hcju/isabelleps huggingface dataset:
intro: fixes Y :: "'c set"   and is_open\<^sub>Y :: "'c set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'c set \<Rightarrow> 'd set"   and \<rho>\<^sub>Y :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and d :: "'d"   and add_str\<^sub>Y :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str\<^sub>Y :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str\<^sub>Y :: "'c set \<Rightarrow> 'd"   and one_str\<^sub>Y :: "'c set \<Rightarrow> 'd"   and X :: "'f set"   and \<O>\<^sub>X :: "'f set \<Rightarrow> 'e set"   and f :: "'f \<Rightarrow> 'c"   and \<rho>\<^sub>X :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'f set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'f set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'f set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'f set \<Rightarrow> 'e"   and \<phi>\<^sub>f :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" assumes "morphism_sheaves_of_rings Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y (im_sheaf.im_sheaf X \<O>\<^sub>X f) (im_sheaf.im_sheaf_morphisms X \<rho>\<^sub>X f) b (im_sheaf.add_im_sheaf X add_str\<^sub>X f) (im_sheaf.mult_im_sheaf X mult_str\<^sub>X f) (im_sheaf.zero_im_sheaf X zero_str\<^sub>X f) (im_sheaf.one_im_sheaf X one_str\<^sub>X f) \<phi>\<^sub>f" shows "morphism_ringed_spaces_axioms X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f"
simps_1: fixes \<O> :: "'a \<Rightarrow> 'b pmf"   and \<S> :: "'a set"   and state_list :: "'a list"   and \<O>\<^sub>s :: "'b set"   and \<K> :: "'a \<Rightarrow> 'a pmf"   and s :: "'a"   and t_end :: "'a" assumes "HMM3 \<O> \<S> state_list \<O>\<^sub>s \<K>" shows "HMM3.viterbi \<O> state_list \<K> s t_end [] = ([], indicator {t_end} s)"
isomorphic_categories_axioms: shows "isomorphic_categories J' J.comp"
in_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> L F h) = (g \<in> L F h)"
conj_cancel_left: fixes x :: "'a" shows "NOT x AND x = (0::'a)"
xa: shows "\<forall>xa\<in>row_space A. orthogonal x xa"
subst_apply_fv_empty: fixes t :: "('d, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "fv t = {}" shows "fv (t \<cdot> \<sigma>) = {}"
cring_fieldI2: fixes R :: "('e, 'f) ring_scheme" assumes "cring R"   and "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub>"   and "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes>\<^bsub>R\<^esub> b = \<one>\<^bsub>R\<^esub>" shows "field R"
P2FQ2: shows "(P2, F, Q2) = Smith_2xn D"
simps: fixes x :: "'a"   and y :: "'b Filters.filter"   and z :: "'a"   and w :: "'b Filters.filter" shows "pairs_less_eq (x, y) (z, w) = (x \<le> z \<and> w \<le> y)"
to_nat_inj: fixes a :: "'b"   and b :: "'b" assumes "Bij_Nat.to_nat a = Bij_Nat.to_nat b" shows "a = b"
admissible_compact_neq: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" assumes "ccpo.compact lub ord k" shows "ccpo.admissible lub ord ((\<noteq>) k)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_monoid_mult_hom hom"   and "zero_hom hom" shows "comm_semiring_1_hom hom"
intro: fixes M :: "'b dfa" assumes "dfa.init M \<in> dfa.states M"   and "dfa.final M \<subseteq> dfa.states M"   and "\<And>q x. q \<in> dfa.states M \<Longrightarrow> dfa.nxt M q x \<in> dfa.states M"   and "finite (dfa.states M)" shows "dfa M"
Int_Diff_disjoint: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B \<inter> (A - B) = {}"
mpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "mpt M T \<equiv> qmpt M T \<and> mpt_axioms M T"
ennreal_pmf_bind: fixes N :: "'c pmf"   and f :: "'c \<Rightarrow> 'b pmf"   and i :: "'b" shows "ennreal (pmf (N \<bind> f) i) = \<integral>\<^sup>+ x. ennreal (pmf (f x) i) \<partial>measure_pmf N"
hom_hd: shows "homotopic_with_canon (\<lambda>z. True) (sphere (0::'a) 1 \<inter> S) (T - {0::'a}) h (\<lambda>x. - d)"
exec_meth_take_xt': fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and ins' :: "'addr instr list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P (ins @ ins') (xt' @ xt) t h (stk, loc, pc, xcp) ta h' s'"   and "pc < length ins"   and "pc \<notin> pcs xt" shows "exec_meth ci P ins xt' t h (stk, loc, pc, xcp) ta h' s'"
simps_2: fixes Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "transfer (Q)\<^sub>\<surd> (cf # cfs) = cf # cfs"
subtype_tranclp_Collection_x: fixes \<tau> :: "'b OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type" assumes "(\<sqsubset>)\<^sup>+\<^sup>+ (Collection \<tau>) \<sigma>" obtains \<rho> :: "'b OCL_Types.type" where "\<sigma> = Collection \<rho>" and "(\<sqsubset>)\<^sup>+\<^sup>+ \<tau> \<rho>"   | "\<sigma> = OclSuper"
PE_lf_rep_def: fixes R :: "(nat \<Rightarrow> ValD) admS dual \<times> (nat \<Rightarrow> ValD) admS" shows "PE_lf_rep R \<equiv> PE_base_lf_rep R \<union> fn_lf_rep R"
lookup_map_default: fixes x :: "'c"   and d :: "'b"   and f :: "'b \<Rightarrow> 'b"   and m :: "('c, 'b) mapping" shows "lookup (map_default x d f m) x = Some (f (lookup_default d m x))"
iMODb_add: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and k :: "nat" shows "[ r, mod m, c ] \<oplus> k = [ r + k, mod m, c ]"
zcount_sum: fixes f :: "'b \<Rightarrow> 'a zmultiset"   and MM :: "'b set"   and t :: "'a" shows "zcount (sum f MM) t = (\<Sum>M\<in>MM. zcount (f M) t)"
SeqCTermP_imp_LstSeqP: fixes vf :: "bool"   and s :: "tm"   and k :: "tm"   and t :: "tm" shows "{SeqCTermP vf s k t} \<turnstile> LstSeqP s k t"
simps_1: fixes x21 :: "'d"   and x22 :: "'d list"   and y21 :: "'d"   and y22 :: "'d list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
binomial_pmf_0: fixes p :: "real" assumes "p \<in> {0..1}" shows "binomial_pmf 0 p = return_pmf 0"
prefs_from_table_wf_def: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list" shows "prefs_from_table_wf agents alts xss = (agents \<noteq> {} \<and> alts \<noteq> {} \<and> distinct (map fst xss) \<and> set (map fst xss) = agents \<and> (\<forall>xs\<in>set (map snd xss). \<Union> (set xs) = alts \<and> is_finite_weak_ranking xs))"
4: fixes init :: "'a list"   and qs :: "'a list"   and x :: "nat" assumes "distinct init"   and "set qs \<subseteq> set init"   and "x < length qs" shows "finite (set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take x qs)))"
widest_spread_axis_def: fixes k :: "'b"   and K :: "'b set"   and ps :: "(real, 'b) vec set" shows "widest_spread_axis k K ps = (\<forall>k'\<in>K. spread k' ps \<le> spread k ps)"
normalize_match_empty: fixes m :: "'b match_expr"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" assumes "normalize_match m = []" shows "\<not> matches \<gamma> m a p"
neq_Nil_conv: fixes xs :: "'d list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
D_gchain1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "D_gchain G n f" shows "inj_on f {i. i \<le> n}"
align\<^sub>d_Na: shows "local.aligned\<^sub>d e Na"
cp: shows "check_precond s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o"
subset_\<Q>: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "\<Q> i \<subseteq> \<Q> j"
hidden: assumes "V \<in> set Vs" shows "ls ! last_index Vs V = ls' ! last_index Vs V"
encode2_lt: fixes i :: "nat"   and dims :: "nat list"   and vars :: "nat set" assumes "i < state_sig.d dims" shows "partial_state.encode2 dims vars i < partial_state.d2 dims vars"
witness_is_deep_model: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "remove_weights (witness Y r rs) = deep_model Y r rs"
exp_eq_zero_iff: fixes n :: "nat" shows "(2 ^ n = 0) = (LENGTH('a) \<le> n)"
takeWhile_append: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "takeWhile P (xs @ ys) = (if \<forall>x\<in>set xs. P x then xs @ takeWhile P ys else takeWhile P xs)"
simps_5: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n\<^sub>s :: "nat"   and n' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "Option.is_none (Blockchain.devBC active cmp bc mining t n nid (n\<^sub>s + Suc n'))" shows "Blockchain.devExt active cmp bc mining t n nid n\<^sub>s (Suc n') = Blockchain.devExt active cmp bc mining t n nid n\<^sub>s n'"
vcard_mono: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "vcard A \<subseteq>\<^sub>\<circ> vcard B"
saturatep_saturate_eq: fixes CC :: "'a literal multiset set" shows "saturatep (\<lambda>x. x \<in> CC) = (\<lambda>x. x \<in> saturate CC)"
Max_mapping_def: fixes Ps :: "('d \<Rightarrow> 'c option) set"   and x :: "'d" shows "Max_mapping Ps x = (if \<forall>P\<in>Ps. P x \<noteq> None then Some (MAX P\<in>Ps. the (P x)) else None)"
pell_solution_exists: fixes D :: "nat" assumes "pell D" shows "\<exists>x y. y \<noteq> 0 \<and> x\<^sup>2 = 1 + D * y\<^sup>2"
esssup_I: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and c :: "'c" assumes "f \<in> borel_measurable M"   and "AE x in M. f x \<le> c" shows "esssup M f \<le> c"
induct: fixes P :: "(Vars.var \<times> exp) list \<Rightarrow> bool"   and a0 :: "(Vars.var \<times> exp) list" assumes "P []"   and "\<And>x e \<Gamma>. (\<And>xa. P \<Gamma>) \<Longrightarrow> P ((x, e) # \<Gamma>)" shows "P a0"
append_take_drop_id: fixes n :: "nat"   and xs :: "'e list" shows "take n xs @ drop n xs = xs"
tensor_prod_uniqueTr: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and N :: "('i, 'e, 'h) Module_scheme"   and MN :: "('j, 'e, 'h) Module_scheme"   and MN1 :: "('j, 'e, 'h) Module_scheme"   and f :: "'g \<times> 'i \<Rightarrow> 'j"   and g :: "'g \<times> 'i \<Rightarrow> 'j" assumes "Ring R"   and "R module M"   and "R module N"   and "R module MN"   and "R module MN1"   and "universal_property R M N MN f"   and "universal_property R M N MN1 g" shows "\<exists>!k. k \<in> mHom R MN1 MN \<and> compose (M \<times>\<^sub>c N) k g = f"
equiv_transI: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> t P = u P" shows "equiv_trans t u"
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>"
assms_2: shows "zb \<noteq> \<infinity>\<^sub>h"
reachable: fixes as :: "'b list"   and r :: "'b rexp" shows "reachable_b as r = {fold delta_b w (init_b r) |w. w \<in> lists (set as)}"
if_distrib_map_spmf: fixes f :: "'b \<Rightarrow> 'a"   and b :: "bool"   and p :: "'b spmf"   and q :: "'b spmf" shows "map_spmf f (if b then p else q) = (if b then map_spmf f p else map_spmf f q)"
trivial: fixes eq :: "'a"   and xs :: "'b list" assumes "trivial eq" shows "I\<^sub>a eq xs"
simps_2: fixes x :: "name"   and xs :: "name list"   and t :: "lam" shows "bind (x # xs) t = Lam [x].bind xs t"
support_preList: fixes upds :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'a) list"   and C1 :: "acom" shows "support (preList upds C1) \<subseteq> lesvars upds"
mset_quicksort: fixes xs :: "'c list" shows "mset (quicksort xs) = mset xs"
diffs_def: fixes c :: "nat \<Rightarrow> 'b" shows "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
add_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {arith_type ADD, arith_type ADDcc, arith_type ADDX}" shows "snd (add_instr i s) = False"
shrK_in_analz_respond: fixes RB :: "msg"   and evs :: "event list"   and B :: "agent" assumes "RB \<in> responses evs"   and "evs \<in> recur" shows "(Key (shrK B) \<in> analz (insert RB (knows Spy evs))) = (B \<in> bad)"
size_mset_heap: fixes ts :: "'b tree list" assumes "invar ts" shows "real (length ts) \<le> log 2 (real (size (mset_heap ts) + 1))"
sbelow_halfspace_def: fixes sctn :: "'b sctn" shows "sbelow_halfspace sctn = Collect (lt_halfspace sctn)"
hom_a_inv: fixes x :: "'a" assumes "x \<in> carrier R" shows "h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
that: fixes d :: "'a"   and E :: "'a set" assumes "f a = g d"   and "f ` B = g ` E"   and "d \<in> C"   and "E \<subseteq> C" shows "thesis"
elem_exact: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> carrier S"   and "a \<in> A" shows "a .\<in> A"
prob_\<mu>_\<Delta>\<^sub>a: shows "prob {a \<in> space M. Y a = 0} \<le> 1 / \<mu> + \<Delta>\<^sub>a / \<mu>\<^sup>2 - 1"
pac_idealI2: fixes p :: "int mpoly"   and A :: "int mpoly set" assumes "p \<in> More_Modules.ideal A" shows "p \<in> pac_ideal A"
split_list: fixes x :: "'a"   and xs :: "'a list" assumes "x mem xs" shows "\<exists>ys zs. xs = ys @ x # zs"
balance_bheight: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "bheight l = bheight r" shows "bheight (balance l k v r) = Suc (bheight l)"
fm: shows "integrable (lebesgue_on {0..a}) (\<lambda>x. f (- x))"
iprev_atMost: fixes t :: "nat"   and n :: "nat" assumes "t \<le> n" shows "iprev t {..n} = t - Suc 0"
case_option_collapse: fixes x :: "'d" shows "case_option x (\<lambda>_. x) = (\<lambda>_. x)"
f6: fixes r :: "real"   and ra :: "real" shows "r + (1 - (r + 1) + ra) = ra"
v_3: shows "v\<^sub>3 \<in> carrier_vec n"
bypass: fixes z :: "'v" assumes "z \<in> set p" shows "z \<notin> TER (current_of_bipartite f)"
freeIn_elimVar: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and A :: "atom" shows "freeIn var (elimVar var L F A)"
opp_fold_A_chamber_system_image_fixespointwise: fixes F :: "'a set" assumes "F \<in> ChamberComplex.chamber_system A" shows "fixespointwise opp_fold_A (opp_fold_A ` F)"
States_def: shows "States = fst \<circ> Rep_seqauto"
vconverse_vcomp: fixes r :: "V"   and s :: "V" shows "(r \<circ>\<^sub>\<circ> s)\<inverse>\<^sub>\<circ> = s\<inverse>\<^sub>\<circ> \<circ>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
ovalidNF_combine: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "ovalid P f Q"   and "no_ofail P f" shows "ovalidNF P f Q"
similar_sq_mtx_matrix: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx" shows "(A \<sim> B) = similar_matrix (to_vec A) (to_vec B)"
hd_prog_last_prog_end: fixes p :: "'b"   and sb :: "'b memref list"   and p\<^sub>s\<^sub>b :: "'b" assumes "p = hd_prog p sb"   and "last_prog p sb = p\<^sub>s\<^sub>b" shows "p = hd_prog p\<^sub>s\<^sub>b sb"
ts_def: fixes c :: "'form" shows "ts c = (SOME t. snd (fst (tree.root t)) = c \<and> fst (fst (tree.root t)) |\<subseteq>| ass_forms \<and> local.wf t \<and> tfinite t)"
dests_bigger_than_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:15..PAodv-:17} \<union> {PPkt-:7..PPkt-:9} \<union> {PRreq-:11..PRreq-:13} \<union> {PRreq-:20..PRreq-:22} \<union> {PRrep-:7..PRrep-:9} \<union> {PRerr-:1..PRerr-:4} \<union> {PRerr-:6} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) ip \<le> the (dests \<xi> ip)))"
cod_char: fixes f :: "'a" shows "hom'.cod f = (if hom'.arr f then cod f else null)"
tt_2: shows "MP_Rel T (of_int_poly (Mp T))"
gpop_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat" shows "gpop_code node_eq_impl node_hash_impl \<equiv> \<lambda>x. let y' = x in case y' of (xa, xb) \<Rightarrow> pop_code node_eq_impl node_hash_impl xb \<bind> (\<lambda>xc. let y'a = as_pop xa in RETURN (y'a, xc))"
fmpred_iff: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "fmpred P m = (\<forall>x y. fmlookup m x = Some y \<longrightarrow> P x y)"
is_Fv_def: fixes "term" :: "Core.term" shows "is_Fv term = (\<exists>x21 x22. term = Fv x21 x22)"
AbstTermP_Var_diff: fixes v :: "tm"   and w :: "tm"   and t :: "tm" shows "{VarP v, VarP w, Neg (v EQ w), OrdP t} \<turnstile> AbstTermP v t w w"
completely_multiplicative_function_def: fixes f :: "nat \<Rightarrow> 'b" shows "completely_multiplicative_function f \<equiv> f 0 = (0::'b) \<and> f (Suc 0) = (1::'b) \<and> (\<forall>a b. 1 < a \<longrightarrow> 1 < b \<longrightarrow> f (a * b) = f a * f b)"
B_def: shows "B = Matrix.mat n n (\<lambda>(i, j). A $$ (p i, p j))"
class_semiring: fixes b :: "'d" shows "semiring (class_semiring TYPE('c) b)"
fmrel_on_fsetD: fixes x :: "'c"   and S :: "'c fset"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'e) fmap" assumes "x |\<in>| S"   and "fmrel_on_fset S P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
Cod_dom: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Cod (local.dom f) = Dom f"
denormal_imp_not_zero: fixes f :: "('a, 'b) IEEE.float" assumes "is_denormal f" shows "\<not> is_zero f"
subset_of_opens: fixes U :: "'c set" assumes "U \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}" shows "is_open\<^sub>Y U"
path_component_subset_connected_component_of: fixes X :: "'c topology"   and x :: "'c" shows "path_component_of_set X x \<subseteq> connected_component_of_set X x"
freshEnv_getEnv1: fixes ys :: "'a"   and y :: "'b"   and rho :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm set option"   and xs :: "'a"   and x :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm set" assumes "freshEnv ys y rho"   and "rho xs x = Some X" shows "ys \<noteq> xs \<or> y \<noteq> x"
delta_Cons: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> bool)"   and a :: "'d"   and w :: "'d list"   and s :: "'c" shows "delta A (a # w) s = delta A w (next A a s)"
is_extensional: fixes f :: "'a" assumes "\<not> F.A.arr f" shows "F f = F.B.null"
hol_pal_lem1: fixes S :: "complex set"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and k :: "real"   and f :: "complex \<Rightarrow> complex" assumes "convex S"   and "open S"   and "a \<in> S"   and "b \<in> S"   and "c \<in> S"   and "d \<noteq> 0"   and "d \<bullet> a \<le> k"   and "d \<bullet> b \<le> k"   and "d \<bullet> c \<le> k"   and "f holomorphic_on {z \<in> S. d \<bullet> z < k}"   and "continuous_on S f" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
span_clauses_1: fixes a :: "'a poly"   and S :: "'a poly set" assumes "a \<in> S" shows "a \<in> poly_vs.span S"
summarize_aforms_def: fixes p :: "nat"   and C :: "(real \<times> real pdevs) list \<Rightarrow> nat \<Rightarrow> real list \<Rightarrow> bool"   and d :: "nat"   and X :: "(real \<times> real pdevs) list" shows "summarize_aforms p C d X = zip (map fst X) (summarize_pdevs_list p (C X) d (map snd X))"
anonymous_profile_permute: fixes \<sigma> :: "'b \<Rightarrow> 'b" assumes "\<sigma> permutes alts"   and "finite agents" shows "anonymous_profile (permute_profile \<sigma> R) = image_mset (map ((`) \<sigma>)) (anonymous_profile R)"
image_set: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "f ` set xs = set (map f xs)"
strict_order_iff: fixes a :: "'d"   and b :: "'d" shows "(a < b) = (a = min a b \<and> a \<noteq> b)"
init_fin_simulation1: fixes t :: "'t"   and s1 :: "(status \<times> 'x1) \<times> 'm1"   and s2 :: "(status \<times> 'x2) \<times> 'm2"   and tl1 :: "('l, 't, status \<times> 'x1, 'm1, 'w, 'o action) thread_action"   and s1' :: "(status \<times> 'x1) \<times> 'm1" assumes "t \<turnstile> s1 \<approx>i s2"   and "r1.init_fin t s1 tl1 s1'"   and "\<not> r1.init_fin_\<tau>move s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. (\<tau>trsys.silent_move (r2.init_fin t) r2.init_fin_\<tau>move)\<^sup>*\<^sup>* s2 s2' \<and> r2.init_fin t s2' tl2 s2'' \<and> \<not> r2.init_fin_\<tau>move s2' tl2 s2'' \<and> t \<turnstile> s1' \<approx>i s2'' \<and> ta_bisim init_fin_bisim tl1 tl2"
residue_simple: fixes s :: "complex set"   and z :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "open s"   and "z \<in> s"   and "f holomorphic_on s" shows "residue (\<lambda>w. f w / (w - z)) z = f z"
lossless_R1: fixes m1 :: "'msg1"   and m2 :: "'msg2" shows "lossless_spmf (R1 m1 m2)"
guha_deterministic1: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and ft :: "('d, 'f) flow_entry_match list"   and p :: "'e"   and x1 :: "'f" assumes "guha_table_semantics \<gamma> ft p (Some x1)" shows "\<not> guha_table_semantics \<gamma> ft p None"
values_set: fixes q :: "('c, 'd) pq" shows "set |q| = fst ` set (pq.alist_of q)"
REFL: fixes r :: "('c \<times> 'c) set" assumes "wo_rel r" shows "Refl r"
v: fixes i :: "nat" shows "vertices (map_graph (g i) X) \<subseteq> vertices Y"
sets_lebesgue_outer_open: fixes S :: "'a set"   and e :: "real" assumes "S \<in> sets lebesgue"   and "0 < e" obtains T :: "'a set" where "open T" and "S \<subseteq> T" and "T - S \<in> lmeasurable" and "emeasure lebesgue (T - S) < ennreal e"
cBall_def: shows "cBall \<equiv> (rcset ---> id) Ball"
stable_pair_on_Ch: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set"   and h :: "'c" assumes "stable_pair_on ds XD_XH" shows "Ch h (snd XD_XH) = match XD_XH \<inter> {x. Xh x = h}"
le_fun_nat_inv: fixes t2 :: "(val \<times> val) fset"   and x1 :: "nat"   and P :: "bool" assumes "VFun t2 \<sqsubseteq> VNat x1" shows "P"
a: shows "- {a} \<subseteq> f ` (M - {z})"
safe_progress_get_and: fixes \<phi> :: "Formula.formula"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "safe_formula \<phi>" shows "(MIN \<phi>\<in>set (get_and_list \<phi>). Monitor.progress \<sigma> P \<phi> j) = Monitor.progress \<sigma> P \<phi> j"
simps_2: fixes T :: "nat set"   and i :: "instr"   and "is" :: "instr list" shows "data_dependency_consistent_instrs T (i # is) = (case i of Read x xa t \<Rightarrow> data_dependency_consistent_instrs (insert t T) is | Write volatile a (D, f) x xa xb xc \<Rightarrow> D \<subseteq> T \<and> D \<inter> load_tmps is = {} \<and> data_dependency_consistent_instrs T is | RMW a t (D, f) cond ret x xa xb xc \<Rightarrow> D \<subseteq> T \<and> D \<inter> load_tmps is = {} \<and> data_dependency_consistent_instrs (insert t T) is | _ \<Rightarrow> data_dependency_consistent_instrs T is)"