_converges: fixes f :: "'b fds"   and s :: "'b" assumes "Bseq (fds_nth f)"   and "1 < s \<bullet> (1::'b)" shows "fds_abs_converges f s"
IH: fixes t :: "'a trm"   and s :: "'a trm" assumes "subterm t q s"   and "q \<noteq> []" shows "(s, t) \<in> trm_ord"
attractor_unfolding: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and W :: "'c set" assumes "ParityGame G" shows "ParityGame.attractor G p W = W \<union> ParityGame.attractor G p W \<union> ParityGame.directly_attracted G p (ParityGame.attractor G p W)"
nv2L: fixes l :: "('a, 'psym) lit"   and c :: "('a, 'psym) lit list"   and \<sigma> :: "'tp" assumes "l \<in> set c"   and "\<sigma> \<turnstile> c" shows "infTp \<sigma> \<or> (\<forall>x\<in>nv2L (gL l). GE.tpOfV x \<noteq> \<sigma>)"
assms_4: fixes exp :: "exp"   and var :: "Vars.var" assumes "P1 exp" shows "P1 (Lam [var]. exp)"
rev_flow_image_eq: fixes x :: "'a"   and S :: "real set" shows "rev.flow0 x ` S = flow0 x ` uminus ` S"
gcd_1_nat: fixes m :: "nat" shows "gcd m 1 = 1"
rank_iarrays_code: fixes A :: "'d iarray iarray" shows "rank_iarray A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (Gauss_Jordan_iarrays A)))"
emeasure_eq_measure: fixes M :: "'c pmf"   and A :: "'c set" shows "emeasure (measure_pmf M) A = ennreal (measure_pmf.prob M A)"
distinct_89: fixes name :: "name"   and pi :: "pi"   and pi' :: "pi" shows "<\<nu>name>pi \<noteq> !pi'"
assoc_match_Type_conds_1: fixes a :: "String.literal"   and Ts :: "typ list"   and b :: "String.literal"   and Us :: "typ list"   and subs :: "((variable \<times> String.literal set) \<times> typ) list"   and subs' :: "((variable \<times> String.literal set) \<times> typ) list" assumes "assoc_match (Ty a Ts) (Ty b Us) subs = Some subs'" shows "a = b"
priority_Min_priorities: fixes q :: "('c, 'd) pq" assumes "\<not> PQ.is_empty q" shows "PQ.priority q (PQ.min q) = Some (Min (set \<parallel>q\<parallel>))"
assms_2: shows "bicategory_of_spans (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
blocks_nempty: fixes bl :: "'a set" assumes "bl \<in># \<B>'" shows "bl \<noteq> {}"
shift_def: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "local.shift qs = shift2 (k + card {q \<in> set qs. m < card (snd q)}) k qs"
intros_2: fixes \<tau> :: "'a OCL_Types.type" shows "element_type (Set \<tau>) \<tau>"
memb_equiv_2: fixes f :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "arr f"   and "t \<in> f"   and "u \<in> f" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"
PreFunctor_def: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" shows "PreFunctor F \<equiv> ((\<forall>f g. f \<approx>>\<^bsub>CatDom F\<^esub> g \<longrightarrow> F ## f ;;\<^bsub>CatDom F\<^esub> g = F ## f ;;\<^bsub>CatCod F\<^esub> (F ## g)) \<and> (\<forall>X. X \<in> obj\<^bsub>CatDom F\<^esub> \<longrightarrow> (\<exists>Y\<in>obj\<^bsub>CatCod F\<^esub>. F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y))) \<and> Category (CatDom F) \<and> Category (CatCod F)"
simps_1: fixes x :: "real" shows "ms_exp_gt x None = True"
orbit_stab_fun_is_well_defined1: fixes x :: "'c"   and R :: "'a set"   and g :: "'a" assumes "x \<in> E"   and "R \<in> rcosets stabilizer G \<phi> x"   and "g \<in> R" shows "\<phi> (inv (SOME h. h \<in> R)) x = \<phi> (inv g) x"
tell_writer_def: fixes w :: "'w"   and m :: "('w, 'a, 'm) writerT" shows "local.tell_writer w m = WriterT (bind (run_writer m) (\<lambda>(a, ws). return (a, w # ws)))"
analz_Un_implConfidSet_aux: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implConfidSet_aux Ag G \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
isolate_variable_sparse_ne_zeroD: fixes mp :: "'a mpoly"   and v :: "nat"   and x :: "nat" assumes "isolate_variable_sparse mp v x \<noteq> 0" shows "x \<le> MPoly_Type.degree mp v"