hcju/isabelleps huggingface dataset:
mat_comp_all_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat" shows "mat_comp_all r A B = (\<forall>i<dim_row A. \<forall>j<dim_col A. r (A $$ (i, j)) (B $$ (i, j)))"
assms: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "(\<Psi>, P, Q) \<in> Rel"
cofactorM_iarrays_def: fixes A :: "'c iarray iarray" shows "cofactorM_iarrays A = tabulate2 (nrows_iarray A) (nrows_iarray A) (cofactor_iarrays A)"
separate_compact_closed: fixes s :: "'b set"   and t :: "'b set" assumes "compact s"   and "closed t"   and "s \<inter> t = {}" shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
WT_gpv_simps: fixes \<Gamma> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, ('c, 'a, 'b) rpv) generat spmf" shows "\<Gamma> \<turnstile>g GPV gpv \<surd> = (\<forall>out c. IO out c \<in> set_spmf gpv \<longrightarrow> out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. \<Gamma> \<turnstile>g c input \<surd>))"
lenlex_trans: fixes x :: "'b list"   and y :: "'b list"   and r :: "('b \<times> 'b) set"   and z :: "'b list" assumes "(x, y) \<in> lenlex r"   and "(y, z) \<in> lenlex r"   and "trans r" shows "(x, z) \<in> lenlex r"
assms_2: shows "(msgs c i \<noteq> [] \<and> hd (msgs c i) = m) \<and> (msgs c' i = [] \<or> hd (msgs c' i) \<noteq> m)"
sum_list_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero (map (\<lambda>x. zero) xs) = zero"
pred_map: fixes Q :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c list" shows "list_all Q (map f x) = list_all (Q \<circ> f) x"
put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes document_ptr :: "(_) document_ptr"   and document :: "('o, 'm, 'n, 'v option) RDocument_scheme" shows "put document_ptr document = put (cast document_ptr) (cast document)"
1: shows "v \<sqinter> w \<sqinter> - (v \<sqinter> y) \<sqinter> - (- v \<sqinter> z) = v \<sqinter> w \<sqinter> - y"
NOT_X: shows "makespan (T(x := T x + y)) \<noteq> (T(x := T x + y)) x"
transp_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp less"
wtE_not_grdLit: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" assumes "Ik.wtE \<xi>"   and "prot (GE.tpOfV x) \<or> protFw (GE.tpOfV x)" shows "\<not> GE.satL \<xi> (grdLit x)"
IDirProd_one_right: fixes A :: "'a set" assumes "A \<subseteq> carrier G" shows "IDirProd G A {\<one>} = generate G A"
seq_bot_right: fixes c :: "'b::refinement_lattice" shows "(c::'b::refinement_lattice) ; \<bottom> \<sqsubseteq> c"
rtrancl_image_advance_rtrancl: fixes q :: "'a"   and R :: "('a \<times> 'a) set"   and Q0 :: "'a set"   and x :: "'a" assumes "q \<in> R\<^sup>* `` Q0"   and "(q, x) \<in> R\<^sup>*" shows "x \<in> R\<^sup>* `` Q0"
simps_2: fixes d :: "bool"   and ds :: "bool list"   and l :: "tree"   and r :: "tree" shows "internal (d # ds) (Branching l r) = (if d then internal ds l else internal ds r)"
subterm_reduction_def: fixes S :: "'a eclause set"   and t :: "'a trm" shows "subterm_reduction S t = trm_rep (trm_rep (lhs t) S \<cdot> trm_rep (rhs t) S) S"
proj_psi_minus_1_mult_psi'_l: fixes l :: "nat" shows "(2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) *\<^sub>v psi'_l l = psi_l (l + 1)"
monoid_def: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d" shows "Groups.monoid f z \<equiv> semigroup f \<and> monoid_axioms f z"
WT_gpv: shows "\<I> \<turnstile>g gpv \<surd>"
cut2: fixes A :: "fm"   and B :: "fm"   and C :: "fm"   and H :: "fm set" assumes "{A, B} \<turnstile> C"   and "H \<turnstile> A"   and "H \<turnstile> B" shows "H \<turnstile> C"
converse_equivclp_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "equivclp r a b"   and "P b"   and "\<And>y z. \<lbrakk>r y z \<or> r z y; equivclp r z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
fds_mangoldt: shows "fds mangoldt = fds moebius_mu * fds (\<lambda>n. of_real (ln (real n)))"
indeps_2: shows "mrg_left \<bowtie> mrg_prior"
rewrite_def: fixes p :: "'b list"   and pc :: "nat"   and i :: "'b" shows "rewrite p pc i = p[pc := i]"
doesnt_modify_def: fixes c :: "'Com"   and x :: "'Var" shows "doesnt_modify c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> mem x = mem' x \<and> dma mem x = dma mem' x)"
distrib_sup_le: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup" shows "less_eq (sup x (inf y z)) (inf (sup x y) (sup x z))"
sup_absorb2: fixes x :: "'a upred"   and y :: "'a upred" assumes "y \<sqsubseteq> x" shows "(x \<or> y) = y"
pbeta_induct: fixes t :: "sterm"   and t' :: "sterm"   and P1 :: "sterm \<Rightarrow> sterm \<Rightarrow> bool"   and P2 :: "sterm \<Rightarrow> sterm \<Rightarrow> bool" assumes "t => t'"   and "\<And>x. P1 (Fvar x) (Fvar x)"   and "\<And>t t' l u u'. \<lbrakk>t => t'; P1 t t'; lc t; u => u'; P1 u u'; lc u\<rbrakk> \<Longrightarrow> P1 (Call t l u) (Call t' l u')"   and "\<And>t t' l u u'. \<lbrakk>t => t'; P1 t t'; lc t; P2 u u'; body u\<rbrakk> \<Longrightarrow> P1 (Upd t l u) (Upd t' l u')"   and "\<And>f f' T t t' l. \<lbrakk>Obj f T => Obj f' T; P1 (Obj f T) (Obj f' T); P2 t t'; l \<in> dom f; lc (Obj f T); body t\<rbrakk> \<Longrightarrow> P1 (Upd (Obj f T) l t) (Obj (f'(l \<mapsto> t')) T)"   and "\<And>f f' T. \<lbrakk>dom f' = dom f; \<forall>l\<in>dom f. body (the (f l)); \<forall>l\<in>dom f. P2 (the (f l)) (the (f' l))\<rbrakk> \<Longrightarrow> P1 (Obj f T) (Obj f' T)"   and "\<And>f f' T l p p'. \<lbrakk>Obj f T => Obj f' T; P1 (Obj f T) (Obj f' T); lc (Obj f T); l \<in> dom f; p => p'; P1 p p'; lc p\<rbrakk> \<Longrightarrow> P1 (Call (Obj f T) l p) (the (f' l)\<^bsup>[Obj f' T,p']\<^esup>)"   and "\<And>L t t'. \<lbrakk>finite L; \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> P1 (t\<^bsup>[Fvar s,Fvar p]\<^esup>) t'' \<and> t' = \<sigma>[s,p] t'')\<rbrakk> \<Longrightarrow> P2 t t'" shows "P1 t t'"
IH_3: shows "b = sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<jj])"
insert_disjoint_2: fixes a :: "'e"   and A :: "'e set"   and B :: "'e set" shows "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
pochhammer_poly_def: fixes n :: "nat" shows "pochhammer_poly n = Poly (map (\<lambda>k. of_nat (stirling n k)) [0..<Suc n])"
Red_I_of_Inf_to_N: fixes \<iota> :: "'g inference"   and N :: "'g set" assumes "\<iota> \<in> Inf_G"   and "concl_of \<iota> \<in> N" shows "\<iota> \<in> Red_I_G N"
d_IN_restrict_current_outside: fixes x :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "x \<in> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_IN (g \<upharpoonleft> \<Gamma> / f) x = 0"
c_eq: shows "card (columns (Gram_Schmidt_matrix A)) = ncols A"
invar_Cons: fixes t :: "'b tree"   and ts :: "'b tree list" shows "invar (t # ts) = (invar_tree t \<and> invar ts \<and> (\<forall>t'\<in>set ts. rank t < rank t'))"
enc_reflects_pred_iff_source_target_rel_preserves_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_reflects_pred Pred = (\<exists>Rel. (\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_preserves_pred Rel Pred)"
map_add_dom_app_simps_1: fixes m :: "'d"   and l2 :: "'d \<Rightarrow> 'e option"   and l1 :: "'d \<Rightarrow> 'e option" assumes "m \<in> dom l2" shows "(l2 \<Oplus> l1) m = l2 m"
complex_rotations_comp: fixes r :: "Vertex \<Rightarrow> Vertex"   and s :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> complex_rotations"   and "s \<in> complex_rotations" shows "r \<circ> s \<in> complex_rotations"
assms_2: shows "S = affine hull S \<inter> \<Inter> F"
simps_19: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b" shows "AExp_Lexorder.less_aexp_aux (Minus e1 e2) (V v) = False"
filter_valid_lists: fixes xs :: "Vars.var list"   and S :: "Vars.var set"   and G :: "CoCalls"   and P :: "Vars.var \<Rightarrow> bool" assumes "xs \<in> valid_lists S G" shows "filter P xs \<in> valid_lists {a \<in> S. P a} G"
2: shows "ipd (\<pi> k) pd\<rightarrow> \<pi> j"
hidden_index: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "hidden (xs @ [x]) (index xs x)"
basis_for_Span_ex: fixes vs :: "'b list" assumes "set vs \<subseteq> V" shows "\<exists>us. set us \<subseteq> RSpan vs \<and> RSpan vs = RSpan us \<and> R_lin_independent us"
element_put_get_preserved: fixes getter :: "('a, 'b, 'c, 'd, 'e option) RElement_scheme \<Rightarrow> 'f"   and setter :: "('g \<Rightarrow> 'g) \<Rightarrow> ('a, 'b, 'c, 'd, 'e option) RElement_scheme \<Rightarrow> ('a, 'b, 'c, 'd, 'e option) RElement_scheme"   and v :: "'g"   and h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and h' :: "(_) heap"   and element_ptr' :: "(_) element_ptr" assumes "\<And>x. getter (setter (\<lambda>_. v) x) = getter x"   and "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' getter) h h'"
Transset_sup: fixes x :: "hf"   and y :: "hf" assumes "Transset x"   and "Transset y" shows "Transset (x \<squnion> y)"
strict_trans1: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b"   and "b \<subset># c" shows "a \<subset># c"
OclIf_def: fixes C :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B\<^sub>1 :: "'a state \<times> 'a state \<Rightarrow> 'b"   and B\<^sub>2 :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(if C then B\<^sub>1 else B\<^sub>2 endif) = (\<lambda>\<tau>. if (\<delta> C) \<tau> = true \<tau> then if C \<tau> = true \<tau> then B\<^sub>1 \<tau> else B\<^sub>2 \<tau> else invalid \<tau>)"
Gromov_product_at_diff2: fixes e :: "'b"   and x :: "'b"   and z :: "'b"   and y :: "'b" shows "\<bar>Gromov_product_at e x z - Gromov_product_at e y z\<bar> \<le> dist x y"
dlex_fun_def: shows "dlex_fun \<equiv> dord_fun lex_fun"
eq: fixes E1 :: "'a set"   and E2 :: "'a set"   and P :: "bool" shows "(openin X E1 \<and> openin X E2 \<and> E1 \<union> E2 = topspace X \<and> E1 \<inter> E2 = {} \<and> P) = (E2 = topspace X - E1 \<and> openin X E1 \<and> openin X E2 \<and> P)"
eq_dfa_accepting: fixes l :: "int"   and m :: "int"   and n :: "nat"   and ks :: "int list"   and bss :: "bool list list" assumes "(l, m) \<in> (succsr (dioph_succs n ks))\<^sup>*"   and "list_all (is_alph n) bss" shows "dfa_accepting (eq_dfa n ks l) (dfa_steps (eq_dfa n ks l) (the (fst (dioph_dfs n ks l) ! int_encode m)) bss) = (eval_dioph ks (nats_of_boolss n bss) = m)"
Skip: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "'b" shows "\<Gamma>\<turnstile> \<langle>Skip,Normal s\<rangle> \<Rightarrow> Normal s"
monoE: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" obtains "f x \<le> f y"
div_dvd_iff_mult: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "b \<noteq> zero"   and "dvd.dvd times b a" shows "dvd.dvd times (divide a b) c = dvd.dvd times a (times c b)"
eye_mv_l: fixes v :: "complex \<times> complex" shows "eye *\<^sub>m\<^sub>v v = v"
matchCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "[a\<frown>b]P \<longmapsto> Rs" obtains P :: "pi" and ba :: "name" where "a = ba \<and> b = ba \<and> Pa__ = P" and "P \<longmapsto> Rs"
permutations_of_multiset_def: fixes A :: "'a multiset" shows "permutations_of_multiset A = {xs. mset xs = A}"
one_p: shows "D1 * S1 + H1 * T1 =m 1"
keys_single: fixes k :: "'b"   and v :: "'c" shows "keys (Poly_Mapping.single k v) = (if v = (0::'c) then {} else {k})"
P1': fixes x :: "'a"   and x' :: "'b" shows "P1 x x' = \<up> (P1' x x')"
pure_app_eq: fixes R :: "('d \<times> 'e) set"   and a :: "'e"   and c :: "'d" shows "pure R a c = \<up> ((c, a) \<in> R)"
ide_in_hom: fixes a :: "'a" shows "H.ide a = H.in_hom a a a"
wf_sub_rel_def: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "wf_sub_rel subs \<equiv> (\<forall>sub. sub \<in> subs \<longrightarrow> fst (subsumee sub) = fst (subsumer sub)) \<and> (\<forall>v\<in>subsumees subs. \<exists>!v'. (v, v') \<in> subs) \<and> subsumers subs \<inter> subsumees subs = {}"
fromMOD_basic_simps_2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model"   and us_s :: "'varSort \<times> 'sort"   and A'A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd" shows "igWlsAbs (fromMOD MOD) us_s A'A = (wlsAbs us_s (fst A'A) \<and> gWlsAbs MOD us_s (snd A'A))"
list_of_oalist_update_by_fun: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalist :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_list :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and k :: "'b"   and f :: "'d \<Rightarrow> 'd"   and xs :: "'c" assumes "oalist_abstract rep_key_order list_of_oalist oalist_of_list" shows "list_of_oalist (oalist_abstract.update_by_fun rep_key_order list_of_oalist oalist_of_list k f xs) = oalist_raw.update_by_fun_raw rep_key_order k f (list_of_oalist xs)"
deduct_if_closure_deduct: fixes M' :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "M' \<turnstile> t"   and "decomp_closure M M'" shows "M \<turnstile> t"
assms_1: shows "OUTfromV x \<noteq> {}"
simps_66: fixes f1 :: "'f \<Rightarrow> 'e"   and f2 :: "nat set \<Rightarrow> 'f \<Rightarrow> 'e"   and f3 :: "nat \<Rightarrow> 'f \<Rightarrow> 'e"   and f4 :: "nat \<Rightarrow> 'e"   and f5 :: "'f \<Rightarrow> 'e"   and f6 :: "nat \<Rightarrow> 'e"   and f7 :: "'f \<Rightarrow> 'e"   and f8 :: "'e"   and x31 :: "nat"   and x32 :: "'f" shows "(case unicast x31 x32 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f3 x31 x32"
less_trans: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" assumes "x <s y"   and "y <s z" shows "x <s z"
isomorphic_implies_hpar_1: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong> f'" shows "ide f"
mix_4: shows "local.general_U p = measure_pmf.expectation p local.ocU"
inf_bot_right: fixes x :: "'d upred" shows "(x \<and> false) = false"
limI: fixes T :: "'b set set"   and F :: "'b set set"   and x :: "'b" assumes "T2 T"   and "F \<in> Filters\<^bsub>T\<^esub>"   and "x \<in> carrier\<^bsub>T\<^esub>"   and "F \<longlongrightarrow>\<^bsub>T\<^esub> x" shows "lim\<^bsub>T\<^esub> F = x"
not_cong_2143: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Cong A B C D" shows "\<not> Cong B A D C"
distinct_79: fixes x4 :: "val"   and x51 :: "expr"   and x52 :: "bop"   and x53 :: "expr" shows "Val x4 \<noteq> x51 \<guillemotleft>x52\<guillemotright> x53"
GB_adds_lt: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and f :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.is_Groebner_basis G"   and "f \<in> local.punit.pmdl G"   and "f \<noteq> 0" obtains g :: "'a \<Rightarrow>\<^sub>0 'c" where "g \<in> G" and "g \<noteq> 0" and "local.punit.adds_term (punit.lt g) (punit.lt f)"
embed: fixes a :: "'a"   and n :: "nat"   and b :: "'a" assumes "a \<in> \<Sigma> n" shows "(b \<in> set (embed a)) = (b \<in> \<Sigma> (Suc n) \<and> project b = a)"
tm3': fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "n x \<cdot> n (n n z \<cdot> n n y) = n (n (n x \<cdot> n y) \<cdot> n (n x \<cdot> n z))"
associated_polynomials_imp_same_length: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list"   and q :: "'c list" assumes "domain R"   and "subring K R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "p \<sim>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> q" shows "length p = length q"
mkM_kripke: fixes T :: "'s Trace set" shows "kripke (Clock.mkM T)"
CARD: shows "\<forall>r\<in>R. Card_order r"
induct: fixes P :: "(('fun, 'c, 'sets) prot_fun, 'd) Term.term set \<Rightarrow> ('sets set \<times> 'sets set) list \<Rightarrow> (('fun, 'c, 'sets) prot_fun, 'd) Term.term \<Rightarrow> bool"   and a0 :: "(('fun, 'c, 'sets) prot_fun, 'd) Term.term set"   and a1 :: "('sets set \<times> 'sets set) list"   and a2 :: "(('fun, 'c, 'sets) prot_fun, 'd) Term.term" assumes "\<And>M TI x. P M TI (Var x)"   and "\<And>M TI f T. (\<And>z. z \<in> set T \<Longrightarrow> P M TI z) \<Longrightarrow> P M TI (Fun f T)" shows "P a0 a1 a2"
simps_3: fixes p :: "int \<times> int"   and \<delta> :: "real"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "t_find_closest p \<delta> (p\<^sub>0 # v # va) = 1 + (if \<delta> \<le> real_of_int (snd p\<^sub>0 - snd p) then 0 else t_find_closest p (min \<delta> (dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))) (v # va))"
pick_is_element: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "pick A \<in> A"
assms_5: shows "h \<turnstile> get_elements_by_class_name ptr name \<rightarrow>\<^sub>h h'"
crel_vs_iter_and_compute: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'd"   and f :: "'a \<Rightarrow> ('b, 'e) state"   and x :: "'a"   and y :: "'a" assumes "((=) ===>\<^sub>T R) g f" shows "crel_vs R (g x) (iter_state f y \<bind> (\<lambda>_. f x))"
exhaust: fixes y :: "varID" obtains "y = stA1"   | "y = stA2"   | "y = stA4"   | "y = stA6"
\<theta>_img: shows "subst_range \<theta> = insert (\<delta> x) (subst_range \<sigma>)"
ModelIkPolMcalc2C_axioms: shows "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf Ik_intT Ik_intF Ik_intP"
homogeneous_setI: fixes A :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "\<And>a n. a \<in> A \<Longrightarrow> hom_component a n \<in> A" shows "homogeneous_set A"
right_neutral: fixes a :: "'a" shows "a AND - (1::'a) = a"
wpo_ns_refl: fixes s :: "('f, 'b) Term.term" shows "wpo_ns s s"
boundedE: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "b + c \<le> a" obtains "b \<le> a" and "c \<le> a"
sgn_eq: fixes z :: "complex" shows "sgn z = z / complex_of_real (cmod z)"
less_restrictive_and_finite: fixes R1 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and R2 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" assumes "less_restrictive R1 R2"   and "derived_clauses_are_finite R1" shows "derived_clauses_are_finite R2"
exp_mult_2: fixes y :: "real" shows "exp (y * 2) = exp y * exp y"
R: shows "(w', w'') \<in> (\<Union> (relations M ` as))\<^sup>+"
assms: shows "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (z, y) \<in> r \<longrightarrow> y \<notin> Q)"
indets_dehomogenize: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "indets (dehomogenize x p) \<subseteq> indets p - {x}"
acc_lengths_eq_Nil_iff: fixes acc :: "nat"   and ls :: "'a list list" shows "(acc_lengths acc ls = []) = (ls = [])"
l_red: shows "is_reducible_lit Ds (C @ remove1 L (M # C')) L"
DiaN: fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and i :: "'c"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> (\<^bold>\<diamond> p)) at a in' set ((ps, a) # branch)"   and "(\<^bold>\<diamond> Nom i) at a in' set ((ps, a) # branch)"   and "new (\<^bold>\<not> (\<^bold>@ i p)) a ((ps, a) # branch)"   and "A, Suc n \<turnstile> ((\<^bold>\<not> (\<^bold>@ i p)) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
new_document_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "ptr \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
not_in_Gromov_boundary': fixes x :: "'c" shows "to_Gromov_completion x \<notin> Gromov_boundary"
psubst_Var: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "goodEnv rho" shows "Var xs x #[rho] = (case rho xs x of None \<Rightarrow> Var xs x | Some X \<Rightarrow> X)"
crel_vs_lookup: fixes param :: "'k" shows "state_dp_consistency.crel_vs (\<lambda>v v'. case v' of None \<Rightarrow> True | Some v' \<Rightarrow> v = v' \<and> v = dp param) (dp param) (lookup_st param)"
set_zip_leftD: fixes x :: "'g"   and y :: "'h"   and xs :: "'g list"   and ys :: "'h list" assumes "(x, y) \<in> set (zip xs ys)" shows "x \<in> set xs"
Inl_in_sum_iff: fixes a :: "V"   and A :: "V"   and B :: "V" shows "(ZFC_Cardinals.Inl a \<in>\<^sub>\<circ> A \<Uplus> B) = (a \<in>\<^sub>\<circ> A)"
a_def: shows "a \<in> carrier R \<and> monom P (f (deg R f)) (deg R f) = monom P a n"
image_group_def: fixes f :: "'b \<Rightarrow> 'a"   and G :: "('b, 'c) monoid_scheme" shows "image_group f G \<equiv> \<lparr>carrier = f ` carrier G, mult = \<lambda>a b. f (inv_into (carrier G) f a \<otimes>\<^bsub>G\<^esub> inv_into (carrier G) f b), one = f \<one>\<^bsub>G\<^esub>\<rparr>"
T_hiding: fixes P :: "'b process"   and A :: "'b set" shows "T (P \ A) = {trace_hide t (ev ` A) |t. (t, ev ` A) \<in> F P} \<union> div_hide P A"
Reco2: assumes "Reco t t' \<and> s = s'"   and "a = Recover id1" shows "thesis"
inorder_delete: fixes t :: "'t"   and a :: "'a" assumes "inv t \<and> sorted1 (inorder t)" shows "inorder (delete a t) = del_list a (inorder t)"
Seq1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and e\<^sub>2 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a;; e\<^sub>2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
finite_induct: fixes F :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (x \<triangleright> F)" shows "P F"
ag_mOp_closed: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> x \<in> carrier A"
upd_obj_def: fixes n :: "vn"   and v :: "val" shows "upd_obj n v = (\<lambda>obj. obj\<lparr>values := values obj(n \<mapsto> v)\<rparr>)"