hcju/isabelleps huggingface dataset:
lemma_termdiff3: fixes h :: "'a"   and z :: "'a"   and K :: "real"   and n :: "nat" assumes "h \<noteq> (0::'a)"   and "norm z \<le> K"   and "norm (z + h) \<le> K" shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \<le> real n * real (n - Suc 0) * K ^ (n - 2) * norm h"
HStartBallot_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv4d s q" shows "HInv4d s' q"
unicastT: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>i\<^sub>p :: "'e \<Rightarrow> nat"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q), unicast (s\<^sub>i\<^sub>p \<xi>) (s\<^sub>m\<^sub>s\<^sub>g \<xi>), \<xi>, p) \<in> seqp_sos \<Gamma>"
mask_eq_iff: fixes w :: "'b word"   and n :: "nat" shows "(w AND mask n = w) = (uint w < 2 ^ n)"
1: shows "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go x) x (\<epsilon>o x)"
a_rcosets_part_G: shows "\<Union> (a_rcosets I) = carrier R"
norm_Pair1: fixes x :: "'d" shows "norm (0::'c, x) = norm x"
SIod_Ssegment_Ssegment: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier (SIod D (Ssegment D b))" shows "SIod (SIod D (Ssegment D b)) (Ssegment (SIod D (Ssegment D b)) a) = SIod D (Ssegment D a)"
conn2: fixes kid :: "'kid"   and k :: "cnf" assumes "\<parallel>kid\<parallel>\<^bsub>k\<^esub>" shows "ksop (\<sigma>\<^bsub>kid\<^esub>k) = (\<Union>bid\<in>{bid. \<parallel>bid\<parallel>\<^bsub>k\<^esub>}. {bbop (\<sigma>\<^bsub>bid\<^esub>k)})"
mem_case_prodI: fixes z :: "'a"   and c :: "'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and a :: "'b"   and b :: "'c" assumes "z \<in> c a b" shows "z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)"
assms_1: shows "fmdom' (fst a) \<subseteq> fmdom' s1"
f_1: fixes x :: "'a" assumes "x \<in> S" shows "g (f x) = x"
infplays_augment_one: fixes \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list" shows "strategy_progression \<sigma> (augment_list \<sigma> p) = strategy_progression \<sigma> p"
emeasure_lborel_cbox_finite: fixes a :: "'b"   and b :: "'b" shows "emeasure lborel (cbox a b) < \<infinity>"
digitToTree_list: fixes d :: "('c, 'd) Digit" shows "toList (digitToTree d) = digitToList d"
reject: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and p :: "'d" assumes "Matching_Ternary.matches \<gamma> m Reject p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m Reject], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalDeny"
maxsimpD_maximal: fixes x :: "'a set"   and z :: "'a set" assumes "chamber x"   and "z \<in> X"   and "order.greater_eq z x" shows "z = x"
newton_poly_impl_def: fixes xs :: "'b list"   and fs :: "'b list" shows "newton_poly_impl xs fs = horner_composition (rev (newton_coefficients xs fs)) xs"
that: fixes a :: "'a \<times> 'b"   and s :: "('a, 'b) FingerTreeStruc" assumes "viewL t = Some (a, s)"   and "ft_invar s"   and "toList t = a # toList s" shows "thesis"
group_homomorphism_axioms: shows "group_homomorphism \<eta> G (\<cdot>) \<one> G' (\<cdot>') \<one>'"
canon_ang_cos: fixes \<alpha> :: "real" shows "cos \<downharpoonright>\<alpha>\<downharpoonleft> = cos \<alpha>"
setVar_cl_inv: fixes g :: "'a gState_scheme"   and p :: "pState"   and v :: "varRef"   and val :: "integer" assumes "cl_inv (g, p)" shows "cl_inv (setVar v val g p)"
h2: shows "\<turnstile>\<^sub>p {adjoint (exexH_k k) * (adjoint (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * P * tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * exexH_k k} hadamard_n (Suc k) {adjoint (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * P * tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)}"
simps_2: fixes k :: "nat"   and x :: "'a"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "ins k x (Node ts t) = (case split ts x of (ls, []) \<Rightarrow> case ins k x t of T\<^sub>i a \<Rightarrow> T\<^sub>i (Node ls a) | Up\<^sub>i l a x \<Rightarrow> node\<^sub>i k (ls @ [(l, a)]) x | (ls, (sub, sep) # rs) \<Rightarrow> if sep = x then T\<^sub>i (Node ts t) else case ins k x sub of T\<^sub>i a \<Rightarrow> T\<^sub>i (Node (ls @ (a, sep) # rs) t) | Up\<^sub>i l a r \<Rightarrow> node\<^sub>i k (ls @ (l, a) # (r, sep) # rs) t)"
OUTfromVCorrect1_data1: shows "OUTfromVCorrect1 data1"
assms: shows "Deliver (i, Insert e n) \<in> set (history j)"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier \<G>"   and "y \<in> carrier \<G>" shows "x \<otimes> y \<in> carrier \<G>"
convergent_minus_iff_ereal: fixes f :: "nat \<Rightarrow> ereal" shows "convergent (\<lambda>x. - f x) = convergent f"
chkargs_def: fixes gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "chkargs gt t s = (\<forall>s'\<in>set (args s). gt t s')"
f2: shows "Deliver (i, Create i e2) \<notin> set (history k)"
nth_default_def: fixes dflt :: "'c"   and xs :: "'c list"   and n :: "nat" shows "nth_default dflt xs n = (if n < length xs then xs ! n else dflt)"
f_Exec_Stream_expand_shrink_last_nth_eq_f_Exec_Comp: fixes k :: "nat"   and n :: "nat"   and xs :: "'d message_af list"   and trans_fun :: "'d message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "0 < k"   and "n < length xs" shows "f_Exec_Comp_Stream trans_fun (xs \<odot> k) c \<div>\<^sub>l k ! n = f_Exec_Comp trans_fun (xs \<down> Suc n \<odot> k) c"
simps_10: fixes f1 :: "agent \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> 'a"   and f2 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and f3 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and x11 :: "agent"   and x12 :: "agent"   and x13 :: "msg" shows "(case Says x11 x12 x13 of Says x xa xb \<Rightarrow> f1 x xa xb | Gets x xa \<Rightarrow> f2 x xa | Notes x xa \<Rightarrow> f3 x xa) = f1 x11 x12 x13"
"4_2": shows "length (filter P3 l''') \<le> length (filter P3 l2)"
subst_Pair: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f"   and a :: "'e \<times> 'f" assumes "P x y"   and "a = (x, y)" shows "P (fst a) (snd a)"
gen_dfs'_def: shows "gen_dfs' \<equiv> gds_init gds \<bind> WHILE gen_cond gen_step'"
that: fixes U :: "'a set" assumes "pairwise orthogonal (S \<union> U)"   and "span (S \<union> U) = span (S \<union> T)" shows "thesis"
simps_17: fixes b :: "nat"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex" shows "Formula.fvi b (formula.MatchP I r) = Regex.fv_regex (Formula.fvi b) r"
in_remove1_mset_neq: fixes a :: "'b"   and b :: "'b"   and C :: "'b multiset" assumes "a \<noteq> b" shows "(a \<in># remove1_mset b C) = (a \<in># C)"
ab_mod_pP: shows "[a = b] (mod p * \<Prod>P)"
euclidean_representation: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and Basis :: "'d set"   and x :: "'d" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "comm_monoid_add.sum plus zero (\<lambda>b. scaleR (inner x b) b) Basis = x"
assms: shows "set \<Phi>\<^sub>1 \<subseteq> set \<Phi>\<^sub>2"
foldl_map: fixes g :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
hyp2_axiom5: shows "\<forall>a b c d a' b' c' d'. a \<noteq> b \<and> B\<^sub>K a b c \<and> B\<^sub>K a' b' c' \<and> a b \<congruent>\<^sub>K a' b' \<and> b c \<congruent>\<^sub>K b' c' \<and> a d \<congruent>\<^sub>K a' d' \<and> b d \<congruent>\<^sub>K b' d' \<longrightarrow> c d \<congruent>\<^sub>K c' d'"
final: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.final M \<subseteq> dfa.states M"
dom_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.dom (B.VVV.dom f) = B.VVV.dom f"
fringe_def: fixes prb :: "('d, 'e, 'f, 'g) pre_RedBlack_scheme" shows "fringe prb \<equiv> {rv \<in> red_vertices prb. rv \<notin> subsumees (subs prb) \<and> \<not> marked prb rv \<and> ui_edge ` out_edges (red prb) rv \<subset> out_edges (black prb) (fst rv)}"
linear_continuous_on: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "bounded_linear f" shows "continuous_on s f"
simps_1: fixes Q :: "hf set" shows "ST.nextl Q [] = ST.epsclo Q"
path_from_toE: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H_x\<^esub> w" shows "Q.path xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
IH1: fixes s :: "'a net_state" assumes "s \<in> init (closed (pnet np p1))"   and "i \<in> net_tree_ips p1" shows "the (netmap s i) \<in> init (np i)"
assms_5: fixes n :: "nat" shows "f n \<in> S"
subcomplex_gallery: fixes Y :: "'a set set"   and Cs :: "'a set list" assumes "folding.ChamberSubcomplex Y"   and "SimplicialComplex.maxsimpchain Y Cs" shows "folding.gallery Cs"
wti_pres: fixes n :: "nat"   and c :: "'a list"   and pc :: "nat"   and s :: "'a" assumes "pres_type step n A"   and "c ! (pc + 1) \<in> A"   and "s \<in> A"   and "pc < n" shows "wti c pc s \<in> A"
unpadl_padl: fixes m :: "nat"   and v :: "'b vec" shows "vardim.unpadl m (0\<^sub>v m @\<^sub>v v) = v"
step: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r x y"   and "star r y z" shows "star r x z"
top: shows "(f \<longlongrightarrow> b) at_top"
inject_2: fixes x3 :: "tnam"   and y3 :: "tnam" shows "(TName x3 = TName y3) = (x3 = y3)"
quot_of_fract_diff: fixes x :: "'b fract"   and y :: "'b fract" shows "quot_of_fract (x - y) = (let (a, b) = quot_of_fract x; (c, d) = quot_of_fract y in normalize_quot (a * d - b * c, b * d))"
finite_extension_is_subfield: fixes K :: "'a set"   and xs :: "'a list" assumes "subfield K R"   and "set xs \<subseteq> carrier R"   and "\<And>x. x \<in> set xs \<Longrightarrow> (algebraic over K) x" shows "subfield (finite_extension K xs) R"
KL_divergence_def: fixes b :: "real"   and M :: "'b measure"   and N :: "'b measure" shows "KL_divergence b M N = integral\<^sup>L N (entropy_density b M N)"
inverts_mat_def: fixes A :: "'a Matrix.mat"   and B :: "'a Matrix.mat" shows "inverts_mat A B \<equiv> A * B = 1\<^sub>m (dim_row A)"
intro: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool" assumes "ltl_equivalence eq"   and "af_congruent_axioms eq" shows "af_congruent eq"
mult_inf_left: fixes z :: "complex_homo" assumes "z \<noteq> 0\<^sub>h" shows "\<infinity>\<^sub>h *\<^sub>h z = \<infinity>\<^sub>h"
IH: assumes "\<Gamma> \<turnstile>g c input \<bind> f \<surd>" shows "\<Gamma> \<turnstile>g f x \<surd>"
has_integral_empty_eq: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c" shows "(f has_integral i) {} = (i = (0::'c))"
assms_2: fixes i :: "nat" assumes "enat i < llength u"   and "enat i < llength v" shows "u ?! i = v ?! i"
nth_map_index: fixes p :: "nat"   and xs :: "'b list"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "p < length xs" shows "map_index f xs ! p = f p (xs ! p)"
fds_const_def: fixes c :: "'b" shows "fds_const c = fds (\<lambda>n. if n = 1 then c else (0::'b))"
convergent_LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'a" shows "convergent X = X \<longlonglongrightarrow> lim X"
oreturn_NF_wp: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b" shows "ovalidNF (P x) (oreturn x) P"
idem: fixes a :: "'b" shows "a \<sqinter> a = a"
facet_ex_third_chamber: fixes C :: "'a set"   and z :: "'a set"   and D :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "\<exists>E\<in>X - {C, D}. z \<lhd> E"
matrix_to_iarray_fst_Gauss_Jordan_PA: fixes A :: "(('b, 'd) vec, 'c) vec" shows "matrix_to_iarray (fst (Gauss_Jordan_PA A)) = fst (Gauss_Jordan_iarrays_PA (matrix_to_iarray A))"
f: shows "atom a \<sharp> subst_type t (Mu a t) a"
RSD_pareto_eqclassD_2: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "RSD_pareto_eqclass agents alts R A" shows "A \<subseteq> alts"
wge: fixes n :: "nat" assumes "0 < n" shows "real n * w 1 \<le> w n"
psimps_3: fixes ks :: "'a list"   and v :: "(real, 'a) vec"   and vb :: "(real, 'a) vec"   and vc :: "(real, 'a) vec list" assumes "build_dom (ks, v # vb # vc)" shows "build ks (v # vb # vc) = (let (k, uu_) = widest_spread ks (v # vb # vc); (l, m, r) = partition_by_median k (v # vb # vc) in Node k m (build ks l) (build ks r))"
zcf_zero: fixes R :: "('c, 'd) ring_scheme" assumes "UP_cring R" shows "zcf \<zero>\<^bsub>UP R\<^esub> = \<zero>\<^bsub>R\<^esub>"
sup_inf_distrib_2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<sqinter> z \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
b_assn_invalid_merge6: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" shows "hn_invalid (b_assn A P) x y \<or>\<^sub>A hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_invalid A x y"
second_vertices_new_path: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and xs :: "'c list" assumes "DisjointPaths G v0 v1 paths"   and "hd (tl xs) \<notin> DisjointPaths.second_vertices paths" shows "xs \<notin> paths"
qAFresh_imp_qFresh: fixes xs :: "'a"   and x :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" assumes "qAFresh xs x X" shows "qFresh xs x X"
E'_V2: fixes x :: "'a"   and y :: "'a" assumes "E' x y" shows "V y"
q_2: fixes x :: "'a set" assumes "x \<in> p" shows "x \<in> q x"
leq: shows "(\<Sum>(r, a)\<leftarrow>C. lec_of_nsc (r *R nsc_of_atom a)) = Le_Constraint Leq_Rel p c"
While: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and T :: "(char list \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> bool"   and c :: "com" assumes "\<And>n. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> T s n} c {\<lambda>s. P s \<and> (\<exists>n'<n. T s n')}" shows "\<turnstile>\<^sub>t {\<lambda>s. P s \<and> (\<exists>n. T s n)} WHILE b DO c {\<lambda>s. P s \<and> \<not> bval b s}"
lincomb_linear_image: fixes A :: "'c set"   and a :: "'e \<Rightarrow> 'a" assumes "inj_on T (carrier V)"   and "A \<subseteq> carrier V"   and "a \<in> T ` A \<rightarrow> carrier K"   and "finite A" shows "W.module.lincomb a (T ` A) = T (V.module.lincomb (a \<circ> T) A)"
f18: shows "s \<otimes> s' \<otimes> (fst x \<otimes> fst x') \<in> carrier R"
acyclic_6_implies_spannable: fixes x :: "'a" assumes "acyclic_6 x" shows "spannable x"
leqS: shows "P \<turnstile> C' \<prec>\<^sub>S D"
exec_stop1: fixes \<sigma> :: "'a control_state_scheme" assumes "break_status \<sigma>" shows "exec_stop \<sigma>"
not_outside_1: fixes x :: "'alt"   and y :: "'alt" assumes "R i x y" shows "x \<in> alts"
cDiffD2: fixes c :: "'f"   and A :: "'f cset"   and B :: "'f cset"   and P :: "bool" assumes "cin c (cDiff A B)"   and "cin c B" shows "P"
disj_nb: fixes p :: "fm"   and q :: "fm" assumes "bound0 p"   and "bound0 q" shows "bound0 (Cooper.disj p q)"
Int_lower1: fixes A :: "'a set"   and B :: "'a set" shows "order.greater_eq A (A \<inter> B)"
distinct_7: fixes x1 :: "nat"   and x51 :: "trm"   and x52 :: "POPLmark.type" shows "Var x1 \<noteq> x51 \<bullet>\<^sub>\<tau> x52"
iso_2: shows "hom_induced p (subtopology X S) {} X {} id \<in> Group.iso (subgroup_generated (homology_group p (subtopology X S)) K) (homology_group p X)"
vsingletonI: fixes x :: "V" shows "x \<in>\<^sub>\<circ> ZFC_in_HOL.set {x}"
I_in_na_def: fixes s :: "'b list"   and t :: "'b list"   and i :: "nat" shows "I_in_na s t i \<equiv> \<lambda>(j, pos). case pos of None \<Rightarrow> j < length s \<and> (\<forall>jj<j. t ! (i + jj) = s ! jj) | Some p \<Rightarrow> sublist_at s t i"
compact_def: fixes S :: "'a set" shows "compact S = (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
filterlim_at_top: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "filterlim f at_top F = (\<forall>Z. \<forall>\<^sub>F x in F. Z \<le> f x)"
simps_1: fixes f :: "'e option \<Rightarrow> 'e"   and v :: "'e option"   and ps :: "('d \<times> ('d, 'e) trie) list" shows "update_with_trie [] f (Trie v ps) = Trie (Some (f v)) ps"
relcomp_distrib2: fixes S :: "('b \<times> 'd) set"   and T :: "('b \<times> 'd) set"   and R :: "('d \<times> 'c) set" shows "(S \<union> T) O R = S O R \<union> T O R"
b1: fixes s1 :: "lam"   and s2 :: "lam"   and t :: "lam" assumes " s1 \<longrightarrow>\<^sub>\<beta> s2" shows " App s1 t \<longrightarrow>\<^sub>\<beta> App s2 t"
pdevs_val_inj_sumI: fixes K :: "'b set"   and g :: "'b \<Rightarrow> nat"   and x :: "'c pdevs"   and f :: "'b \<Rightarrow> 'c"   and e :: "nat \<Rightarrow> real" assumes "finite K"   and "inj_on g K"   and "pdevs_domain x \<subseteq> g ` K"   and "\<And>i. \<lbrakk>i \<in> K; g i \<notin> pdevs_domain x\<rbrakk> \<Longrightarrow> f i = (0::'c)"   and "\<And>i. \<lbrakk>i \<in> K; g i \<in> pdevs_domain x\<rbrakk> \<Longrightarrow> f i = e (g i) *\<^sub>R pdevs_apply x (g i)" shows "pdevs_val e x = sum f K"
igSubstIGAbs_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstIGAbs MOD \<equiv> \<forall>ys y Y xs x s X. isInBar (xs, s) \<and> igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
polymap_plus: fixes p :: "'f poly"   and q :: "'f poly" shows "polymap (p + q) = polymap p + polymap q"
sub2: fixes xs :: "'a vec list"   and n :: "nat"   and us :: "'a vec list" shows "rev xs @ snd (gram_schmidt.sub2_wit n xs us) = rev (gram_schmidt_sub n xs us)"
plus_mat0: fixes nr :: "nat"   and nc :: "nat"   and m :: "'a list list" assumes "mat nr nc m" shows "mat_plus m (mat0 nr nc) = m"
quot_dbfm_coding: fixes fm :: "dbfm" shows "coding_tm (quot_dbfm fm)"
nth_append_length_plus: fixes xs :: "'g list"   and ys :: "'g list"   and n :: "nat" shows "(xs @ ys) ! (length xs + n) = ys ! n"
map_closed: fixes a :: "'b" assumes "a \<in> M'" shows "restrict (inv_into M \<eta>) M' a \<in> M"
length_modify: fixes tp :: "(action \<times> nat) list" shows "length (modify_tprog tp) = 2 * length tp"
runit_char_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b"   and f :: "'b" assumes "right_hom_with_unit V H \<a> \<iota> a"   and "partial_magma.ide (subcategory.comp V (weak_composition.right V H a)) f" shows "weak_composition.H\<^sub>R H a (right_hom_with_unit.runit V H \<a> \<iota> a f) = subcategory.comp V (weak_composition.right V H a) (H f \<iota>) (\<a> f a a)"
openin_Un: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "openin U S"   and "openin U T" shows "openin U (S \<union> T)"
initialized_def: fixes A :: "int \<Rightarrow> two_fields"   and i :: "int" shows "initialized A i = (\<forall>j\<in>{0..<i}. initialized2 (Field1 (A j)) 10 \<and> Field2 (A j) = 0)"
simps: fixes a :: "int"   and n :: "int" shows "jacobi_code a n = (if n = 0 then 0 else if n = 1 then 1 else if a = 1 then 1 else if n < 0 then jacobi_code a (- n) else if even n then if even a then 0 else jacobi_code a (n div 2) else if a < 0 then (if n mod 4 = 1 then 1 else - 1) * jacobi_code (- a) n else if a = 0 then 0 else if n \<le> a then jacobi_code (a mod n) n else if even a then (if n mod 8 \<in> {1, 7} then 1 else - 1) * jacobi_code (a div 2) n else if coprime a n then (if n mod 4 = 3 \<and> a mod 4 = 3 then - 1 else 1) * jacobi_code n a else 0)"
prems_12: shows "\<nexists>\<nu> \<mu>. \<psi> = \<nu> and\<^sub>r \<mu> \<or> \<psi> = X\<^sub>r \<nu>"
invmfun_l_inv: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and m :: "'a" assumes "R module N"   and "bijec\<^bsub>M,N\<^esub> f"   and "m \<in> carrier M" shows "invmfun R M N f (f m) = m"
suminf_cong: fixes f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" assumes "\<And>n. f n = g n" shows "suminf f = suminf g"
fresh_def: fixes a :: "atom"   and x :: "'b" shows "a \<sharp> x \<equiv> a \<notin> supp x"
quotient_map_from_composition: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "continuous_map X X' f"   and "continuous_map X' X'' g"   and "quotient_map X X'' (g \<circ> f)" shows "quotient_map X' X'' g"
eventually_bot: fixes P :: "'c \<Rightarrow> bool" shows "eventually P bot"
cases: fixes x :: "'a \<times> nat" obtains c :: "'a" and n :: "nat" where "x = (c, n)"
sgn_greater: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (sgn a) = less zero a"
floor_eq2: fixes n :: "int"   and x :: "real" assumes "real_of_int n \<le> x"   and "x < real_of_int n + 1" shows "\<lfloor>x\<rfloor> = n"
subgame_valid_path: fixes P :: "'a llist"   and V' :: "'a set" assumes "valid_path P"   and "lset P \<subseteq> V'" shows "Digraph.valid_path (subgame V') P"
finsert_iff: fixes a :: "'x"   and b :: "'x"   and A :: "'x fset" shows "(a |\<in>| finsert b A) = (a = b \<or> a |\<in>| A)"
row_c: shows "dmu_array_row_cost dmus i = (dmus', c_row)"
app_term_def: fixes t :: "Term_Class.term"   and u :: "Term_Class.term" shows "app t u = t $ u"
inv_inv: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "inv\<^bsub>DirProds G I\<^esub> (inv\<^bsub>DirProds G I\<^esub> x) = x"
classes_above_fields: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "fields P C = fields P' C"
rep_eq: fixes x :: "(real \<times> real) pdevs" shows "pdevs_apply (nlex_pdevs x) = (\<lambda>n. if Counterclockwise_2D_Arbitrary.lex 0 (pdevs_apply x n) then - pdevs_apply x n else pdevs_apply x n)"
iso_lunit: fixes a :: "'a" assumes "ide a" shows "local.iso (MC.lunit a)"
fsubset_fempty: fixes A :: "'c fset" shows "(A |\<subseteq>| {||}) = (A = {||})"
map_of_delete_iff: fixes x :: "'e"   and \<Gamma> :: "('e \<times> 'd) list"   and xa :: "'e"   and e :: "'d" shows "(map_of (delete x \<Gamma>) xa = Some e) = (map_of \<Gamma> xa = Some e \<and> xa \<noteq> x)"
Suc_dim: fixes v :: "'a"   and E :: "'a set"   and n :: "nat"   and K :: "'a set" assumes "v \<in> carrier R"   and "v \<notin> E"   and "dimension n K E" shows "dimension (Suc n) K (line_extension K v E)"
finite_cone_def: fixes Vs :: "'a vec set" shows "next_dim.finite_cone Vs = {b. \<exists>c. next_dim.nonneg_lincomb c (if finite Vs then Vs else {}) b}"
process_prop_5_def: fixes P :: "('a list \<times> 'a set) set \<times> 'a list set" shows "process_prop_5 P \<equiv> \<forall>xs x. xs \<in> snd P \<longrightarrow> xs @ [x] \<in> snd P"