hcju/isabelleps huggingface dataset:
Integral_mult_const: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a \<le> b" shows "Integral (a, b) (\<lambda>x. c) (c * (b - a))"
wf_nFOR: fixes idx :: "'n"   and \<phi>s :: "('a, 'k) aformula list" shows "local.wf idx (nFOR \<phi>s) = (\<forall>\<phi>\<in>set \<phi>s. local.wf idx \<phi>)"
r_smn: fixes n :: "nat"   and cs :: "nat list"   and m :: "nat"   and p :: "nat" assumes "0 < n"   and "length cs = m" shows "eval (r_smn n m) (p # cs) \<down>= smn n p cs"
SUP_le_iff: fixes f :: "'d \<Rightarrow> 'b \<times> 'c"   and A :: "'d set"   and u :: "'b \<times> 'c" shows "less_eq_prod (Sup_prod (f ` A)) u = (\<forall>i\<in>A. less_eq_prod (f i) u)"
simps: fixes ts :: "'b tree list" shows "T_list_fast_rec ts = (let us = filter (\<lambda>t. t \<noteq> \<langle>\<rangle>) ts in length ts + (if us = [] then 0 else 5 * length us + T_list_fast_rec (map left us @ map right us)))"
routing_ipassmt_wi: fixes tbl :: "'b routing_rule list"   and k :: "'b word"   and output_port :: "char list" assumes "valid_prefixes tbl" shows "(output_iface (routing_table_semantics tbl k) = output_port) = (\<exists>ip_range. k \<in> wordinterval_to_set ip_range \<and> (output_port, ip_range) \<in> set (routing_ipassmt_wi tbl))"
lins2_def: fixes S :: "node fset"   and G :: "diagram" shows "lins2 S G \<equiv> {\<pi>. distinct \<pi> \<and> set \<pi> = (fset G^V - fset S) <+> set G^E \<and> (\<forall>i j v e. i < length \<pi> \<and> j < length \<pi> \<and> \<pi> ! i = Inl v \<and> \<pi> ! j = Inr e \<and> v |\<in>| fst3 e \<longrightarrow> i < j) \<and> (\<forall>j k w e. j < length \<pi> \<and> k < length \<pi> \<and> \<pi> ! j = Inr e \<and> \<pi> ! k = Inl w \<and> w |\<in>| thd3 e \<longrightarrow> j < k)}"
dom_mkArr: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and B :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and F :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr \<Rightarrow> (('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.arr (S.mkArr A B F)" shows "S.dom (S.mkArr A B F) = S.mkIde A"
one_neq_zero: fixes one :: "'a"   and zero :: "'a" assumes "class.zero_neq_one one zero" shows "one \<noteq> zero"
subseqs_mem_dbproj: fixes Di :: "('a \<times> 'b) list"   and D :: "('a \<times> 'b) list"   and i :: "'a" assumes "Di \<in> set (subseqs D)"   and "list_all (\<lambda>d. fst d = i) Di" shows "Di \<in> set (subseqs (dbproj i D))"
integrable_setdiff: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and S :: "'a set"   and j :: "'b"   and T :: "'a set" assumes "(f has_integral i) S"   and "(f has_integral j) T"   and "negligible (T - S)" shows "f integrable_on S - T"
partition_on_empty: fixes P :: "'b set set" shows "partition_on {} P = (P = {})"
Bad_meas: fixes U :: "'a set"   and n :: "nat" assumes "U \<in> sets M" shows "Bad U n \<in> sets M"
Hash: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth H" shows "Hash X \<in> synth H"
that: fixes Bs :: "'a tensor list"   and a :: "'a" assumes "\<And>B. B \<in> set Bs \<Longrightarrow> order B = 1"   and "a \<cdot> prod_list Bs = A" shows "thesis"
le_INF_iff: fixes u :: "'b \<times> 'c"   and f :: "'d \<Rightarrow> 'b \<times> 'c"   and A :: "'d set" shows "less_eq_prod u (Inf_prod (\<P> f A)) = (\<forall>i\<in>A. less_eq_prod u (f i))"
subsetD: fixes U :: "'c set"   and V :: "'c set"   and x :: "'c" assumes "Subspace.subspace U V"   and "x \<in> U" shows "x \<in> V"
simps_2: fixes z :: "gauss_int" shows "Im (gauss2complex z) = real_of_int (ImZ z)"
boruvka_outer_invariant_def: fixes f :: "'a"   and g :: "'a" shows "boruvka_outer_invariant f g \<equiv> symmetric g \<and> forest f \<and> f \<le> - - g \<and> regular f \<and> (\<exists>w. minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T)"
valid_dbm_cases: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "valid_dbm M" obtains "[M]\<^bsub>v,n\<^esub> \<subseteq> V" and "dbm_int M n"
safe_max_Some_dest_in: fixes X :: "'a set"   and x :: "'a" assumes "finite X"   and "safe_max X = Some x" shows "x \<in> X"
ptrm_alpha_equiv_fvs_transfer: fixes A :: "'b ptrm"   and a :: "'b"   and b :: "'b"   and B :: "'b ptrm" assumes "A \<approx> [a \<leftrightarrow> b] \<bullet> B"   and "a \<notin> ptrm_fvs B" shows "b \<notin> ptrm_fvs A"
igWlsInp_def: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'h option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
assms: fixes i :: "nat" assumes "n < i" shows "f i = (0::'a)"
l\<^sub>2': shows "l\<^sub>2' = [this \<mapsto> Ref (a, Ds), pns [\<mapsto>] vs']"
accp_induct_rule: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "Wellfounded.accp r a"   and "\<And>x. \<lbrakk>Wellfounded.accp r x; \<And>y. r y x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
cnf2bhv_mono: fixes n :: "nat"   and n' :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'" shows "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n \<le> \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n'"
a: shows "(f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> Hom (Dom f) (Cod f)) \<and> a = MkIde (Dom f)"
simps_3: fixes i :: "nat"   and P :: "pol"   and c :: "int" shows "Pinj i P \<langle>*\<rangle> Pc c = (if c = 0 then Pc 0 else mkPinj i (P \<langle>*\<rangle> Pc c))"
cnf_of_encode_problem_structure_2: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf ((\<Phi>\<^sub>G \<Pi>) t) \<subseteq> cnf (\<Phi> \<Pi> t)"
snd_set_zip_map_snd: fixes xs :: "'a list"   and ys :: "('b \<times> 'c) list" assumes "length xs = length ys" shows "snd ` set (zip xs (map snd ys)) = snd ` set ys"
length_Suc_conv: fixes xs :: "'e list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
disc_2: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "\<not> is_TNil (unfold_tllist p g1 g21 g22 a)"
abs_eq: fixes xa :: "'b"   and x :: "nat" shows "Polynomial.monom xa x = Abs_poly (\<lambda>n. if x = n then xa else (0::'b))"
pref_lcp_pref: fixes w :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "w \<le>p u \<and>\<^sub>p v" shows "w \<le>p u"
open_segment_bound1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> {a<--<b}" shows "norm (x - a) < norm (b - a)"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a option" assumes "\<And>x. Ra x x" shows "rel_option Ra x x"
all: shows "\<forall>as pex. n -as\<rightarrow>\<^sub>\<iota>* pex \<and> method_exit pex \<longrightarrow> n' \<in> set (sourcenodes as)"
apply_rsp': fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and x :: "'g"   and y :: "'h" assumes "(R1 ===> R2) f g"   and "R1 x y" shows "R2 (f x) (g y)"
kmax_def: fixes f :: "nat \<Rightarrow> real"   and P :: "nat set" shows "kmax f P = (SOME S. S \<subseteq> P \<and> card S = khl \<and> (\<forall>i\<in>S. \<forall>j\<in>P - S. f j \<le> f i))"
uexpr_eq_iff: fixes e :: "('e, 'f) uexpr"   and f :: "('e, 'f) uexpr" shows "(e = f) = (\<forall>b. \<lbrakk>e\<rbrakk>\<^sub>e b = \<lbrakk>f\<rbrakk>\<^sub>e b)"
insert_not_elem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A"   and "A \<noteq> {}" shows "Min (insert x A) = min x (Min A)"
concat_is_inverse_of_embed: fixes xs :: "'c list" shows "concat (List_Supplement.embed xs) = xs"
CCexp_pre_simps_2: fixes x :: "Vars.var"   and e :: "exp"   and n :: "Arity" shows "ccExp_syn n (Lam [x]. e) = predCC (fv (Lam [x]. e)) (CCexp e)\<cdot>n"
g_def: fixes out :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and GL :: "'b set"   and s :: "('a, 'd, 'e) state"   and s' :: "('a, 'f, 'g) state" shows "GM_sec_model.g out GL s s' \<equiv> \<forall>u1. u1 \<in> GL \<longrightarrow> out (local.getGMState s) u1 = out (local.getGMState s') u1"
apd_d_def: fixes x :: "'a" shows "d x = ad (ad x)"
le_ivl_def: fixes i1 :: "ivl"   and i2 :: "ivl" shows "(i1 \<sqsubseteq> i2) = (if is_empty i1 then True else if is_empty i2 then False else le_aux i1 i2)"
map_snd_enumerate: fixes n :: "nat"   and xs :: "'b list" shows "map snd (enumerate n xs) = xs"
arr_char: fixes f :: "'a" shows "S.arr f = Arr f"
alphabet_huffman: fixes ts :: "'a tree list" assumes "ts \<noteq> []" shows "alphabet (huffman ts) = alphabet\<^sub>F ts"
T_vrestr_same_measure_2: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> events" shows "prob ((T ^^ n) --` A) = prob A"
BadWrite_def: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs"   and v :: "Vals" shows "BadWrite mm rs p l v \<equiv> \<not> (#l \<in> #MemLoc \<and> #v \<in> #MemVal) \<and> (rs!p)$ = #BadArg \<and> unchanged (mm!l)"
cnf2bhv_ge_llength: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "dynamic_component.lActive active c t \<le> n" shows "the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1 \<le> dynamic_component.cnf2bhv tCMP active c t n"
rep_eq: fixes x :: "('a, 'b) alist" shows "AssocList.set x = list.set (alist.impl_of x)"
below_monicI: fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. f x \<sqsubseteq> f y \<Longrightarrow> x \<sqsubseteq> y" shows "below_monic f"
adapt: shows "P \<subseteq> {s. s \<in> P' s}"
bin_cat_cong: fixes n :: "nat"   and m :: "nat"   and a :: "int"   and c :: "int"   and b :: "int"   and d :: "int" assumes "n = m"   and "a = c"   and "take_bit m b = take_bit m d" shows "concat_bit n b a = concat_bit m d c"
add_nonneg_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (plus a b)"
map_poly_monom: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c"   and n :: "nat" assumes "f (0::'c) = (0::'b)" shows "map_poly f (monom c n) = monom (f c) n"
set_mult_hom: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme"   and I :: "'e set"   and J :: "'e set" assumes "h \<in> hom G H"   and "I \<subseteq> carrier G"   and "J \<subseteq> carrier G" shows "h ` (I <#>\<^bsub>G\<^esub> J) = h ` I <#>\<^bsub>H\<^esub> h ` J"
l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs"
bot': fixes f :: "'b \<Rightarrow> 'c" shows "\<Omega>[bot](f) = UNIV"
Ix: shows "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
residuated_pair_def: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "residuated_pair f g \<equiv> \<forall>x y. (f x \<le> y) = (x \<le> g y)"
isPREFIX_sublist: fixes x :: "'c list"   and y :: "'c list" assumes "prefix x y" shows "subseq x y"
fold_Cons_rev: fixes xs :: "'k list" shows "fold (#) xs = (@) (rev xs)"
limsup_obtain: fixes c :: "'a"   and u :: "nat \<Rightarrow> 'a"   and N :: "nat" assumes "c < limsup u" shows "\<exists>n\<ge>N. c < u n"
qtable_l\<epsilon>_strict: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and A :: "nat set"   and n :: "nat"   and R :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat"   and rels :: "event_data option list set list" assumes "safe_regex Futu Strict (from_mregex mr \<phi>s)"   and "ok (length \<phi>s) mr"   and "A = fv_regex (from_mregex mr \<phi>s)"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s rels" shows "qtable n A (mem_restr R) (\<lambda>v. Regex.eps (Formula.sat \<sigma> V (map the v)) i (from_mregex mr \<phi>s)) (l\<epsilon>_strict n rels mr)"
assms_2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)]))"
gstep_succI: fixes gc :: "('c, 'ls, 'gs) global_config"   and lc :: "('c, 'ls) local_config"   and lcs :: "('c, 'ls) local_config multiset"   and a :: "'a"   and c' :: "'c"   and ls' :: "'ls"   and gs' :: "'gs" assumes "global_config.processes gc = {#lc#} + lcs"   and "cstep (command lc) a c'"   and "en (local_config.state lc, global_config.state gc) a"   and "ex (local_config.state lc, global_config.state gc) a = (ls', gs')" shows "\<lparr>global_config.processes = {#\<lparr>command = c', state = ls'\<rparr>#} + lcs, state = gs'\<rparr> \<in> gstep_succ gc"
1: shows "fmdom' (snd a) \<inter> vs = {}"
emeasure_density: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and A :: "'b set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure (density M f) A = \<integral>\<^sup>+ x. f x * indicator A x \<partial>M"
scale_left_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "(a + b) * x = a * x + b * x"
associative: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
eval_trm_cong: fixes t :: "'b trm"   and v :: "'b list"   and v' :: "'b list" assumes "\<forall>x\<in>fv_trm t. v ! x = v' ! x" shows "eval_trm v t = eval_trm v' t"
assms_2: assumes "\<alpha> = \<tau>"   and "P = P'" shows "Prop (\<tau>) P"
Sign: fixes a :: "cmsg"   and b :: "cmsg"   and c :: "cmsg"   and d :: "cmsg" assumes "eq a b"   and "eq c d" shows "eq (cSign a c) (cSign b d)"
positive_p: fixes \<Omega> :: "'a set"   and p :: "'a \<Rightarrow> real"   and b :: "real"   and x :: "'a" assumes "finite_information \<Omega> p b" shows "0 \<le> p x"
representsI: fixes T :: "'s Trace set" assumes "jkbpT T = T" shows "represents T"
intros: fixes v :: "hf"   and x :: "hf"   and x0 :: "hf"   and w :: "hf"   and xw :: "hf"   and xevw :: "hf"   and allw :: "hf"   and allvw :: "hf"   and ax :: "hf" assumes "SubstForm v 0 x x0"   and "SubstForm v w x xw"   and "SubstForm v (q_Eats v w) x xevw"   and "AbstForm w 0 (q_Imp x (q_Imp xw xevw)) allw"   and "AbstForm v 0 (q_All allw) allvw"   and "AbstForm v 0 x ax"   and "v \<noteq> w"   and "VarNonOccForm w x" shows "q_Imp x0 (q_Imp (q_All allvw) (q_All ax)) \<in> Induction_ax"
modular_1: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "f x \<cdot> y = f (x \<cdot> g y) \<cdot> y"
simps_1: fixes a :: "'b"   and uu :: "'b tree" shows "sift_down \<langle>\<rangle> a uu = \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>\<rangle>"
unrest_usubst_def: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d \<Rightarrow> 'd" shows "(x \<sharp> \<sigma>) = (\<forall>\<rho> v. \<sigma> (put\<^bsub>x\<^esub> \<rho> v) = put\<^bsub>x\<^esub> (\<sigma> \<rho>) v)"
one: fixes xs :: "'b com option list"   and s :: "'b" assumes "xs \<noteq> []" shows "par_cp xs s = {c. \<exists>clist. length clist = length xs \<and> (\<forall>i<length clist. clist ! i \<in> cp (xs ! i) s) \<and> c \<propto> clist}"
allAllocationsIntersectionSubset: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> allocationsUniverse \<inter> {a. Domain a \<subseteq> N \<and> \<Union> (Range a) = G}"
S4: shows "(LEAST z. 1 < 2 ^ (z + 1)) \<le> 0"
Plus_OneL: fixes r :: "'a language" assumes "\<oo> r" shows "Plus One r = r"
simps_2: fixes n :: "nat" shows "s_A (Suc n) = step (s_A n) (qs ! n) (free_A ! n, paid_A ! n)"
d_approx: fixes fs :: "int vec list"   and ii :: "nat" assumes "LLL_invariant_weak fs"   and "g_bound fs"   and "ii < m" shows "rat_of_int (d fs ii) \<le> rat_of_nat (N ^ ii)"
step: fixes I :: "nat set" assumes "I \<in> valid_I" shows "\<exists>I'. (I', I) \<in> step_rel"
disc_1: fixes p :: "'b \<Rightarrow> bool"   and a :: "'b"   and g21 :: "'b \<Rightarrow> 'c"   and g22 :: "'b \<Rightarrow> 'b" assumes "p a" shows "lnull (unfold_llist p g21 g22 a)"
root_def: fixes u :: "'b list"   and r :: "'b list" shows "u \<in> r* = (\<exists>k. r \<^sup>@ k = u)"
nat_term_order_eq_sym: fixes to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order"   and dg :: "bool"   and ps :: "bool" shows "nat_term_order_eq to1 to2 dg ps = nat_term_order_eq to2 to1 dg ps"
assms_3: shows "ks \<notin> specKeysSecrets PQ"
Trancl: assumes "(a, b) \<in> r\<^sup>+" shows "thesis"
induct: fixes P1 :: "fml \<Rightarrow> bool"   and P2 :: "game \<Rightarrow> bool"   and fml :: "fml" assumes "\<And>x1 x2. P1 (Pred x1 x2)"   and "\<And>x1 x2. P1 (Geq x1 x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (! x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 && x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Exists x1 x2)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (\<langle> x1 \<rangle> x2)"   and "\<And>x. P2 (Game x)"   and "\<And>x1 x2. P2 (x1 := x2)"   and "\<And>x. P1 x \<Longrightarrow> P2 (? x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 \<union>\<union> x2)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 ;; x2)"   and "\<And>x. P2 x \<Longrightarrow> P2 x**"   and "\<And>x. P2 x \<Longrightarrow> P2 x^d"   and "\<And>x1 x2. P2 (ODE x1 x2)" shows "P1 fml"
chop_emb_step_at: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "chop t = emb_step_at (replicate (num_args (fun t)) dir.Left) dir.Right t"
steps_to_trace: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'a \<Rightarrow> 'a \<times> instr list \<Rightarrow> bool"   and x :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and y :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" assumes "ini.direct_concurrent_steps program_step x y" shows "\<exists>c k. ini.trace program_step c 0 k \<and> c 0 = x \<and> c k = y"
inj_image_subset_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj f" shows "(f ` A \<subseteq> f ` B) = (A \<subseteq> B)"
SkipE: fixes s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(SKIP, s) \<Rightarrow> t" obtains "t = s"
div_le_mono: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m \<le> n" shows "m div k \<le> n div k"
divide_self_if: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a a = (if a = zero then zero else one)"
fun_relD: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'f \<Rightarrow> 'g"   and A :: "('d \<times> 'f) set"   and B :: "('e \<times> 'g) set"   and x :: "'d"   and x' :: "'f" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
map_of_filter_distinct: fixes xs :: "('c \<times> 'd) list"   and P :: "'c \<times> 'd \<Rightarrow> bool"   and x :: "'c" assumes "distinct (map fst xs)" shows "map_of (filter P xs) x = (case map_of xs x of None \<Rightarrow> None | Some y \<Rightarrow> if P (x, y) then Some y else None)"
shiftr_eq_drop_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.shiftr drop_bit a n = drop_bit n a"
assms_4: shows "r ; x \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
assms: assumes "nofail m" shows "<\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t"
\<chi>_in_terms_of_\<Phi>: shows "\<chi> = \<o> (\<Phi>.FUN a ((S.UP \<circ> inl) a))"
div_mult_self3: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus (times c b) a) b = plus c (divide a b)"
card_Diff_subset_Int: fixes A :: "'b set"   and B :: "'b set" assumes "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
fun_upd_twist: fixes a :: "'d"   and c :: "'d"   and m :: "'d \<Rightarrow> 'e"   and b :: "'e"   and d :: "'e" assumes "a \<noteq> c" shows "m(a := b, c := d) = m(c := d, a := b)"
intros_1: fixes dims :: "nat list"   and P :: "complex mat" assumes "state_sig.is_quantum_predicate dims P" shows "state_sig.hoare_partial dims P SKIP P"
VV: shows "S \<inter> f -` f ` (S \<inter> V) = S \<inter> V"
all: shows "wtl ins c 0 s\<^sub>0 \<noteq> \<top>"
the_def: fixes option :: "'b option" shows "the option = (case option of Some x2 \<Rightarrow> x2)"
divides_mult_r: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "comm_monoid_cancel G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<otimes>\<^bsub>G\<^esub> c divides\<^bsub>G\<^esub> b \<otimes>\<^bsub>G\<^esub> c = a divides\<^bsub>G\<^esub> b"
\<A>: shows "\<A> = (\<A>1, \<A>2)"
leg0_commutes: shows "src.cod.leg0 \<cdot> src.chine = src.leg0"
scale_cancel_left: fixes a :: "complex"   and x :: "'b"   and y :: "'b" shows "(a *\<^sub>C x = a *\<^sub>C y) = (x = y \<or> a = 0)"
Im3_quat_of_real: fixes a :: "real" shows "Im3 (quat_of_real a) = 0"
2: shows "(\<tau> |\<noteq> \<delta> x) = (\<tau> \<Turnstile> x \<triangleq> invalid \<or> \<tau> \<Turnstile> x \<triangleq> null)"
berlekamp_basis_length_factorization: fixes us :: "'a mod_ring poly list" assumes "f = prod_list us"   and "\<And>u. u \<in> set us \<Longrightarrow> 0 < degree u" shows "length us \<le> n"