hcju/isabelleps huggingface dataset:
comp_of_presheaves: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and \<phi> :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f"   and \<FF>'' :: "'d set \<Rightarrow> 'g set"   and \<rho>'' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'g \<Rightarrow> 'g"   and b'' :: "'g"   and add_str'' :: "'d set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str'' :: "'d set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str'' :: "'d set \<Rightarrow> 'g"   and one_str'' :: "'d set \<Rightarrow> 'g"   and \<phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'g" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<phi>"   and "morphism_presheaves_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<FF>'' \<rho>'' b'' add_str'' mult_str'' zero_str'' one_str'' \<phi>'" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>'' \<rho>'' b'' add_str'' mult_str'' zero_str'' one_str'' (\<lambda>U. \<phi>' U \<circ> \<phi> U \<down> \<FF> U)"
induct: fixes P :: "('b word \<times> 'b word) list \<Rightarrow> bool"   and a0 :: "('b word \<times> 'b word) list" assumes "P []"   and "\<And>s ss. \<lbrakk>Ball (set ss) (disjoint_intervals s) \<Longrightarrow> P ss; \<not> Ball (set ss) (disjoint_intervals s) \<Longrightarrow> P (merge_overlap s ss)\<rbrakk> \<Longrightarrow> P (s # ss)" shows "P a0"
assms: fixes x :: "nat" shows "eval r_phi [i, x] \<up>"
partList_sub_equi0: fixes s :: "'b set"   and ts :: "'b set list" shows "set (partList0 s ts) = set (difference_list s ts) \<union> set (intersection_list s ts)"
intros_2: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and u :: "'e \<Rightarrow> 'f"   and d :: "'f"   and l' :: "'d"   and u' :: "'e \<Rightarrow> 'f" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsup>d\<^esup> \<langle>l', u'\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"
suffix_def: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "x [k..] \<equiv> \<lambda>n. x (k + n)"
height_balR: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "hbt l"   and "hbt r"   and "height r = height l + m + 1" shows "height (balR l a r) \<in> {height l + m + 1, height l + m + 2}"
zassenhaus_reconstruction_generic_def: fixes sl_impl :: "(int poly, int \<times> int poly list, 'a) subseqs_foldr_impl"   and vs :: "int poly list"   and p :: "int"   and n :: "nat"   and f :: "int poly" shows "zassenhaus_reconstruction_generic sl_impl vs p n f = (let lf = lead_coeff f; pn = p ^ n; (uu_, state) = subseqs_foldr sl_impl (lf, []) vs 0 in poly_mod.reconstruction pn sl_impl (pn div 2) state f (Polynomial.smult lf f) lf 0 (length vs) vs [] [])"
ord_p_minimum_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and x :: "'t \<Rightarrow>\<^sub>0 'c"   and Q :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "x \<in> Q"   and "Q \<subseteq> dgrad_p_set d m" obtains q :: "'t \<Rightarrow>\<^sub>0 'c" where "q \<in> Q" and "\<And>y. y \<prec>\<^sub>p q \<Longrightarrow> y \<notin> Q"
3: shows "runiq a \<and> runiq (a\<inverse>) \<and> is_non_overlapping (Range a) \<and> Range (a - (X \<union> {i}) \<times> Range a) \<subseteq> Range a \<and> a `` (X \<union> {i}) \<subseteq> Range a"
reachable_mode_states_def: fixes gc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "reachable_mode_states gc = {mdss. \<exists>cms' mem'. gc \<rightarrow>\<^sup>* (cms', mem') \<and> map snd cms' = mdss}"
cases: fixes a :: "'d \<Rightarrow> real"   and X :: "'d set"   and I :: "'d \<Rightarrow> Regions.intv"   and r :: "('d \<times> 'd) set" assumes "a \<in> Regions.region X I r" obtains u :: "'d \<Rightarrow> real" and X\<^sub>0 :: "'d set" where "a = u" and "\<forall>x\<in>X. 0 \<le> u x" and "\<forall>x\<in>X. Regions.intv_elem x u (I x)" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d}" and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))"
cont_times1: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" assumes "class.times_cpo below" shows "cont (\<lambda>x. x * y)"
is_extensional: fixes f :: "'b" assumes "\<not> D.arr f" shows "trg\<^sub>D f = D.null"
mod_by_Suc_0: fixes m :: "nat" shows "m mod Suc 0 = 0"
eq_set_inc: fixes a :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "a \<in> A"   and "A = B" shows "a \<in> B"
inf_add_of_complex: fixes x :: "complex" shows "\<infinity>\<^sub>h +\<^sub>h of_complex x = \<infinity>\<^sub>h"
symP_def: fixes \<phi> :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "symP \<phi> \<equiv> \<forall>x y. \<phi> x y \<longrightarrow> \<phi> y x"
eventually_floor_eq: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<notin> \<int>" shows "\<forall>\<^sub>F x in F. \<lfloor>f x\<rfloor> = \<lfloor>l\<rfloor>"
bsok_simps_5: fixes e1 :: "'a expr1"   and bop :: "bop"   and e2 :: "'a expr1"   and n :: "nat" shows "bsok (e1 \<guillemotleft>bop\<guillemotright> e2) n = (bsok e1 n \<and> bsok e2 n)"
positive_def: fixes M :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "positive M \<mu> = (\<mu> {} = 0)"
is_tame_def: fixes g :: "graph" shows "is_tame g \<equiv> tame10 g \<and> tame11a g \<and> tame12o g \<and> is_tame13a g"
max_v_ev_2: shows "cmod local.max_ev = HMA_Connect.spectral_radius A"
sum_fun_image_sup: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and z :: "'d" assumes "finite A" shows "(\<Sum> (f ` A) \<le> z) = (\<forall>a\<in>A. f a \<le> z)"
surj_fun_eq: fixes f :: "'c \<Rightarrow> 'b"   and X :: "'c set"   and g1 :: "'b \<Rightarrow> 'd"   and g2 :: "'b \<Rightarrow> 'd" assumes "f ` X = UNIV"   and "\<forall>x\<in>X. (g1 \<circ> f) x = (g2 \<circ> f) x" shows "g1 = g2"
insert_union: fixes x :: "'b"   and X :: "'b set" shows "insert x X = X \<union> {x}"
f_shrink_last_def: fixes xs :: "'b list"   and k :: "nat" shows "xs \<div>\<^sub>l k \<equiv> f_aggregate xs k last"
tendsto_EM_remainder: fixes a :: "int"   and G :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and n :: "nat"   and f :: "real \<Rightarrow> 'a" assumes "\<And>y. real_of_int a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {real_of_int a..})"   and "\<And>a' b. \<lbrakk>real_of_int a \<le> a'; a' \<le> b\<rbrakk> \<Longrightarrow> (\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on {a'..b}"   and "convergent (\<lambda>y. G (real y))"   and "\<forall>\<^sub>F x in at_top. norm (f x) \<le> g x" shows "(\<lambda>b. EM_remainder' n f (real_of_int a) (real b)) \<longlonglongrightarrow> EM_remainder n f a"
nat_pow_modify_carrier: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "nat" shows "a [^]\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> b = a [^]\<^bsub>G\<^esub> b"
aval''_sound: fixes s :: "char list \<Rightarrow> int"   and S :: "'av Abs_State.st option"   and a :: "aexp" assumes "s \<in> \<gamma>\<^sub>o S" shows "aval a s \<in> \<gamma> (aval'' a S)"
simps_2: shows "PX4d D2 = linord_of_list [X4.Xd2, Xd2']"
ik\<^sub>s\<^sub>s\<^sub>t_def: fixes A :: "('f, 'g) stateful_strand_step list" shows "ik\<^sub>s\<^sub>s\<^sub>t A \<equiv> {t. receive\<langle>t\<rangle> \<in> set A}"
MM6: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<or> \<box>G \<longrightarrow> \<box>(\<box>F \<or> \<box>G)"
liminf_subseq_mono: fixes r :: "nat \<Rightarrow> nat"   and X :: "nat \<Rightarrow> 'b" assumes "strict_mono r" shows "liminf X \<le> liminf (X \<circ> r)"
lcoset_closed: fixes a :: "'g"   and A :: "'g set" assumes "a \<in> G"   and "order.greater_eq G A" shows "order.greater_eq G (a +o A)"
right_impl_times: fixes a :: "'b"   and b :: "'b" shows "(a l\<rightarrow> b) * a = (b l\<rightarrow> a) * b"
record_simps_7: fixes x :: "'a state" shows "update_clock (\<lambda>_. clock x) x = x"
equiv_lsbis1: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" assumes "coalg B1 B2 s1 s2" shows "equiv B1 (lsbis1 B1 B2 s1 s2)"
SUP_escape_nonneg: shows "0 \<le> \<Squnion> range escape_rate"
finsert_iff: fixes a :: "'b"   and b :: "'b"   and A :: "'b fset" shows "(a |\<in>| finsert b A) = (a = b \<or> a |\<in>| A)"
refl: shows "refl_on {x \<in> X. \<exists>c. I x = Intv c} r"
assms_3: shows "(a2, b2) \<in> getRel l G"
Pm_fmap_of_list_eq_zero_iff: fixes xs :: "('c \<times> 'd) list" shows "(Pm_fmap (fmap_of_list xs) = 0) = (filter (\<lambda>(k, v). v \<noteq> (0::'d)) (AList.clearjunk xs) = [])"
holo: fixes X :: "complex set"   and p :: "nat" assumes "X \<subseteq> {s. 1 / 2 < Re s}" shows "(\<lambda>s. if prime p then B p s else 0) holomorphic_on X"
scaleR: fixes r :: "real"   and b :: "'c" shows "g (r *\<^sub>R b) = r *\<^sub>R g b"
inters_fin_fund: fixes s :: "'c event list"   and A :: "'c event set" shows "finite {(t, u). s setinterleaves ((t, u), A)}"
lfinite_code_2: fixes x :: "'b"   and xs :: "'b llist" shows "lfinite (LCons x xs) = lfinite xs"
simps_18: fixes x3 :: "('b, 'c) fm"   and x2 :: "'c" shows "(\<^bold>\<not> x3) \<noteq> Nom x2"
FSwSbImorph_def: fixes h :: "'d \<Rightarrow> 'e"   and hA :: "'f \<Rightarrow> 'g"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'd, 'f, 'i) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'e, 'g, 'j) model_scheme" shows "FSwSbImorph h hA MOD MOD' \<equiv> FSwImorph h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
78: fixes x :: "'a" shows "top \<squnion> - x = top"
Max_ge_iff: fixes A :: "'a Word.word set"   and x :: "'a Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le>s signed.Max A) = (\<exists>a\<in>A. x \<le>s a)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) local.map \<Pi>o\<Delta>.map dup"
x_axis_gauge_integrable: fixes x :: "'a" shows "(\<lambda>y. f (x, y)) integrable_on UNIV"
children_eq2_h: fixes ptr' :: "(_) object_ptr" assumes "ptr' \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr" shows "|h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r"
identical_until_bad: fixes bad :: "'b \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'b spmf"   and f :: "'b \<Rightarrow> 'c"   and A :: "'c set" assumes "map_spmf bad p = map_spmf bad q"   and "Sigma_Algebra.measure (measure_spmf (map_spmf (\<lambda>x. (f x, bad x)) p)) (A \<times> {False}) = Sigma_Algebra.measure (measure_spmf (map_spmf (\<lambda>x. (f x, bad x)) q)) (A \<times> {False})" shows "\<bar>Sigma_Algebra.measure (measure_spmf (map_spmf f p)) A - Sigma_Algebra.measure (measure_spmf (map_spmf f q)) A\<bar> \<le> spmf (map_spmf bad p) True"
length_prefix_ex': fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "\<exists>ys zs. xs = ys @ xs ! n # zs \<and> length ys = n"
unfold_zipWithStep: fixes f :: "'e \<rightarrow> 'f \<rightarrow> 'd"   and ha :: "'b \<rightarrow> ('e, 'b) Step"   and hb :: "'c \<rightarrow> ('f, 'c) Step" shows "(\<forall>sa sb. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(zipWithStep\<cdot>f\<cdot>ha\<cdot>hb)\<cdot>(sa :!: sb :!: Nothing) = zipWithL\<cdot>f\<cdot>(unfold\<cdot>ha\<cdot>sa)\<cdot>(unfold\<cdot>hb\<cdot>sb)) \<and> (\<forall>sa sb a. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(zipWithStep\<cdot>f\<cdot>ha\<cdot>hb)\<cdot>(sa :!: sb :!: Just\<cdot>(L\<cdot>a)) = zipWithL\<cdot>f\<cdot>(LCons\<cdot>a\<cdot>(unfold\<cdot>ha\<cdot>sa))\<cdot>(unfold\<cdot>hb\<cdot>sb))"
comm_groupI: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "comm_group G"
\<R>\<^sub>1_sym: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "sym (\<R>\<^sub>1 \<Gamma>' \<S>' P')"
star_n_less: fixes X :: "nat \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'c" shows "(star_n X < star_n Y) = (\<forall>\<^sub>F n in \<U>. X n < Y n)"
shift_one: fixes R :: "('c, 'd) ring_scheme" assumes "UP_cring R" shows "shift (Suc 0) = poly_shift"
fimage_vempty: fixes A :: "V" shows "[]\<^sub>\<circ> `\<^sub>\<bullet> A = []\<^sub>\<circ>"
set_mset_eq_empty_iff: fixes M :: "'d multiset" shows "(set_mset M = {}) = (M = {#})"
complete_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "complete_on r A = (SN_on r A \<and> CR_on r A)"
GT: fixes x :: "'c"   and y :: "'c" shows "(cmp x y = GT) = (y < x)"
update_convs_7: fixes dwrite' :: "(int \<times> word32 \<times> CPU_register) list \<Rightarrow> (int \<times> word32 \<times> CPU_register) list"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "dwrite_update dwrite' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite' dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
urel32_minus: fixes x :: "uint32"   and y :: "int"   and x' :: "uint32"   and y' :: "int" assumes "urel32 x y"   and "urel32 x' y'" shows "urel32 (minus_p32 pp x x') (minus_p p y y')"
False: shows "\<not> (y = (0::'a) \<or> x = (0::'a))"
IO: fixes x :: "'d"   and c :: "('b, 'd, 'e) rpv"   and gpv :: "('b, 'd, 'e) gpv"   and \<I> :: "('d, 'e) \<I>" assumes "IO x c \<in> set_spmf (the_gpv gpv)" shows "x \<in> outs_gpv \<I> gpv"
suc: shows "Suc i_end - i_begin = Suc (i_end - i_begin)"
rangeHeuristic: fixes n :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and var :: "nat"   and step :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm" assumes "heuristicPicker n L F = Some (var, step)" shows "var \<le> n"
lex_pm_plus_monotone: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd"   and u :: "'c \<Rightarrow>\<^sub>0 'd" assumes "lex_pm s t" shows "lex_pm (s + u) (t + u)"
induct: fixes x1 :: "v sem_env"   and x2 :: "exp0"   and x3 :: "(v, v) result"   and P :: "v sem_env \<Rightarrow> exp0 \<Rightarrow> (v, v) result \<Rightarrow> bool" assumes "cupcake_evaluate_single x1 x2 x3"   and "\<And>env cn es rs vs v0. \<lbrakk>do_con_check (c env) cn (length es); list_all2_shortcircuit (\<lambda>x1 x2. cupcake_evaluate_single env x1 x2 \<and> P env x1 x2) (rev es) rs; sequence_result rs = Rval vs; build_conv (c env) cn (rev vs) = Some v0\<rbrakk> \<Longrightarrow> P env (Con cn es) (Rval v0)"   and "\<And>env cn es. \<not> do_con_check (c env) cn (length es) \<Longrightarrow> P env (Con cn es) (Rerr (Rabort Rtype_error))"   and "\<And>env cn es rs err. \<lbrakk>do_con_check (c env) cn (length es); list_all2_shortcircuit (\<lambda>x1 x2. cupcake_evaluate_single env x1 x2 \<and> P env x1 x2) (rev es) rs; sequence_result rs = Rerr err\<rbrakk> \<Longrightarrow> P env (Con cn es) (Rerr err)"   and "\<And>env n v0. nsLookup (sem_env.v env) n = Some v0 \<Longrightarrow> P env (Var n) (Rval v0)"   and "\<And>env n. nsLookup (sem_env.v env) n = None \<Longrightarrow> P env (Var n) (Rerr (Rabort Rtype_error))"   and "\<And>env n e. P env (Fun n e) (Rval (Closure env n e))"   and "\<And>env es rs vs env' e bv. \<lbrakk>list_all2_shortcircuit (\<lambda>x1 x2. cupcake_evaluate_single env x1 x2 \<and> P env x1 x2) (rev es) rs; sequence_result rs = Rval vs; do_opapp (rev vs) = Some (env', e); cupcake_evaluate_single env' e bv; P env' e bv\<rbrakk> \<Longrightarrow> P env (App Opapp es) bv"   and "\<And>env es rs vs. \<lbrakk>list_all2_shortcircuit (\<lambda>x1 x2. cupcake_evaluate_single env x1 x2 \<and> P env x1 x2) (rev es) rs; sequence_result rs = Rval vs; do_opapp (rev vs) = None\<rbrakk> \<Longrightarrow> P env (App Opapp es) (Rerr (Rabort Rtype_error))"   and "\<And>env es rs err op0. \<lbrakk>list_all2_shortcircuit (\<lambda>x1 x2. cupcake_evaluate_single env x1 x2 \<and> P env x1 x2) (rev es) rs; sequence_result rs = Rerr err\<rbrakk> \<Longrightarrow> P env (App op0 es) (Rerr err)"   and "\<And>env e v0 pes e' uu_ env' bv. \<lbrakk>cupcake_evaluate_single env e (Rval v0); P env e (Rval v0); cupcake_match_result (c env) v0 pes Bindv = Rval (e', uu_, env'); cupcake_evaluate_single (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) e' bv; P (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) e' bv\<rbrakk> \<Longrightarrow> P env (Mat e pes) bv"   and "\<And>env e v0 pes err. \<lbrakk>cupcake_evaluate_single env e (Rval v0); P env e (Rval v0); cupcake_match_result (c env) v0 pes Bindv = Rerr err\<rbrakk> \<Longrightarrow> P env (Mat e pes) (Rerr err)"   and "\<And>env e err pes. \<lbrakk>cupcake_evaluate_single env e (Rerr err); P env e (Rerr err)\<rbrakk> \<Longrightarrow> P env (Mat e pes) (Rerr err)" shows "P x1 x2 x3"
not_less_Ord_Least: fixes k :: "V"   and P :: "V \<Rightarrow> bool" assumes "k < (LEAST x. Ord x \<and> P x)"   and "Ord k" shows "\<not> P k"
progressing_rev_path: fixes p :: "('b \<times> 'b) list" shows "progressing (rev_path p) = progressing p"
function_nat_pow_eval: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and s :: "nat \<Rightarrow> int"   and n :: "nat" assumes "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "s \<in> carrier Zp" shows "(f [^]\<^bsub>function_ring (carrier Zp) Zp\<^esub> n) s = f s [^] n"
tm1: shows "\<tt>\<tt> (1::'a) = (1::'a)"
cancel_suffix: fixes a :: "'b list"   and c :: "'b list"   and b :: "'b list"   and d :: "'b list" assumes "a @ c = b @ d"   and "length c \<le> length d" shows "a = b @ take (length d - length c) d"
ord_isom_restricted: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and D1 :: "'a set" assumes "Order E"   and "ord_isom D E f"   and "D1 \<subseteq> carrier D" shows "ord_isom (Iod D D1) (Iod E (f ` D1)) (restrict f D1)"
a6: shows "generate_valid_topology M (fully_connected G) = \<lparr>nodes = V, edges = E\<rparr>"
ipresOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'e, 'f, 'opSym, 'g, 'c, 'd, 'h) Iteration.model_scheme" shows "ipresOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = igOp MOD delta (lift h inp) (lift hA binp)"
set0_3: fixes x1 :: "('f \<times> 'e) pmf" shows "set2_var0 (PMF x1) = \<Union> (Basic_BNFs.snds ` set_pmf x1)"
RPDi_safe: fixes g :: "safety"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex"   and n :: "nat" assumes "safe_regex Past g (from_mregex mr \<phi>s)"   and "ms \<in> RPDi n mr" shows "safe_regex Past g (from_mregex ms \<phi>s)"
PiE_restrict: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "f \<in> Pi\<^sub>E A B" shows "restrict f A = f"
complies_BLP: shows "SecurityInvariant_complies_formal_def BLP_m (the BLP_m_spec)"
vunion_assoc: fixes A :: "V"   and B :: "V"   and C :: "V" shows "A \<union>\<^sub>\<circ> (B \<union>\<^sub>\<circ> C) = A \<union>\<^sub>\<circ> B \<union>\<^sub>\<circ> C"
exp_cf6_pos: fixes x :: "real" shows "0 < exp_cf6 x"
DISJNT: shows "\<Union> (A ` {1..m}) \<inter> A (Suc m) = {}"
fpxs_val_compose_power: fixes r :: "rat"   and f :: "'b fpxs" assumes "0 < r" shows "fpxs_val (fpxs_compose_power f r) = fpxs_val f * r"
function_smult_assoc2: fixes a :: "'a"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier F"   and "g \<in> carrier F" shows "a \<odot>\<^bsub>F\<^esub> f \<otimes>\<^bsub>F\<^esub> g = a \<odot>\<^bsub>F\<^esub> (f \<otimes>\<^bsub>F\<^esub> g)"
pure_returns_heap_eq: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and h' :: "'x" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "pure f h" shows "h = h'"
intros_8: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and ins :: "'c list"   and outs :: "'c list"   and V :: "'c"   and x :: "nat"   and n :: "'b SDG_node"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "(p, ins, outs) \<in> set procs"   and "V = outs ! x"   and "x < length outs"   and "n = Formal_out (sourcenode a, x)"   and "n' = Actual_out (targetnode a, x)" shows "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (Some V) (Some (p, False)) n'"