hcju/isabelleps huggingface dataset:
sup_left_commute: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "sup_prod x (sup_prod y z) = sup_prod y (sup_prod x z)"
valid_prefix_implies_ltl: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool"   and \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "ltl_equivalence eq"   and "eq (af \<phi> (prefix i w)) true\<^sub>n" shows "w \<Turnstile>\<^sub>n \<phi>"
sees_field_is_type: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list"   and wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "P \<turnstile> C sees F,b:T in D"   and "wf_prog wf_md P" shows "is_type P T"
p: shows "p \<noteq> 0 \<or> x \<noteq> 1"
scf_cond_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and p :: "'e"   and d :: "nat"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "scf_cond R S X p d c = (pol_coeff S c \<and> fst c = d \<and> p = polyn_expr R X d c)"
pps_to_list_sorted_wrt: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_term_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "sorted_wrt ord_term_strict\<inverse>\<inverse> (ordered_term.pps_to_list ord_term S)"
subst_subst_heap_sumC_def: shows "subst_subst_heap_sumC \<equiv> \<lambda>x. THE_default (case x of Inl x \<Rightarrow> Inl undefined | Inr x \<Rightarrow> Inr undefined) (subst_subst_heap_graph x)"
is_lasso_prpl_def: fixes G :: "('e, 'f) igb_graph_rec_scheme"   and L :: "'e list \<times> 'e list" assumes "igb_graph G" shows "igb_graph.is_lasso_prpl G L \<equiv> graph.is_lasso_prpl_pre G L \<and> (\<forall>i<igbg_num_acc G. \<exists>q\<in>set (snd L). i \<in> igbg_acc G q)"
cdf_bounded_prob: fixes x :: "real" shows "C x \<le> 1"
ccProd_mono: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and T :: "Vars.var set"   and T' :: "Vars.var set" assumes "S \<subseteq> S'"   and "T \<subseteq> T'" shows "S G\<times> T \<sqsubseteq> S' G\<times> T'"
simps_16: fixes mmu' :: "MMU_state \<Rightarrow> MMU_state"   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 "mmu_update mmu' \<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' mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
eq_fin_alt_def: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" shows "(w\<^sub>1 =\<^sub>F w\<^sub>2) = (w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2 \<and> w\<^sub>2 \<preceq>\<^sub>F w\<^sub>1)"
pseudominimal_inD2: fixes P :: "'a set"   and x :: "'a"   and z :: "'a" assumes "domain.pseudominimal_in P x"   and "z \<in> P"   and "x \<succ> z" shows "z = domain.bottom P"
map_fusion_def: fixes f :: "'e \<Rightarrow> 'd"   and g :: "('e, 'f) generator"   and s :: "'f" shows "map_fusion f g s = map f (list.unfoldr g s)"
theorem12: fixes g :: "'b \<rightarrow> 'a set"   and R :: "'b \<rightarrow> 'b set"   and x :: "'b" shows "(\<mu> f. \<Lambda> x. g\<cdot>x \<union> \<Union> (Rep_cfun f ` R\<cdot>x))\<cdot>x = \<^ps> g\<cdot>(\<Union>i. (\<^ps> R)\<^bsup>i\<^esup>\<cdot>{x})"
index_mod_eq: fixes i :: "nat"   and a :: "nat"   and b :: "nat" assumes "i \<in> {a * b..<(a + 1) * b}" shows "i mod b = i - a * b"
distinct_19: fixes x2 :: "('c, 'd) Term.term"   and x61 :: "poscheckvariant"   and x62 :: "('c, 'd) Term.term"   and x63 :: "('c, 'd) Term.term" shows "receive\<langle>x2\<rangle> \<noteq> \<langle>x61: x62 \<in> x63\<rangle>"
red2_in_Hom: fixes a :: "'c local.term"   and b :: "'c local.term" assumes "Ide a"   and "Diag a"   and "Ide b"   and "Diag b" shows "a \<^bold>\<Down> b \<in> Hom (a \<^bold>\<otimes> b) \<^bold>\<lfloor>a \<^bold>\<otimes> b\<^bold>\<rfloor>"
induct: fixes P :: "value \<Rightarrow> value \<Rightarrow> bool"   and a0 :: "value"   and a1 :: "value" assumes "\<And>n s. P (Num n) (Str s)"   and "\<And>s n. P (Str s) (Num n)"   and "\<And>s1 s2. P (Str s1) (Str s2)"   and "\<And>n1 n2. P (Num n1) (Num n2)" shows "P a0 a1"
healthy_inf: fixes A :: "'a hrel set" assumes "A \<subseteq> \<lbrakk>\<H>\<rbrakk>\<^sub>H" shows "\<^bold>\<Sqinter>A = \<H> (\<Or>A)"
multi_shift_simps_5: fixes c :: "nat"   and b :: "nat"   and a :: "'b word" assumes "c \<le> b" shows "a >> b << c = (a >> b - c) AND NOT (mask c)"
cSup_upper: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "preorder.bdd_above less_eq X" shows "less_eq x (Sup X)"
sum0: shows "(\<Sum>(c, \<alpha>)\<leftarrow>xs. c * exp \<alpha>) = 0"
segment_to_rel_frontier_aux: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "convex S"   and "bounded S"   and "x \<in> rel_interior S"   and "y \<in> S"   and "x \<noteq> y" obtains z :: "'a" where "z \<in> rel_frontier S" and "y \<in> closed_segment x z" and "open_segment x z \<subseteq> rel_interior S"
remove_term_coeff: fixes m0 :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (remove_term m0 p) m = (MPoly_Type.coeff p m when m \<noteq> m0)"
of_nat_1_eq_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(one = semiring_1.of_nat one plus zero n) = (n = 1)"
simps: fixes gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "sym_preproc_aux gs ks (vs, fs) = (if vs = [] then (ks, fs) else let v = ord_term_lin.max_list vs; vs' = removeAll v vs in sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v))"
LstSeq_append: fixes s1 :: "hf"   and k1 :: "hf"   and y1 :: "hf"   and s2 :: "hf"   and k2 :: "hf"   and y2 :: "hf" assumes "LstSeq s1 k1 y1"   and "LstSeq s2 k2 y2" shows "LstSeq (seq_append (succ k1) s1 s2) (succ (k1 @+ k2)) y2"
echelon_form_upt_k_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "echelon_form_upt_k A k = ((\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)) \<and> (\<forall>i j. i < j \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k j k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ j $ n \<noteq> (0::'b))))"
b_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "b_aux a b c = a_aux b c a"
lemma_3_10_i_ii: fixes H :: "'a set"   and a :: "'a" assumes "H \<in> normalfilters" shows "{a} ** H = H ** {a}"
not_eventually_sequentiallyD: fixes P :: "nat \<Rightarrow> bool" assumes "\<not> eventually P sequentially" shows "\<exists>r. strict_mono r \<and> (\<forall>n. \<not> P (r n))"
rootI: fixes r :: "'b list"   and k :: "nat" shows "r \<^sup>@ k \<in> r*"
conductor_eq_1_iff_principal: shows "(principal.conductor = 1) = (principal_dchar n = principal_dchar n)"
reachable_s_def: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" shows "reachable_s PROB s \<equiv> {exec_plan s as |as. as \<in> valid_plans PROB}"
elements_mat_uminus: fixes A :: "'b mat" shows "elements_mat (- A) = uminus ` elements_mat A"
FL_valid_Tree_eqvt': fixes P :: "'state"   and t :: "('b, 'pred, 'act, 'effect) Tree"   and p :: "perm" shows "FL_valid_Tree P t = FL_valid_Tree (p \<bullet> P) (p \<bullet> t)"
gf_U'_tendsto_U': fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" assumes "MC_syntax.recurrent K x"   and "(x, y) \<in> (SIGMA x:UNIV. set_pmf (K x))\<^sup>*" shows "((\<lambda>z. ennreal (MC_syntax.gf_U' K x y z)) \<longlongrightarrow> MC_syntax.U' K x y) (at_left 1)"
add_ac_2: fixes a :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool" shows "(a \<and>* b) = (b \<and>* a)"
star_subdist: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> \<le> (x + y)\<^sup>\<star>"
set_rec: fixes xs :: "'d list" shows "set xs = rec_list {} (\<lambda>x _. insert x) xs"
INF_insert: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` insert a A) = inf (f a) (Inf (f ` A))"
prems_3: fixes j :: "nat" assumes "j < Suc i" shows "suffix n w \<Turnstile>\<^sub>n af \<phi> (w [j \<rightarrow> n])[X]\<^sub>\<nu>"
infsetsum_Times: fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "countable A"   and "countable B"   and "f abs_summable_on A \<times> B" shows "infsetsum f (A \<times> B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B. f (x, y))"
simps_2: fixes i :: "nat"   and P :: "pol" shows "sqr (Pinj i P) = mkPinj i (sqr P)"
fds_deriv_add: fixes f :: "'b fds"   and g :: "'b fds" shows "fds_deriv (f + g) = fds_deriv f + fds_deriv g"
isomorphic_def: fixes a :: "'c"   and a' :: "'c" shows "D.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f)"
gauss_jordan_check_invertable: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat"   and b :: "'b" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n nc" shows "(A \<in> Units (ring_mat TYPE('a) n b)) = (fst (gauss_jordan A B) = 1\<^sub>m n)"
len_as: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list" assumes "MTF_Off as rs s0" shows "length as = length rs"
uniqHeadTailSet: fixes l :: "'a list" assumes "uniq l" shows "set (tl l) = set l - {hd l}"
alphaAbs_ex_distinct_qAFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_ex_distinct_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
AMax: fixes A :: "ant set"   and z :: "int" assumes "A \<subseteq> UBset (ant z)"   and "A \<noteq> {}" shows "AMax A \<in> A \<and> (\<forall>x\<in>A. x \<le> AMax A)"
elims: fixes x :: "Unboxed.type"   and y :: "('a, 'ubx1, 'ubx2) unboxed \<Rightarrow> 'dyn option" assumes "cast_and_box x = y" obtains "x = Ubx1" and "y = map_option box_ubx1 \<circ> cast_Ubx1"   | "x = Ubx2" and "y = map_option box_ubx2 \<circ> cast_Ubx2"
simps_2: fixes s :: "'a"   and ss :: "'a list" shows "state_set (s # ss) = insert [s] ((#) s ` state_set ss)"
list_all2_nthD: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'f list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length xs" shows "P (xs ! p) (ys ! p)"
rep_eq: fixes x :: "'b ratfps" shows "Rep_ratfps (inverse x) = (let (a, b) = quot_of_fract (Rep_ratfps x) in if constant_term a = (0::'b) then 0 else inverse (Rep_ratfps x))"
holomorphic_onD: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and x :: "complex" assumes "f holomorphic_on s"   and "x \<in> s" shows "f field_differentiable at x within s"
longest_common_suffix: fixes xs :: "'g list"   and ys :: "'g list" shows "\<exists>ss xs' ys'. xs = xs' @ ss \<and> ys = ys' @ ss \<and> (xs' = [] \<or> ys' = [] \<or> last xs' \<noteq> last ys')"
fBall_cong: fixes A :: "'d fset"   and B :: "'d fset"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x" shows "fBall A P = fBall B Q"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>Adj.\<epsilon> f : Adj.FG.map a \<rightarrow>\<^sub>D D.map b\<guillemotright>"
I\<^sub>a_aneg: fixes a :: "'a"   and xs :: "'b list" shows "I (aneg a) xs = (\<not> I\<^sub>a a xs)"
Always_Constrains_pre: fixes F :: "'b program"   and INV :: "'b set"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> Always INV" shows "(F \<in> INV \<inter> A Co A') = (F \<in> A Co A')"
ex_y': fixes x :: "'a" assumes "x \<in># X + Ya" shows "\<exists>y. y \<in># Y - Ya + {#z#} \<and> R x y"
distinct_1: fixes x21 :: "'c"   and x22 :: "'c list" shows "[] \<noteq> x21 # x22"
G_def: shows "G = {f \<in> F'. \<forall>f'\<in>F'. lt f' adds\<^sub>t lt f \<longrightarrow> f' = f}"
LLq: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and Lq :: "'c"   and t1 :: "'b"   and t2 :: "'b" assumes "Syntax_PseudoOrder var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num Lq"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "Syntax_PseudoOrder.LLq var Var FvarsT Fvars subst Lq t1 t2 \<in> fmla"
expr_typing_valE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and v :: "val"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> Val v : t" obtains "t = val_type v"
divmod_field_poly_def: fixes p :: "'b poly"   and q :: "'b poly" shows "divmod_field_poly p q = (p div q, p mod q)"
rank_None_Suc: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n"   and "rank x n = None" shows "rank x (Suc n) = None"
antisym_conv3: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
mon: fixes n :: "nat" shows "c n \<le> c (Suc n)"
semicategory_smc_0: fixes \<alpha> :: "V" assumes "\<Z> \<alpha>" shows "semicategory \<alpha> smc_0"
yun_wrel_minus: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly"   and G :: "int poly"   and g :: "rat poly" assumes "yun_wrel F c f"   and "yun_wrel G c g" shows "yun_wrel (F - G) c (f - g)"
not_neutral_contains_not_neutral: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "F g A \<noteq> \<^bold>1" obtains a :: "'c" where "a \<in> A" and "g a \<noteq> \<^bold>1"
tl_replicate: fixes n :: "nat"   and x :: "'a" shows "tl (replicate n x) = replicate (n - 1) x"
consistent_def: fixes S :: "('c, 'd) fm set" shows "consistent S \<equiv> \<nexists>S'. set S' \<subseteq> S \<and> (S' \<turnstile> \<^bold>\<bottom>)"
subint_antisym: fixes G :: "prog" assumes "ws_prog G" shows "antisym ((subint1 G)\<^sup>*)"
max_0_1_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq zero one = one"
permutation_disjoint_dom: fixes i :: "'b"   and S :: "'b set"   and i' :: "'b"   and j :: "'b" assumes "i \<in> S"   and "i' \<in> S"   and "j \<in> S"   and "i \<noteq> i'" shows "{p. p permutes S \<and> p i = j} \<inter> {p. p permutes S \<and> p i' = j} = {}"
subst_sequent: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<Gamma> :: "('sf, 'sc, 'sz) formula list"   and \<Delta> :: "('sf, 'sc, 'sz) formula list"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "Sadmit \<sigma> (\<Gamma>, \<Delta>)"   and "Ssafe (\<Gamma>, \<Delta>)"   and "ssafe \<sigma>" shows "(\<nu> \<in> seq_sem I (Ssubst (\<Gamma>, \<Delta>) \<sigma>)) = (\<nu> \<in> seq_sem (local.adjoint I \<sigma> \<nu>) (\<Gamma>, \<Delta>))"
linear_power_nonzero: fixes a :: "'b"   and n :: "nat" shows "[:a, 1::'b:] ^ n \<noteq> 0"
axioms_1: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "Set_Theory.map \<eta> R R'"
real_power_add: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and d :: "real"   and a :: "'a"   and b :: "'a" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < d" shows "real_power d (plus a b) = times (real_power d a) (real_power d b)"
prems_1: shows "inv_max_hi \<langle>l, (v, m), r\<rangle>"
disj_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_disj [x, y] = (if x = 0 \<and> y = 0 then 0 else 1)"
mat_max_mono: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'b mat" assumes "class.non_strict_order less_eq"   and "ord.mat_ge less_eq A B"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "ord.mat_ge less_eq (ord.mat_max less_eq C A) (ord.mat_max less_eq C B)"
R0: fixes Y :: "('b, 'c) ipurge_rec" shows "Y \<in> ipurge_fail_aux_t_set Y"
composite_pseudofunctor_axioms: shows "composite_pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H F \<Phi>\<^sub>F"
OptionOverride_def: fixes P :: "'b option \<times> 'b" shows "OptionOverride P = (if fst P = None then snd P else the (fst P))"
simps_1: shows "get_axiom AloopIter = loop_iterate_axiom"
delta': fixes S :: "'d set"   and a :: "'d"   and b :: "'d \<Rightarrow> 'e" assumes "finite S" shows "(\<Sum>k\<in>S. if a = k then b k else (0::'e)) = (if a \<in> S then b a else (0::'e))"
expunge_id_valid: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i1 :: "'a"   and e :: "'b"   and mo :: "'a"   and i2 :: "'a" assumes "xs prefix of j"   and "Deliver (i1, Expunge e mo i2) \<in> set xs" shows "i1 = i2"
Cod_dom: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Cod (local.dom f) = Dom f"
hyps_2: shows "P'' \<longmapsto> \<tau> \<prec> P'"
rep_transfer: shows "rel_fun cr_uint64 (=) (\<lambda>x. x) Rep_uint64"
supports_fresh: fixes S :: "atom set"   and x :: "'a"   and a :: "atom" assumes "S supports x"   and "finite S"   and "a \<notin> S" shows "a \<sharp> x"
simps_10: fixes x41 :: "'d"   and x42 :: "('b, 'c, 'd) Tree"   and x1 :: "('b, 'c, 'd) Tree set['b]" shows "tAct x41 x42 \<noteq> tConj x1"
colossless_gpv_sub_gpvs: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and gpv' :: "('j, 'h, 'i) gpv" assumes "colossless_gpv \<I> gpv"   and "gpv' \<in> sub_gpvs \<I> gpv" shows "colossless_gpv \<I> gpv'"
hyps_3: fixes T :: "ty" assumes "P,E,s'' \<turnstile> e'' : T \<surd>" shows "P,E,hp s' \<turnstile> e' :\<^bsub>NT\<^esub> T"
enat_of_option_simps_4: fixes x :: "nat option" shows "(\<infinity> = Dijkstra_Abstract.enat_of_option x) = (x = None)"
dom_simp: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "A1xA2.dom f = (A1.dom (fst f), A2.dom (snd f))"
eval_init_return: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and C' :: "char list"   and v :: "val"   and a :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "iconf (shp s) e"   and "(\<exists>Cs b. e = INIT C' (Cs,b) \<leftarrow> unit) \<or> (\<exists>C e\<^sub>0 Cs e\<^sub>i. e = RI (C,e\<^sub>0) ; Cs @ [C'] \<leftarrow> unit) \<or> (\<exists>e\<^sub>0. e = RI (C',e\<^sub>0) ; [] \<leftarrow> unit)" shows "(val_of e' = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>sfs i. shp s' C' = \<lfloor>(sfs, i)\<rfloor> \<and> (i = Done \<or> i = Processing))) \<and> (throw_of e' = \<lfloor>a\<rfloor> \<longrightarrow> (\<exists>sfs i. shp s' C' = \<lfloor>(sfs, Error)\<rfloor>))"
divide_right_mono_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" 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 c zero" shows "less_eq (divide b c) (divide a c)"
51: shows "bprv (all yy (imp (SS \<langle>\<chi>\<rangle> (Var yy)) (imp (subst \<phi> (Var yy) xx) (subst \<phi> \<langle>\<psi>\<rangle> xx))))"
tcsigs_translate: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "exe_osig_conds oss" shows "tcsigs (translate_osig oss) = translate_ars (snd oss)"