hcju/isabelleps huggingface dataset:
assms_3: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "x = y"   and "l = l1 @ y # l2" shows "P"
cases: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> write_actions E" obtains aa :: "nat" where "a = aa" and "aa \<in> actions E" and "is_write_action (action_obs E aa)"
c: fixes x :: "complex" assumes "x \<in> S" shows "exp (g x) / f x = c"
ab_2: shows "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) u s"
simps_31: fixes vd :: "char list"   and ve :: "aexp"   and vf :: "'a"   and v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a" shows "(vd ::= ve {vf} \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
b_1: shows "v\<^sub>b \<in> defs g b"
aType_nonempty: fixes a :: "nat \<Rightarrow> aType list"   and t :: "nat" assumes "msg (Suc 0) a"   and "a t \<noteq> []" shows "a t = [sc_ack]"
domain_def: fixes g :: "('c \<times> real) set" shows "domain g = {x. \<exists>y. (x, y) \<in> g}"
eval_elim_cases_23: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (Throw e)\<succ>\<rightarrow> (x, s')" obtains a' :: "val" and s1 :: "state" where "x = \<diamondsuit>" and "s' = abupd (throw a') s1" and "G\<turnstile>Norm s \<midarrow>e-\<succ>a'\<rightarrow> s1"
has_bochner_integral_erf_aux: shows "has_bochner_integral lborel (\<lambda>x. indicat_real {0..} x *\<^sub>R exp (- x\<^sup>2)) (sqrt pi / 2)"
substitute_contains_arg: fixes t :: "'a ttree"   and f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set" shows "paths t \<subseteq> paths (substitute f T t)"
emeasure_eq_measure: fixes A :: "bool stream set" shows "emeasure M A = ennreal (prob A)"
iFROM_cut_ge: fixes n :: "nat"   and t :: "nat" shows "[n\<dots>] \<down>\<ge> t = (if n \<le> t then [t\<dots>] else [n\<dots>])"
assms_1: shows "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) (otherwith S (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) (oarrivemsg I)) (other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)))"
index_free_def: fixes P :: "'d multiset \<Rightarrow> 'a"   and i :: "'d" shows "index_free P i = (\<forall>m. i \<in># m \<longrightarrow> P m = \<zero>)"
hcmod_hrcis: fixes r :: "real star"   and a :: "real star" shows "hcmod (hrcis r a) = \<bar>r\<bar>"
f_props_3: shows "\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {}"
synAssignSC_def: fixes x :: "'c"   and e :: "('c, 'd) Expr" shows "synAssignSC x e \<equiv> \<exists>D. \<turnstile>\<^bsub>\<E>\<^esub> e : D \<and> (\<forall>d\<in>D. d \<le> DA x)"
drop_map: fixes n :: "nat"   and f :: "'l \<Rightarrow> 'k"   and xs :: "'l list" shows "drop n (map f xs) = map f (drop n xs)"
Ring_def: fixes R :: "('c, 'd) Ring_scheme" shows "Ring R \<equiv> (((\<plusminus>\<^bsub>R\<^esub>) \<in> carrier R \<rightarrow> carrier R \<rightarrow> carrier R \<and> (\<forall>a b c. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> c \<in> carrier R \<longrightarrow> a \<plusminus>\<^bsub>R\<^esub> b \<plusminus>\<^bsub>R\<^esub> c = a \<plusminus>\<^bsub>R\<^esub> (b \<plusminus>\<^bsub>R\<^esub> c)) \<and> (\<forall>a b. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> a \<plusminus>\<^bsub>R\<^esub> b = b \<plusminus>\<^bsub>R\<^esub> a)) \<and> mop R \<in> carrier R \<rightarrow> carrier R \<and> (\<forall>a. a \<in> carrier R \<longrightarrow> -\<^sub>a\<^bsub>R\<^esub> a \<plusminus>\<^bsub>R\<^esub> a = \<zero>\<^bsub>R\<^esub>) \<and> \<zero>\<^bsub>R\<^esub> \<in> carrier R) \<and> ((\<forall>a. a \<in> carrier R \<longrightarrow> \<zero>\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> a = a) \<and> (\<cdot>\<^sub>r\<^bsub>R\<^esub>) \<in> carrier R \<rightarrow> carrier R \<rightarrow> carrier R \<and> (\<forall>a b c. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> c \<in> carrier R \<longrightarrow> a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<cdot>\<^sub>r\<^bsub>R\<^esub> c = a \<cdot>\<^sub>r\<^bsub>R\<^esub> (b \<cdot>\<^sub>r\<^bsub>R\<^esub> c))) \<and> ((\<forall>a b. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = b \<cdot>\<^sub>r\<^bsub>R\<^esub> a) \<and> 1\<^sub>r\<^bsub>R\<^esub> \<in> carrier R) \<and> (\<forall>a b c. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> c \<in> carrier R \<longrightarrow> a \<cdot>\<^sub>r\<^bsub>R\<^esub> (b \<plusminus>\<^bsub>R\<^esub> c) = a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<plusminus>\<^bsub>R\<^esub> a \<cdot>\<^sub>r\<^bsub>R\<^esub> c) \<and> (\<forall>a. a \<in> carrier R \<longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>r\<^bsub>R\<^esub> a = a)"
mechanism_stable_def: fixes ds :: "'b set"   and \<phi> :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('c \<Rightarrow> 'a set \<Rightarrow> 'a set) \<Rightarrow> 'b set \<Rightarrow> 'a set" shows "Bossy''.mechanism_stable ds \<phi> = (\<forall>Pd Ch. ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch \<longrightarrow> Contracts.stable_on Pd Ch ds (\<phi> Pd Ch ds))"
simps_10: fixes n :: "nat"   and p :: "fm"   and q :: "fm" shows "boundslt n (Imp p q) = (boundslt n p \<and> boundslt n q)"
RedDynCastNull: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>Cast C null,s\<rangle> \<rightarrow> \<langle>null,s\<rangle>"
nat_pow_pow: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n) [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n * m)"
f: shows "f = (\<Prod>a\<in>P. a ^ Suc (n a))"
arr_Par_incl_ParI: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>"   and "A \<subseteq>\<^sub>\<circ> B" shows "arr_Par \<alpha> (incl_Rel A B)"
mbind_try: fixes a :: "'g"   and S :: "'g list"   and F :: "'g \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'd) option"   and M :: "'f list \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e (a # S) F) M = _bind_SE a' (try\<^sub>S\<^sub>E (F a)) (if a' = None then M [] else _bind_SE x (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e S F) (M (the a' # x)))"
even_square_cong_4_nat: fixes x :: "nat" assumes "even x" shows "[x\<^sup>2 = 0] (mod 4)"
diff: shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
vimage2p_map_list_all2: fixes f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "vimage2p (map f) (map g) (list_all2 A) = list_all2 (vimage2p f g A)"
groupWIs_same_fw_not: fixes A :: "'b wordinterval list"   and c :: "parts_connection"   and rs :: "'b simple_rule list"   and B :: "'b wordinterval list" assumes "A \<in> set (groupWIs c rs)"   and "B \<in> set (groupWIs c rs)"   and "A \<noteq> B" shows "\<forall>aw\<in>set (map wordinterval_to_set A). \<forall>bw\<in>set (map wordinterval_to_set B). \<forall>a\<in>aw. \<forall>b\<in>bw. \<not> same_fw_behaviour_one a b c rs"
not_present_alloc: fixes h :: "heap"   and xs :: "'a list" shows "\<not> Array.present h (fst (Array.alloc xs h))"
disc_iff: fixes xs :: "'a msstream" shows "msllist_of_msstream xs \<noteq> MSLNil"
elims_3: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('e, 'i, 'j, 'k) SLin_state_scheme" assumes "\<not> P11 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('e, 'i, 'j, 'k) SLin_state_scheme" where "x = (s1, s2)" and "initVals s2 \<noteq> abortVals s1"
retraction_idempotent: fixes S :: "'b set"   and T :: "'b set"   and r :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "retraction S T r"   and "x \<in> S" shows "r (r x) = r x"
sums_integral: fixes M :: "'c measure"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>i. integrable M (f i)"   and "AE x in M. summable (\<lambda>i. norm (f i x))"   and "summable (\<lambda>i. LINT x|M. norm (f i x))" shows "(\<lambda>i. integral\<^sup>L M (f i)) sums (LINT x|M. (\<Sum>i. f i x))"
cmp_in_hom_1: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b"   and "src a = trg b" shows "\<guillemotleft>S.cmp\<^sub>U\<^sub>P (a, b) : UP.map\<^sub>0 (src b) \<rightarrow>\<^sub>S UP.map\<^sub>0 (trg a)\<guillemotright>"
po_eq_conv: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.po below" shows "(x = y) = (below x y \<and> below y x)"
sign_cases: fixes x :: "('a, 'b) IEEE.float" obtains "sign x = 0"   | "sign x = 1"
simps_1: fixes c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and \<sigma> :: "'c" shows "foldli [] c f \<sigma> = \<sigma>"
inj_congruence: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex"   and H' :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0"   and "congruence M H = congruence M H'" shows "H = H'"
neq0_conv: fixes n :: "nat" shows "(n \<noteq> 0) = (0 < n)"
assms: shows "local.mute q k l (r, A) = (r', A')"
Special_ax_imp_hfthm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> Special_ax" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e" and "{} \<turnstile> A"
induct: fixes x1 :: "'a list"   and x2 :: "'a list"   and x3 :: "'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "app x1 x2 x3"   and "\<And>ys. P [] ys ys"   and "\<And>xs ys zs x. \<lbrakk>app xs ys zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) ys (x # zs)" shows "P x1 x2 x3"
complete\<^sub>N_def: fixes N :: "'a quasinorm" shows "complete\<^sub>N N = (\<forall>u. cauchy_ine\<^sub>N N u \<longrightarrow> (\<exists>x. tendsto_ine\<^sub>N N u x))"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>\<epsilon> f : (F \<circ> G) a \<rightarrow>\<^sub>C C.map b\<guillemotright>"
tendsto_at_topI: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e" shows "(f \<longlongrightarrow> l) at_top"
RSpan_smult_closed: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and ms :: "'b list"   and n :: "'b" assumes "RModule R smult M"   and "r \<in> R"   and "set ms \<subseteq> M"   and "n \<in> R_scalar_mult.RSpan R smult ms" shows "smult r n \<in> R_scalar_mult.RSpan R smult ms"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (Y a f) = Cop.arr f"
assms: fixes h :: "real" assumes "0 < h"   and "t + h \<le> T" shows "dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \<le> B * h ^ (p + 1)"
is_processT1: fixes P :: "'b process" shows "([], {}) \<in> F P"
simps_2: fixes c :: "('h, 'g, 'i, 'j) com"   and uu :: "('h, 'g, 'i, 'j) com list" shows "atCs (c # uu) = atC c"
uminus_def: fixes x :: "'a" shows "- x = (if x = bot then top else bot)"
simps: fixes a :: "'b"   and h :: "'b tree" shows "T_insert a h = 1"
same_length_def: fixes c :: "('b com option list \<times> 'b) list"   and clist :: "('b com option \<times> 'b) list list" shows "same_length c clist \<equiv> \<forall>i<length clist. length (clist ! i) = length c"
assms: fixes x :: "'a" assumes "x \<in> topspace X" shows "f x = g x"
coeff_0_reflect_poly: fixes p :: "'b poly" shows "coeff (reflect_poly p) 0 = lead_coeff p"
head_in_verts: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs (subdivide G e w)" shows "snd e \<in> pverts (subdivide G e w)"
ord0relD: fixes x :: "ord0"   and y :: "ord0" assumes "(x, y) \<in> ord0rel" shows "x \<le> y \<and> y \<le> x"
exp_bound: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "exp x \<le> 1 + x + x\<^sup>2"
conv_no_start_points: fixes x :: "'a" shows "no_start_points x = no_end_points (x\<^sup>T)"
of_int_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (numeral k) = numeral.numeral one plus k"
trans: fixes b :: "'h"   and a :: "'h"   and c :: "'h" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
Cons_2: fixes f :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "(f, g) \<in> set (p # ps) \<or> (g, f) \<in> set (p # ps)"   and "h = monom_mult ((1::'b) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f)" shows "thesis"
eval_subst_fmla: fixes e :: "(name, nat) finfun"   and fmla :: "fmla"   and i :: "name"   and t :: "trm" shows "eval_fmla e (fmla(i::=t)) = eval_fmla (finfun_update e i (eval_trm e t)) fmla"
set_zip_replicate_right: fixes xs :: "'b list"   and y :: "'c" shows "set (zip xs (replicate (length xs) y)) = (\<lambda>x. (x, y)) ` set xs"
cdf_interval_measure: fixes F :: "real \<Rightarrow> real" assumes "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"   and "\<And>a. continuous (at_right a) F"   and "(F \<longlongrightarrow> 0) at_bot" shows "cdf (interval_measure F) = F"
cases: fixes x :: "('f, 'a) Term.term" obtains xa :: "'a" where "x = Var xa"   | f :: "'f" and ts :: "('f, 'a) Term.term list" where "x = Fun f ts"
lift_Sup_distrib: fixes Y :: "V set"   and x :: "V" assumes "small Y" shows "lift x (\<Squnion> Y) = \<Squnion> (lift x ` Y)"
path_targetnode: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" shows "last (CFG.targetnodes targetnode as) = n'"
silent_moves_split: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and as' :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as @ as'\<Rightarrow>\<^sub>\<tau> (ms',s')" obtains ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')" and "S,f \<turnstile> (ms'',s'') =as'\<Rightarrow>\<^sub>\<tau> (ms',s')"
sorted_insort_key: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f (linorder.insort_key less_eq f x xs)) = linorder.sorted less_eq (map f xs)"
order_pos: fixes x :: "complex" shows "order x (charpoly cA) = order (x * cis (arg \<alpha>)) (charpoly cA)"
fun_upd_other: fixes z :: "'f"   and x :: "'f"   and f :: "'f \<Rightarrow> 'g"   and y :: "'g" assumes "z \<noteq> x" shows "(f(x := y)) z = f z"
gen_sqrt_lower: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "sqrtl x n \<le> sqrt x"
simps_9: fixes X :: "WS1S_Nameful.so"   and \<phi> :: "WS1S_Nameful.ws1s" shows "sos (All2 X \<phi>) = remove1 X (sos \<phi>)"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> analz H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>X Y. a = X \<and> \<lbrace>X, Y\<rbrace> \<in> analz H) \<or> (\<exists>X Y. a = Y \<and> \<lbrace>X, Y\<rbrace> \<in> analz H) \<or> (\<exists>K X. a = X \<and> Crypt K X \<in> analz H \<and> Key (invK K) \<in> analz H) \<or> (\<exists>x y. a = Pri_AgrK x \<and> Auth_Data x y \<in> analz H \<and> Pri_AgrK y \<in> analz H) \<or> (\<exists>x y. a = Pri_AgrK y \<and> Auth_Data x y \<in> analz H \<and> Pri_AgrK x \<in> analz H))"
double_complement: fixes A :: "'m set" shows "- (- A) = A"
to_precondition_def: fixes op :: "'b strips_operator" shows "to_precondition op \<equiv> map (\<lambda>v. (v, True)) (precondition_of op)"
Entry_target: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "trg a = NewEntry" shows "False"
assms_1: shows "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
set_insort: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "set (FWNormalisationCore.insort x xs l) = insert x (set xs)"
distinct_1: fixes x11 :: "nat"   and x12 :: "nat"   and x21 :: "nat"   and x22 :: "nat" shows "State x11 x12 \<noteq> Operator x21 x22"
sub_rel_of_def: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "sub_rel_of g subs \<equiv> SubRel.vertices subs \<subseteq> Graph.vertices g"
CCfix_nonrec_eq: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and x :: "Vars.var"   and e :: "exp"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "CoCallArityAnalysis.CCfix_nonrec cccExp x e\<cdot>i = CoCallAnalysis.ccBind (CoCallArityAnalysis.CCexp cccExp) x e\<cdot>(Aheap_nonrec x e\<cdot>i, snd i) \<squnion> fv e G\<times> (ccNeighbors x (snd i) - (if isVal e then {} else {x})) \<squnion> snd i"
is_aligned_neg_mask_eq: fixes p :: "'a word"   and n :: "nat" assumes "is_aligned p n" shows "p && ~~ mask n = p"
diff_eq_diff_less_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a - b = c - d" shows "(a \<le> b) = (c \<le> d)"
measurable_component_update: fixes x :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and i :: "'d" assumes "x \<in> space (Pi\<^sub>M I M)"   and "i \<notin> I" shows "fun_upd x i \<in> M i \<rightarrow>\<^sub>M Pi\<^sub>M (insert i I) M"
w_def: fixes a :: "int \<Rightarrow> nat"   and b :: "int \<Rightarrow> nat"   and i :: "int"   and j :: "int" shows "w a b i j = sum a {i..j + 1} + sum b {i..j}"
that: fixes x :: "'a" assumes "(THE x on U. P x) = Some x"   and "P x" shows "thesis"
index_eq_iff: fixes xs :: "'a list"   and x :: "'a"   and i :: "nat" shows "(index xs x = i) = (i \<le> length xs \<and> (\<forall>j<i. xs ! j \<noteq> x) \<and> (i < length xs \<longrightarrow> xs ! i = x))"
append_Cons: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(x # xs) @ ys = x # xs @ ys"
sat_models_finite_image: fixes P :: "'c ltl set" assumes "finite P" shows "finite (sat_models ` {Abs \<phi> |\<phi>. nested_propos \<phi> \<subseteq> P})"
neq_if_length_neq: fixes xs :: "'b list"   and ys :: "'b list" assumes "\<^bold>|xs\<^bold>| \<noteq> \<^bold>|ys\<^bold>|" shows "xs = ys \<equiv> False"
0: shows "(map_poly hom p = 0) = (p = 0)"
measure_pmf_bind: fixes M :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf" shows "measure_pmf (M \<bind> f) = measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x))"
mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> l F g1"   and "f2 \<in> l F g2" shows "(\<lambda>x. f1 x * f2 x) \<in> l F (\<lambda>x. g1 x * g2 x)"
I2: shows "callee_invariant_on oracle2 I2 \<I>"
A1id: shows "A1 = mat 1 1 (\<lambda>_. e)"
0: assumes "Arr t" shows "Diag \<^bold>\<lfloor>t\<^bold>\<rfloor> \<and> Dom \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Dom t\<^bold>\<rfloor> \<and> Cod \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
strong_induct: fixes P :: "'a \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a"   and trm :: "trm" assumes "\<And>z. P z Unit"   and "\<And>name z. P z (Var name)"   and "\<And>name trm z. \<lbrakk>name \<sharp> z; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (Lam [name].trm)"   and "\<And>trm1 trm2 z. \<lbrakk>\<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (App trm1 trm2)"   and "\<And>nat z. P z (Const nat)" shows "P z trm"
induct: fixes P :: "(('b \<rightarrow> tr) \<rightarrow> ['b] \<rightarrow> ['b]) \<Rightarrow> bool" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> P v. Fixrec.run\<cdot>(match_Nil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>[]) +++ match_Cons\<cdot>v\<cdot>(\<Lambda> xa xs. Fixrec.succeed\<cdot>(If P\<cdot>xa then xa : x\<cdot>P\<cdot>xs else x\<cdot>P\<cdot>xs))))" shows "P filter"
list_sorted_max: fixes list :: "'d list"   and x :: "'d"   and xs :: "'d list" assumes "sorted list"   and "list = x # xs" shows "fold max xs x = last list"
join_cont': fixes Y :: "nat \<Rightarrow> 'b"   and y :: "'b" assumes "chain Y" shows "(\<Squnion>i. Y i) \<squnion> y = (\<Squnion>i. Y i \<squnion> y)"
PM_def: shows "PM = (\<lambda>i. measure_pmf (bernoulli_pmf p))"
Times_mset_single_left: fixes a :: "'a"   and B :: "'b multiset" shows "{#a#} \<times># B = image_mset (Pair a) B"
FunctorId: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and C :: "('a, 'c, 'e) Category_scheme"   and D :: "('b, 'd, 'f) Category_scheme"   and X :: "'a" assumes "Ftor F : C \<longrightarrow> D"   and "X \<in> obj\<^bsub>C\<^esub>" shows "F ## id\<^bsub>C\<^esub> X = id\<^bsub>D\<^esub> (F @@ X)"
random_dictatorship_auxdef: fixes agents :: "'b set"   and alts :: "'a set"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "random_dictatorship agents alts R = pmf_of_set agents \<bind> (\<lambda>i. pmf_of_set (Max_wrt_among (R i) alts))"
I_3: fixes x :: "'a" assumes "x \<in> carrier" shows "finite {i \<in> N. W i \<inter> I x \<noteq> {}}"
enum_mono: fixes i :: "nat" shows "enum i < enum (Suc i)"
attach_wiring_def: shows "attach_wiring = (\<lambda>(f, g). map_fun id (map_fun f (map_gpv (map_prod g id) id)))"
assms: shows "(\<LM> A  \<RM>) \<oplus> B + \<Gamma> = \<LM> C  \<RM>"
annotateI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and anno :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P anno Q,A"   and "c = anno" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
YMapYoneda: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "YFtor C ## f = YMapInv C X (YFtor C @@ Y) (m2z\<^bsub>C\<^esub>f)"
take_nths_00: fixes xs :: "'b list" shows "take_nths 0 0 xs = xs"
simps_2: fixes \<psi> :: "fml option" shows "Ando undeff \<psi> = undeff"
SIGMAF_1: shows "\<Sigma>h = (Qh, Wh, rcmh)"
r_code_constn_def: fixes n :: "nat" shows "r_code_constn n \<equiv> if n = 1 then r_code_const1 else r_triple_encode (r_const 3) (r_const n) (Cn 1 r_prod_encode [r_code_const1, Cn 1 r_singleton_encode [r_triple_encode (r_const 2) (r_const n) Z]])"
Joints_nthE: fixes zs :: "'b list"   and ZS :: "('b \<times> 'b pdevs) list" assumes "zs \<in> Joints ZS" obtains e :: "nat \<Rightarrow> real" where "\<And>i. i < length zs \<Longrightarrow> zs ! i = aform_val e (ZS ! i)" and "\<And>i. e i \<in> {- 1..1}"
except_fun_cong1: fixes s :: "'a \<Rightarrow> 'b"   and V :: "'a set"   and U :: "'a set" assumes "supp_fun s \<inter> (V - U \<union> (U - V)) \<subseteq> {}" shows "except_fun s V = except_fun s U"