hcju/isabelleps huggingface dataset:
ins_edges_def: shows "ins_edges \<equiv> \<lambda>(l\<^sub>1, l\<^sub>2) m. ins_edge (negate l\<^sub>1) l\<^sub>2 (ins_edge (negate l\<^sub>2) l\<^sub>1 m)"
domain_and_range_permutation_def: fixes A :: "'a set"   and B :: "'b set" shows "domain_and_range_permutation A B = {(f, f'). (f, f') \<in> (A \<rightarrow>\<^sub>E B) \<times> (A \<rightarrow>\<^sub>E B) \<and> (\<exists>p\<^sub>A p\<^sub>B. p\<^sub>A permutes A \<and> p\<^sub>B permutes B \<and> (\<forall>x\<in>A. f x = p\<^sub>B (f' (p\<^sub>A x))))}"
Scan_elem_in_Gen: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and I :: "item set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat"   and T :: "(symbol \<times> character list) set"   and x :: "item" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "I \<subseteq> LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc (LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u)"   and "k \<le> length Doc"   and "T \<subseteq> LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u"   and "x \<in> Scan T k I" shows "x \<in> LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc (LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u)"
build_rqrm_correct_2: fixes \<delta> :: "('a, 'b) ta_rule dlist" assumes "ls.invar \<delta>" shows "rqrm_prop (ls.\<alpha> \<delta>) (build_rqrm \<delta>)"
WT_gpv_left_gpv: fixes \<I>1 :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>2 :: "('d, 'e) \<I>" assumes "\<I>1 \<turnstile>g gpv \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<turnstile>g left_gpv gpv \<surd>"
measure\<^sub>s\<^sub>t_def: shows "measure\<^sub>s\<^sub>t \<equiv> measures [\<lambda>(S, \<theta>). card (fv\<^sub>s\<^sub>t S), \<lambda>(S, \<theta>). size\<^sub>s\<^sub>t S]"
cases: fixes P :: "'b prog"   and a1 :: "char list"   and a2 :: "char list" assumes "subcls1p P a1 a2" obtains C :: "char list" and D :: "char list" and rest :: "fdecl list \<times> 'b mdecl list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(D, rest)\<rfloor>" and "C \<noteq> Object"
0: shows "prv (imp (dsj \<phi>1 (dsj \<phi> (ldsj (\<phi>1s @ \<phi>2s)))) (dsj \<phi> (dsj \<phi>1 (ldsj (\<phi>1s @ \<phi>2s)))))"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg f f.tab (tab\<^sub>0 f) (tab\<^sub>1 f) f f.tab (tab\<^sub>0 f) (tab\<^sub>1 f) f"
add_increasing2: fixes c :: "'d multiset"   and b :: "'d multiset"   and a :: "'d multiset" assumes "{#} \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
RSD_pareto_eqclass_def: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" shows "RSD_pareto_eqclass agents alts R A = (A \<noteq> {} \<and> A \<subseteq> alts \<and> (\<forall>x\<in>A. \<forall>y\<in>alts. (y \<in> A) = (\<forall>i\<in>agents. R i x y)))"
Union_def: fixes gs :: "('a, 'b) pre_digraph set" shows "local.Union gs = \<lparr>verts = \<Union> (verts ` gs), arcs = \<Union> (arcs ` gs), tail = tail G, head = head G\<rparr>"
abs_less_iff: fixes a :: "'a"   and b :: "'a" shows "(\<bar>a\<bar> < b) = (a < b \<and> - a < b)"
cases: fixes a1 :: "real \<Rightarrow> real"   and a2 :: "real msstream" assumes "local.asymp_powser a1 a2" obtains f :: "real \<Rightarrow> real" and cs :: "real msstream" and f' :: "real \<Rightarrow> real" and c :: "real" where "a1 = f'" and "a2 = MSSCons c cs" and "local.asymp_powser f cs" and "f' \<in> O[F](\<lambda>_. 1)" and "\<forall>\<^sub>F x in F. f' x = c + f x * h x"
times_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and G :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "is_expansion_aux F f basis"   and "is_expansion_aux G g basis" shows "is_expansion_aux (times_ms_aux F G) (\<lambda>x. f x * g x) basis"
g_def: shows "\<forall>x. {z. (x, z) \<in> S} = {z. g x \<le> ereal z}"
conf_clinit_Called_Throwing: fixes P :: "jvm_prog"   and sh :: "sheap"   and stk' :: "val list"   and loc' :: "val list"   and C' :: "char list"   and pc' :: "nat"   and ics' :: "init_call_status"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and fs :: "frame list"   and xcp :: "nat" assumes "conf_clinit P sh ((stk', loc', C', clinit, pc', ics') # (stk, loc, C, M, pc, Called Cs) # fs)" shows "conf_clinit P sh ((stk, loc, C, M, pc, Throwing (C' # Cs) xcp) # fs)"
add_le_imp_le_diff2: fixes k :: "nat"   and i :: "nat"   and j :: "nat" assumes "k + i \<le> j" shows "i \<le> j - k"
lipschitz_on_cmult_upper: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "real"   and D :: "real" assumes "C-lipschitz_on U f"   and "\<bar>a\<bar> \<le> D" shows "(D * C)-lipschitz_on U (\<lambda>x. a *\<^sub>R f x)"
fixespointwise_subset: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "fixespointwise f A"   and "order.greater_eq A B" shows "fixespointwise f B"
monom_altdef: fixes c :: "'a"   and n :: "nat" shows "monom c n = Polynomial.smult c ([:0::'a, 1::'a:] ^ n)"
divide_right_mono: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (divide a c) (divide b c)"
count_filter_mset: fixes P :: "'k \<Rightarrow> bool"   and M :: "'k multiset"   and a :: "'k" shows "count (filter_mset P M) a = (if P a then count M a else 0)"
gr_implies_not_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'e"   and n :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less m n" shows "n \<noteq> zero"
nth_default_Cons: fixes dflt :: "'a"   and x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "nth_default dflt (x # xs) n = (case n of 0 \<Rightarrow> x | Suc n' \<Rightarrow> nth_default dflt xs n')"
design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" shows "design point_set block_collection \<equiv> finite_incidence_system point_set block_collection \<and> design_axioms block_collection"
moebius_similarity_preserve_orientation': fixes a :: "complex"   and H :: "ocircline"   and b :: "complex" assumes "a \<noteq> 0"   and "\<infinity>\<^sub>h \<notin> ocircline_set H"   and "pos_oriented H" shows "pos_oriented (moebius_ocircline (moebius_similarity a b) H)"
nres_rel_def: fixes R :: "('b \<times> 'd) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
sup_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x + y + z = x + (y + z)"
simps_1: fixes f :: "'c \<rightarrow> 'b" shows "map\<cdot>f\<cdot>[] = []"
diff_prv_eql_Num: fixes m :: "nat"   and n :: "nat" assumes "m \<noteq> n" shows "prv (neg (eql (Num m) (Num n)))"
is_sig_red_singleton_monom_multD: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'b"   and t :: "'a"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail {monom_mult c t f} p" shows "is_sig_red sing_reg top_tail {f} p"
simps_6: fixes V :: "char list" shows "fv (Var V) = {V}"
invf_one: assumes "1\<^sub>r \<noteq> \<zero>" shows "1\<^sub>r\<^bsup>\<hyphen> K\<^esup> = 1\<^sub>r"
elims_3: fixes x :: "ty" assumes "\<not> is_Array x" obtains "x = Void"   | "x = Boolean"   | "x = Integer"   | "x = NT"   | v :: "String.literal" where "x = Class v"
jb_imp_diff_ev: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "local.jb A i j" shows "local.diff_ev A i j"
HEndPhase0_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv5_inner s q"   and "Inv1 s" shows "HInv5_inner s' q"
rep_eq: fixes x :: "lit" shows "comp_fun_idem_apply (succs_loop_body x) = succs_of_clause x"
priK_neq_shrK: fixes A :: "agent"   and b :: "keymode"   and C :: "agent" shows "shrK A \<noteq> privateKey b C"
perm_rem_permutes: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "f permutes S \<union> {x}" shows "perm_rem x f permutes S"
Free: fixes \<alpha> :: "freeRes"   and Q' :: "pi" assumes "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
sel_2: fixes x1 :: "'a fset"   and x2 :: "'a"   and x3 :: "'b set" shows "a_conc (Antecedent x1 x2 x3) = x2"
simps_2: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "Leaf \<noteq> Node x21 x22 x23 x24 x25"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_wt a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt s < wt t)"
Substitution: fixes M :: "('fun, 'var) Term.term \<Rightarrow> bool"   and t :: "('fun, 'var) Term.term"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "SMPp M t"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "SMPp M (t \<cdot> \<delta>)"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(SourceTerm x1 = SourceTerm y1) = (x1 = y1)"
Pi_I: fixes A :: "'ag set"   and f :: "'ag \<Rightarrow> 'ah"   and B :: "'ag \<Rightarrow> 'ah set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
residual_network_sel_1: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "edge (residual_network f) x y = (edge \<Delta> x y \<or> edge \<Delta> y x \<and> y \<noteq> source \<Delta>)"
bind_funionM: fixes f :: "'a fset \<Rightarrow> 'm"   and m :: "'m"   and m' :: "'m" assumes "\<And>A B. f (A |\<union>| B) = bind (f A) (\<lambda>x. bind (f B) (\<lambda>y. return (x |\<union>| y)))" shows "bind (funionM m m') f = funionM (bind m f) (bind m' f)"
combine_prepend_replicate: fixes Ks :: "'a list"   and Us :: "'a list"   and n :: "nat" assumes "set Ks \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "combine (replicate n \<zero> @ Ks) Us = combine Ks (drop n Us)"
da_Skip: fixes A :: "assigned"   and B :: "lname set"   and Env :: "env" assumes "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>" shows "Env\<turnstile> B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> A"
card_PiE_dflt: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and d :: "'c" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (B x)" shows "card (PiE_dflt A d B) = (\<Prod>x\<in>A. card (B x))"
minimal_def: fixes A :: "'g set"   and P :: "'g set \<Rightarrow> bool" shows "minimal A P = (P A \<and> (\<forall>B. B \<subseteq> A \<and> P B \<longrightarrow> B = A))"
WT_bodies_def: shows "WT_bodies = (\<forall>(pn, b)\<in>set bodies. WT b)"
cmp_fun: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set"   and g :: "'i \<Rightarrow> 'j"   and C :: "'j set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "cmp g f \<in> A \<rightarrow> C"
rec_not_def: shows "rec_not = Cn 1 rec_minus [constn 1, recf.id 1 0]"
image_empty: fixes f :: "'c \<Rightarrow> 'b" shows "f ` {} = {}"
f_def: shows "f = (\<lambda>n. real_of_int (b n) / real_of_int (prod a {..n}))"
that: fixes a :: "'a" assumes "A = \<lfloor>a\<rfloor>" shows "thesis"
the_apartment_iso_def: fixes A :: "'a set set"   and A' :: "'a set set" shows "the_apartment_iso A A' \<equiv> THE f. ChamberComplexIsomorphism A A' f \<and> fixespointwise f (\<Union> (A \<inter> A')) \<and> fixespointwise f (- \<Union> A)"
strictly_preserves_\<iota>: shows "F \<iota>\<^sub>C = \<iota>\<^sub>D"
ts_ok_into_ts_inv_const: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and ts :: "'g \<Rightarrow> ('h \<times> 'j \<Rightarrow>f nat) option"   and m :: "'i" assumes "ts_ok P ts m" obtains I :: "'g \<Rightarrow> 'k option" where "ts_inv (\<lambda>_. P) I ts m"
Mf: shows "f \<in> M \<rightarrow>\<^sub>M embed_measure M f"
check_mult_l_check_mult: fixes A :: "(nat, (char list list \<times> int) list) fmap"   and B :: "(nat, int mpoly) fmap"   and r :: "(char list list \<times> int) list"   and r' :: "int mpoly"   and q :: "(char list list \<times> int) list"   and q' :: "int mpoly"   and p :: "nat"   and p' :: "nat"   and i :: "nat"   and i' :: "nat"   and \<V> :: "char list set"   and \<V>' :: "nat set"   and spec :: "(char list list \<times> int) list" assumes "(A, B) \<in> \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(r, r') \<in> sorted_poly_rel O mset_poly_rel"   and "(q, q') \<in> sorted_poly_rel O mset_poly_rel"   and "(p, p') \<in> nat_rel"   and "(i, i') \<in> nat_rel"   and "(\<V>, \<V>') \<in> \<langle>var_rel\<rangle>set_rel" shows "check_mult_l spec A \<V> p q i r \<le> \<Down> {(st, b). (\<not> is_cfailed st) = b \<and> (is_cfound st \<longrightarrow> spec = r)} (check_mult B \<V>' p' q' i' r')"
ex_finite_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" obtains G :: "('t \<Rightarrow>\<^sub>0 'c) set" where "G \<subseteq> dgrad_p_set d m" and "finite G" and "is_Groebner_basis G" and "pmdl G = pmdl F"
l_def: fixes tree :: "'b tree" shows "l tree = (case tree of Leaf \<Rightarrow> 0 | Node x21 x22 x23 x24 x25 \<Rightarrow> x21)"
bf_restrict_def: fixes i :: "'b"   and val :: "bool"   and f :: "('b \<Rightarrow> bool) \<Rightarrow> bool" shows "bf_restrict i val f \<equiv> \<lambda>v. f (v(i := val))"
def_valid_then_def: fixes x :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<upsilon> x->including\<^sub>S\<^sub>e\<^sub>t(y) = \<delta> x->including\<^sub>S\<^sub>e\<^sub>t(y)"
good_ruleset_normalize_rules_dnf: fixes rs :: "'c rule list" assumes "good_ruleset rs" shows "good_ruleset (normalize_rules_dnf rs)"
poly_normalized_nil: fixes p :: "'a list"   and x :: "'a" shows "pnormalize p = [] \<longrightarrow> poly p x = (0::'a)"
mult_assoc: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> V" shows "(a * b) \<cdot> x = a \<cdot> b \<cdot> x"
order_iff: fixes b :: "'b word"   and a :: "'b word" shows "(b \<le>s a) = (a = signed.max a b)"
normalize_1_iff: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(normalize a = one) = dvd.dvd times a one"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> A"   and "r\<lparr>a\<rparr> = b" shows "thesis"
smult_minus_1_back: fixes v :: "'e" assumes "v \<in> carrier N" shows "\<ominus>\<^bsub>N\<^esub> v = \<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>N\<^esub> v"
set_iterator_image_alt_def: fixes g :: "'g \<Rightarrow> 'f"   and it :: "('e \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" shows "set_iterator_image g it = (\<lambda>c f. it c (\<lambda>x. f (g x)))"
spmf_restrict_spmf_outside: fixes x :: "'c"   and A :: "'c set"   and p :: "'c spmf" assumes "x \<notin> A" shows "spmf (p \<upharpoonleft> A) x = 0"
minimal: fixes x :: "'c"   and A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>y\<in>A. size y \<le> size x \<and> P y \<and> (\<forall>z\<in>A. size z < size y \<longrightarrow> \<not> P z)"
g_def: shows "g \<equiv> \<lambda>xs. f (lhd xs) (ltl xs) (LCons (lhd xs) (ltl xs))"
retl: shows "(\<pi> @\<^bsup>k\<^esup> \<pi>l) (k + n) = local.return"
1: shows "\<forall>f. (top, up_filter (Rep_lifted_distrib_lattice_top xf f)) \<in> triple.pairs (Rep_phi f)"
TESL_interpretation_stepwise_fixpoint: fixes \<Phi> :: "'b TESL_atomic list"   and n :: "nat" shows "\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = (\<Inter>\<phi>\<in>set \<Phi>. \<lbrakk> \<phi> \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>)"
C_def: shows "C = (\<lambda>c. sqrt (2 / e) / 3 * (2 * e + 3 * c))"
cases: fixes a1 :: "com \<times> (char list \<Rightarrow> val option)"   and a2 :: "com \<times> (char list \<Rightarrow> val option)" assumes "red a1 a2" obtains V :: "char list" and e :: "expr" and s :: "char list \<Rightarrow> val option" where "a1 = (V:=e, sa__)" and "a2 = (Skip, sa__(V := \<lbrakk>e\<rbrakk>sa__))"   | c\<^sub>1 :: "com" and s :: "char list \<Rightarrow> val option" and c\<^sub>1' :: "com" and s' :: "char list \<Rightarrow> val option" and c\<^sub>2 :: "com" where "a1 = (c\<^sub>1__;; c\<^sub>2__, sb__)" and "a2 = (c\<^sub>1';; c\<^sub>2__, s')" and "\<langle>c\<^sub>1__,sb__\<rangle> \<rightarrow> \<langle>c\<^sub>1',s'\<rangle>"   | c\<^sub>2 :: "com" and s :: "char list \<Rightarrow> val option" where "a1 = (Skip;; c\<^sub>2'__, sc__)" and "a2 = (c\<^sub>2'__, sc__)"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "a1 = (if (b__) c\<^sub>1''__ else c\<^sub>2''__, sd__)" and "a2 = (c\<^sub>1''__, sd__)" and "\<lbrakk>b__\<rbrakk>sd__ = Some true"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "a1 = (if (ba__) c\<^sub>1 else c\<^sub>2, se__)" and "a2 = (c\<^sub>2, se__)" and "\<lbrakk>ba__\<rbrakk>se__ = Some false"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c :: "com" where "a1 = (while (bb__) ca__, sf__)" and "a2 = (ca__;; while (bb__) ca__, sf__)" and "\<lbrakk>bb__\<rbrakk>sf__ = Some true"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c :: "com" where "a1 = (while (b) c, s)" and "a2 = (Skip, s)" and "\<lbrakk>b\<rbrakk>s = Some false"
list_slice2_mod_0: fixes xs :: "'b list"   and k :: "nat" assumes "length xs mod k = 0" shows "list_slice2 xs k = list_slice xs k"
infs_mono: fixes w :: "'a stream"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>a. \<lbrakk>a \<in> sset w; P a\<rbrakk> \<Longrightarrow> Q a"   and "infs P w" shows "infs Q w"
f_range_on_antimono: fixes B :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set" assumes "f_range_on B f"   and "A \<subseteq> B" shows "f_range_on A f"
pconj_rzero: fixes b :: "real" assumes "b \<le> 1" shows "b .& 0 = 0"
df': fixes k :: "nat" assumes "k < n" shows "f' k \<in> carrier_mat d d"
order_zorder: fixes p :: "complex poly"   and z :: "complex" assumes "p \<noteq> 0" shows "order z p = nat (zorder (poly p) z)"
declassified_proj_eq: fixes A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and I :: "'var \<Rightarrow> ('fun, 'a) Term.term"   and n :: "'lbl" shows "declassified\<^sub>l\<^sub>s\<^sub>t A I = declassified\<^sub>l\<^sub>s\<^sub>t (proj n A) I"
alpha_qFreshEnv_qSwap_qPsubst_commute: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGoodEnv rho"   and "qGood X"   and "qFreshEnv zs z1 rho"   and "qFreshEnv zs z2 rho" shows "X #[[z1 \<and> z2]]_zs #[[rho]] #= X #[[rho]] #[[z1 \<and> z2]]_zs"
abs_eq: fixes xc :: "complex"   and xb :: "complex"   and xa :: "complex"   and x :: "complex" shows "mk_moebius xc xb xa x = abs_moebius (mk_moebius_mat xc xb xa x)"
prems_6: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
action_needed_vars_def: fixes a :: "('d, 'e) fmap \<times> 'f"   and s :: "('d, 'e) fmap" shows "action_needed_vars a s \<equiv> {v \<in> fmdom' s. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s v}"
rel_simps_86: fixes n :: "num" shows "(- (1::'a) = - numeral n) = (n = num.One)"
assms_2: fixes r :: "ereal" assumes "0 < r" shows "\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r"
eq_LConsD: fixes xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" assumes "xs = LCons y ys" shows "xs \<noteq> LNil \<and> lhd xs = y \<and> ltl xs = ys"
scalar_prod_def: fixes v :: "'b Matrix.vec"   and w :: "'b Matrix.vec" shows "v \<bullet> w \<equiv> \<Sum>i = 0..<dim_vec w. v $v i * w $v i"
is_inf_lower: fixes x :: "'a"   and y :: "'a"   and i :: "'a" assumes "is_inf x y i" obtains "i \<sqsubseteq> x" and "i \<sqsubseteq> y"
eq_on_refl: fixes s :: "'a \<Rightarrow> 'b"   and X :: "'a set" shows "s = s on X"
keys_monom_of_set: fixes X :: "nat set" shows "keys (monom_of_set X) = (if finite X then X else {})"
span_clauses_2: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale" shows "(0::'c) \<in> module.span scale S"
assms_2: fixes r :: "'r"   and m :: "'m" assumes "r \<in> R"   and "m \<in> M" shows "T (r \<cdot> m) = smult' r (T m)"
residue_primroot_cong: fixes x :: "nat"   and x' :: "nat"   and n :: "nat" assumes "[x = x'] (mod n)" shows "residue_primroot n x = residue_primroot n x'"
rel_fset_aux: fixes a :: "'d fset"   and b :: "'e fset"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "((\<forall>t\<in>fset a. \<exists>u\<in>fset b. R t u) \<and> (\<forall>u\<in>fset b. \<exists>t\<in>fset a. R t u)) = ((BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} ((|`|) fst))\<inverse>\<inverse> OO BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} ((|`|) snd)) a b"
invertible_implies_rref_id: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "invertible A" shows "Gauss_Jordan A = mat (1::'a)"
symmetric: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E" shows "(b, a) \<in> E"
w_def: shows "w = (\<lambda>r. winding_number (poly q \<circ> g r) 0)"
not_irreducible_zero: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.irreducible times one zero zero"
prime_for_berlekamp_zassenhaus_exists: fixes f :: "int poly" assumes "square_free f" shows "\<exists>p. prime p \<and> comm_monoid_mult_class.coprime (lead_coeff f) p \<and> separable_impl p f"
eint_ord_code_6: fixes q :: "eint" shows "(\<infinity> < q) = False"
am2: fixes y :: "'a"   and x :: "'a" shows "ar y \<cdot> ar x = ar x \<cdot> ar y"
assms: shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k)"
assms_1: shows "a{b}.P \<longmapsto> \<alpha> \<prec> P'"
conjugate_id: fixes a :: "'a" shows "conjugate (conjugate a) = a"