hcju/isabelleps huggingface dataset:
l5_6: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" assumes "A B Le C D"   and "Cong A B A' B'"   and "Cong C D C' D'" shows "A' B' Le C' D'"
is_coext_def: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" shows "x\<epsilon>.is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow>\<^sub>D Go x\<guillemotright> \<and> f = \<epsilon>o x \<cdot>\<^sub>C F g"
verts_H: shows "verts (with_proj H) = verts (with_proj I') \<union> {w}"
rel_compp_Grp: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_pmf R = (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
nonempty_path_indices: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u"   and "p \<noteq> []" shows "0 < k \<or> 0 < u"
NoSmallSeparationsInduct_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and sep_size :: "nat" shows "NoSmallSeparationsInduct G v0 v1 sep_size \<equiv> v0_v1_Digraph G v0 v1 \<and> NoSmallSeparationsInduct_axioms G v0 v1 sep_size"
basis_in_card: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and B\<^sub>1 :: "'b set"   and B\<^sub>2 :: "'b set" assumes "matroid carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.basis_in indep \<E> B\<^sub>1"   and "indep_system.basis_in indep \<E> B\<^sub>2" shows "card B\<^sub>1 = card B\<^sub>2"
null_measure_def: fixes M :: "'c measure" shows "null_measure M = sigma (space M) (sets M)"
smallomega_1_iff: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real" assumes "eventually_nonneg F f" shows "(f \<in> \<omega>[F](\<lambda>_. 1)) = filterlim f at_top F"
partition_on_singletons: fixes A :: "'b set" shows "partition_on A ((\<lambda>x. {x}) ` A)"
simps_3: fixes c :: "poly"   and e :: "tm" shows "minusinf (Eq (CNP 0 c e)) = Parametric_Ferrante_Rackoff.conj (eq (CP c)) (eq e)"
productF_simps_4: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM" assumes "productF A B FAIL AB" shows "initial AB = (initial A, initial B)"
length_append_singleton: fixes xs :: "'b list"   and x :: "'b" shows "\<^bold>|xs \<cdot> [x]\<^bold>| = Suc \<^bold>|xs\<^bold>|"
assms_1: shows "strategy_attracts_via p \<sigma> v0 A W"
simps_2: fixes v :: "int \<times> int" shows "headn (C v) = (\<lambda>m. C v)"
CBasis_prod_def: shows "CBasis = (\<lambda>u. (u, 0::'d)) ` CBasis \<union> Pair (0::'c) ` CBasis"
size_mset: fixes xs :: "'b list" shows "size (mset xs) = length xs"
homeomorphic_finite: fixes T :: "'a set"   and S :: "'b set" assumes "finite T" shows "S homeomorphic T = (finite S \<and> finite T \<and> card S = card T)"
Max_wrt_among_cong: fixes A :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "restrict_relation A R = restrict_relation A R'" shows "Max_wrt_among R A = Max_wrt_among R' A"
rtrancl_reflcl: fixes R :: "('b \<times> 'b) set" shows "(R\<^sup>=)\<^sup>* = R\<^sup>*"
R_def: fixes h :: "'pub_input"   and w :: "'witness"   and e :: "'challenge" shows "R h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
Par1B: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<parallel> Q"
insert_iff: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" shows "(a \<in> Set.insert b A) = (a = b \<or> a \<in> A)"
a: shows "base \<le> a \<and> a \<le> base OR mask (LENGTH('a) - len)"
finite_valid_length: fixes n :: "nat" shows "finite {l. length l = n \<and> valid l}"
compat_low_eq: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and x :: "'Var" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "dma mem\<^sub>1 x = Low"   and "x \<in> \<C> \<or> (\<forall>i<length cms\<^sub>1. x \<notin> snd (cms\<^sub>1 ! i) AsmNoReadOrWrite)" shows "mem\<^sub>1 x = mem\<^sub>2 x"
that: assumes "vsv f"   and "f\<lparr>a\<rparr> \<in>\<^sub>\<circ> A a"   and "f\<lparr>b\<rparr> \<in>\<^sub>\<circ> A b"   and "\<D>\<^sub>\<circ> f = set {a, b}"   and "f = set {\<langle>a, f\<lparr>a\<rparr>\<rangle>, \<langle>b, f\<lparr>b\<rparr>\<rangle>}" shows "thesis"
bind_return: fixes m :: "'b \<Rightarrow> ('c \<times> 'b) \<times> bool" shows "m >>= return = m"
Rep_binding_cases: fixes y :: "binding_Rep" assumes "y \<in> Collect binding_Rep_set" obtains x :: "binding" where "y = Rep_binding x"
intro: fixes intT :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>\<sigma>. \<exists>a. intT \<sigma> a" shows "M.Tstruct intT"
length_concat_negGorbit_list: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> 'v"   and as :: "'a list" shows "length (concat (BaseRep.negGorbit_list gs T as)) = length gs * length as"
set_fmap_def: fixes F :: "'g \<Rightarrow> 'h option" shows "set_fmap F = {(x, y). x \<in> dom F \<and> F x = Some y}"
path_components_of_subset: fixes C :: "'c set"   and X :: "'c topology" assumes "C \<in> path_components_of X" shows "C \<subseteq> topspace X"
inner_left_distrib: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   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 x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (plus x y) z = inner x z + inner y z"
union_lcomm: fixes M :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset" shows "M + (N + K) = N + (M + K)"
sup_cases: fixes x :: "'a"   and y :: "'a" shows "- x = - x \<sqinter> - y \<squnion> - x \<sqinter> - - y"
assms_10: shows "InvariantClCurrentLevel (getCl state) (getM state)"
comp_quasi_inverse_1: fixes f :: "'a" assumes "equivalence_map f" shows "f\<^sup>~ \<star> f \<cong> src f"
m_4: shows "\<forall>t\<in>S_Max_Lt. m < t"
process_rule_2: fixes xs :: "'c list"   and x :: "'c"   and X :: "'c set"   and P :: "'c process" assumes "(xs @ [x], X) \<in> failures P" shows "(xs, {}) \<in> failures P"
simple_integral_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M"   and "simple_function (restrict_space M \<Omega>) f" shows "integral\<^sup>S (restrict_space M \<Omega>) f = \<integral>\<^sup>S x. f x * indicator \<Omega> x \<partial>M"
mkarr_extensionality: fixes t :: "'c local.term" assumes "\<not> Arr t" shows "mkarr t = {}"
pullback_algebra_def: fixes M :: "'c measure"   and f :: "'b \<Rightarrow> 'c"   and \<Omega>' :: "'b set" shows "sets.pullback_algebra M f \<Omega>' = sigma_sets \<Omega>' {f -` A \<inter> \<Omega>' |A. A \<in> sets M}"
choose_v__in_short: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "short_cycles G k \<noteq> {}" shows "\<exists>p. p \<in> short_cycles G k \<and> choose_v G k \<in> set p"
uniformity_sq_mtx_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
valid_join_mmsaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list"   and X :: "'a option list set" assumes "valid_mmsaux args cur aux auxlist"   and "table (args_n args) (args_L args) X" shows "valid_mmsaux args cur (join_mmsaux args X aux) (map (\<lambda>(t, rel). (t, join rel (args_pos args) X)) auxlist)"
intros_5: fixes F :: "'b formula"   and G :: "'b formula" shows "(F \<^bold>\<and> G) \<^bold>\<rightarrow> F \<in> AX10"
LPDs_Star: fixes r :: "mregex" shows "LPDs (MStar r) \<subseteq> {MStar r} \<union> MTimesR (LPDs r) (MStar r)"
invertibleMtx2: shows "invertible_mat (M_mat signs2 subsets2)"
level: shows "2 ^ (lv p d - lv b d) = 2 * 2 ^ (lv p d - lv (child b dir d) d)"
real_cond_exp_mono: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "AE x in M. f x \<le> g x"   and "integrable M f"   and "integrable M g" shows "AE x in M. real_cond_exp M F f x \<le> real_cond_exp M F g x"
subst_elimI: fixes v :: "'c"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "\<And>t. v \<notin> fv (t \<cdot> \<sigma>)" shows "subst_elim \<sigma> v"
mklr_Inf: fixes A :: "'a admS set" shows "unlr (Inf A) = \<Inter> (unlr ` A)"
subsubformulae: fixes G :: "'b formula"   and F :: "'b formula"   and H :: "'b formula" assumes "G \<in> set (subformulae F)"   and "H \<in> set (subformulae G)" shows "H \<in> set (subformulae F)"
dim: shows "dimension (n + m - k) K (Span K (Us @ Vs @ Bs))"
elims: fixes x :: "('c \<times> ('a, 'b, 'c) psi) list"   and y :: "('a, 'b, 'c) psiCase" assumes "psiCases x = y" obtains "x = []" and "y = \<bottom>\<^sub>c"   | \<Phi> :: "'c" and P :: "('a, 'b, 'c) psi" and xs :: "('c \<times> ('a, 'b, 'c) psi) list" where "x = (\<Phi>, P) # xs" and "y = \<box> \<Phi> \<Rightarrow> P psiCases xs "
apartments_have_many_foldings1_1: fixes A :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<sim> D"   and "C \<noteq> D"   and "C \<in> A"   and "D \<in> A" shows "restrict1 (canonical_retraction A D \<circ> canonical_retraction (supapartment C (some_third_chamber C D (C \<inter> D))) C) (\<Union> A) ` D = C"
ordered_semiring_record_simps_12: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
exec_1_def: shows "exec_1 \<equiv> \<lambda>P. {(xa, x). exec_1p P xa x}"
simps_1: fixes start :: "'a word"   and "end" :: "'a word" shows "wordinterval_to_set (WordInterval start end) = {start..end}"
in_interval_realI: fixes a :: "float"   and I :: "float interval" assumes "real_of_float a \<in>\<^sub>r I" shows "a \<in>\<^sub>i I"
divide_complex_def: fixes x :: "complex"   and y :: "complex" shows "x div y = x * inverse y"
take_but_one: fixes x :: "'a list" shows "x \<noteq> [] \<longrightarrow> lastn (length x - 1) x = tl x"
carrier_submanifold: shows "sub.carrier = domain \<psi> \<inter> carrier"
Aexp_Let_simp2: fixes x :: "Vars.var"   and e :: "exp"   and exp :: "exp"   and a :: "Arity" assumes "x \<notin> fv e" shows "Aexp_syn' a (let x be e in exp ) = env_delete x (Aexp_bot_syn' (ABind_nonrec x e\<cdot>(Aexp_syn' a exp, ccExp_syn a exp)) e \<squnion> Aexp_syn' a exp)"
list_slice_1: fixes xs :: "'b list" shows "list_slice xs (Suc 0) = map (\<lambda>x. [x]) xs"
IH: assumes "i1 \<le> i2" shows "(r i1, r i2) \<in> E\<^sup>*"
prems_2: fixes x :: "real" assumes "x \<in> {a<..<b} - insert x__ S" shows "(f has_vector_derivative f' x) (at x)"
simps_3: fixes \<sigma> :: "(char list \<times> 'c list) trace"   and v :: "'c list"   and i :: "nat"   and \<phi> :: "'c MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Neg \<phi>) = (\<not> MFOTL.sat \<sigma> v i \<phi>)"
scale_spmf_def: fixes r :: "real"   and p :: "'b spmf" shows "scale_spmf r p = embed_spmf (\<lambda>x. min (inverse (weight_spmf p)) (max 0 r) * spmf p x)"
simps_91: fixes x14 :: "'a exp"   and x3 :: "val" shows "throw x14 \<noteq> Val x3"
simps_4: fixes k :: "name"   and e :: "(name, nat) finfun"   and A :: "fmla" assumes "atom k \<sharp> e" shows "eval_fmla e (exi k A) = (\<exists>x. eval_fmla (finfun_update e k x) A)"
prems_6: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"
eval: shows "prg Env\<turnstile>s0 \<midarrow>c\<rightarrow> s1"
acyclic_5c_def: fixes x :: "'a" shows "acyclic_5c x \<equiv> \<forall>y\<le>x. coreflexive (y\<^sup>+ \<sqinter> (x \<sqinter> - y)\<^sup>\<star>)"
scaleR_diff_left: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
eq: shows "f \<equiv> C (ccpo.fixp (fun_lub Sup) (fun_ord (\<le>)) (\<lambda>f. U (F (C f))))"
quadratic_shape2a: fixes a :: "real"   and x :: "real"   and y :: "real"   and b :: "real"   and c :: "real"   and z :: "real" assumes "a < 0"   and "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"   and "x < z \<and> z < y" shows "0 < a * z\<^sup>2 + b * z + c"
second_summand_overlap: fixes z :: "i"   and y :: "i"   and x :: "i" assumes "O z y" shows "O z (x \<oplus> y)"
fmlookup_restrict_set_id': fixes N :: "('a, 'b) fmap"   and A :: "'a set" assumes "set_mset (dom_m N) \<subseteq> A" shows "fmrestrict_set A N = N"
length_weave: fixes A :: "nat set"   and xs :: "'m list"   and ys :: "'m list" shows "length (weave A xs ys) = length xs + length ys"
f: shows "(\<lambda>x. a x \<cdot>\<^sub>v x) \<in> X \<rightarrow> carrier_vec n"
args_mk: fixes F :: "('c \<times> nat) set"   and mk :: "'c \<Rightarrow> 'd list \<Rightarrow> 'd"   and root :: "'d \<Rightarrow> 'c \<times> nat"   and args :: "'d \<Rightarrow> 'd list"   and trees :: "'d set"   and f :: "'c"   and ts :: "'d list" assumes "kruskal_tree F mk root args trees"   and "(f, length ts) \<in> F" shows "args (mk f ts) = ts"
a: shows "r1 = (p # ps, c) \<and> r1 \<in> upRules"
sel_2: fixes x1 :: "real"   and x2 :: "real" shows "Im (Complex x1 x2) = x2"
h_simps: fixes x :: "'v"   and y :: "'v" shows "h (x, y) = (if reachable x \<and> reachable y then g (x, y) else 0)"
simps: fixes a :: "'l"   and r :: "('l \<times> 'm) set" shows "(a \<in> Domain r) = (\<exists>aa b. a = aa \<and> (aa, b) \<in> r)"
set_fst_sinter_with_subset2: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list" shows "set (map fst (sinter_with f xs ys)) \<subseteq> set (map fst ys)"
WCFG_WhileFalseSkip: fixes b :: "expr"   and c' :: "cmd" shows "while (b) c' \<turnstile> (_ 1 _) -\<Up>id\<rightarrow> (_Exit_)"
rel_fun_def: fixes A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "A ===> B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
assms_3: shows "g piecewise_differentiable_on {a..b}"
pow_zero: fixes n :: "nat" assumes "0 < n" shows "\<zero> [^] n = \<zero>"
matrix_vector_mult_basis: fixes A :: "((real, 'b) vec, 'a) vec"   and k :: "'b" shows "A *v axis k 1 = column k A"
simps_3: fixes m :: "nat"   and n :: "nat" shows "encode (recf.Id m n) = triple_encode 2 m n"
d\<mu>: fixes j :: "nat"   and ii :: "nat" assumes "j < length fs_init"   and "ii < length fs_init" shows "rat_of_int (fs_init.d\<mu> ii j) = rat_of_int (fs_init.d fs_init (Suc j)) * gs.\<mu> ii j"
PSuf_def: fixes s :: "'b list" shows "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
add_top: fixes x :: "'c" assumes "(0::'c) \<le> x" shows "x + \<top> = \<top>"
pause_writer_def: fixes pause :: "'d \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c"   and out :: "'d"   and c :: "'e \<Rightarrow> ('a, 'b, 'c) writerT" shows "pause_writer pause out c = WriterT (pause out (\<lambda>input. run_writer (c input)))"
dgrad_p_set_le_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and G :: "('t \<Rightarrow>\<^sub>0 'e) set"   and m :: "nat" assumes "dgrad_p_set_le d F G"   and "G \<subseteq> dgrad_p_set d m" shows "F \<subseteq> dgrad_p_set d m"
xt1_8: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z < y" shows "z < x"
crit_pair_same: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "fst (crit_pair p p) = snd (crit_pair p p)"
\<G>_subformulas\<^sub>\<nu>: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G> \<phi> w \<subseteq> subformulas\<^sub>\<nu> \<phi>"
assms_2: shows "ring_hom_ring (K [X]) R h"
vsubset_vempty: fixes A :: "V" shows "0 \<subseteq>\<^sub>\<circ> A"
wf_prog_def: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" shows "wf_prog wf_md P = (wf_syscls P \<and> distinct_fst (classes P) \<and> (\<forall>c\<in>set (classes P). wf_cdecl wf_md P c))"
assms: shows "worklist_tree_aux succs f (ws, s) = Some (ws', s')"
lift_nil: fixes f :: "'e \<Rightarrow> 'a" shows "G.lift f [] = \<one>\<^bsub>G\<^esub>"
hyps: assumes "finals es" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
simps_2: shows "X3d X3.Xd1' = D1"
iterative_HSem: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "x \<notin> domA \<Gamma>" shows "\<N>\<lbrace>(x, e) # \<Gamma>\<rbrace>\<rho> = (\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<N>\<lbrace>\<Gamma>\<rbrace>xa))(x := \<N>\<lbrakk> e \<rbrakk>\<^bsub>xa\<^esub>))"
lmeasurable_ball: fixes a :: "'a"   and r :: "real" shows "ball a r \<in> lmeasurable"
normalize_src_ports_def: fixes m :: "'b common_primitive match_expr" shows "normalize_src_ports m = normalize_ports_generic normalize_positive_src_ports rewrite_negated_src_ports m"
strand_fv_card_rm_fun_le: fixes S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list"   and f :: "'b"   and X :: "('b, 'a) Term.term list" shows "card (fv\<^sub>s\<^sub>t (S @ S')) \<le> card (fv\<^sub>s\<^sub>t (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S'))"
wdD_2: assumes "wd TYPE('c)" shows "max_Var_floatariths ode_e \<le> DIM('c)"
restrictionConservation: fixes R :: "('c \<times> 'd) set"   and x :: "'c"   and Y :: "'d set" shows "Outside' {x} (R +< {x} \<times> Y) = Outside' {x} R"
SMP_D_aux3: fixes t' :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "t' \<sqsubseteq> t"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "is_Fun t'"   and "(\<exists>f. t = Fun f []) \<and> (\<exists>m\<in>set M. \<exists>\<delta>. 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>) \<and> t = m \<cdot> \<delta>) \<or> (\<exists>m\<in>set M. \<exists>\<delta>. 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>) \<and> t = m \<cdot> \<delta> \<and> is_Fun m)"   and "finite_SMP_representation arity Ana \<Gamma> M" shows "(\<exists>f. t' = Fun f []) \<and> (\<exists>m\<in>set M. \<exists>\<delta>. 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>) \<and> t' = m \<cdot> \<delta>) \<or> (\<exists>m\<in>set M. \<exists>\<delta>. 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>) \<and> t' = m \<cdot> \<delta> \<and> is_Fun m)"
one_dim_iso_inj: fixes x :: "'b"   and y :: "'b" assumes "one_dim_iso x = one_dim_iso y" shows "x = y"
Times_Int_Times: fixes A :: "'a set"   and B :: "'b set"   and C :: "'a set"   and D :: "'b set" shows "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
assms_2: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
red_rtranclp_diff_in_pmdl: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "(red F)\<^sup>*\<^sup>* p q" shows "p - q \<in> pmdl F"
left_neutral: fixes a :: "'d \<Rightarrow> ('e, 'd) Term.term" shows "Var \<circ>\<^sub>s a = a"
success_returnI: fixes x :: "'a"   and h :: "heap" shows "success (return x) h"
bij_betw_finite: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "finite A = finite B"
restrict_extensional: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set" shows "restrict f A \<in> extensional A"
simps_24: fixes x5 :: "char list" shows "Call x5 \<noteq> Drop"
Dup_omit_branch_mem_same_opening: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i branch xs"   and "p at i in' set branch" shows "p at i in' set (omit_branch xs branch)"
arity_hd_ne_infinity_if_\<delta>_gt_0: fixes \<zeta> :: "('s, 'v) hd" assumes "0 < \<delta>" shows "arity_hd \<zeta> \<noteq> \<infinity>"
uv2: fixes X :: "'b program set" assumes "\<forall>GG. finite GG \<and> GG \<subseteq> X \<and> OK GG (\<lambda>G. G) \<longrightarrow> (\<Squnion>G\<in>GG. G) \<in> X" shows "uv_prop X"
assms: fixes d :: "'a" shows "monotone (fun_ord ordA) ordB (\<lambda>x. F x d)"
erfc_real_gt_0_iff: fixes x :: "real" shows "(1 < erfc x) = (x < 0)"
zmset_elem_nonneg: fixes x :: "'a"   and M :: "'a zmultiset" assumes "x \<in>#\<^sub>z M"   and "\<And>x. x \<in>#\<^sub>z M \<Longrightarrow> 0 \<le> zcount M x" shows "0 < zcount M x"
bounded_linear_divide: fixes y :: "'a" shows "bounded_linear (\<lambda>x. x / y)"
0: shows "map (Ik.int \<xi>) Tl = map (GE.int \<xi> \<circ> gT) Tl"
VectorSpaceHomSet_is_fmaps_in_GroupHomSet: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and W :: "'c set" assumes "VectorSpace smult V" shows "VectorSpaceHomSet smult V smult' W = GroupHomSet V W \<inter> {T. \<forall>a. \<forall>v\<in>V. T (smult a v) = smult' a (T v)}"
norm_some_CBasis: shows "norm (SOME i. i \<in> CBasis) = 1"
intro: fixes R :: "('e, 'g) ring_scheme"   and M :: "('e, 'f, 'h) module_scheme" assumes "\<And>a x. \<lbrakk>a \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"   and "\<And>a b x. \<lbrakk>a \<in> carrier R; b \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> (a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x"   and "\<And>a x y. \<lbrakk>a \<in> carrier R; x \<in> carrier M; y \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y"   and "\<And>a b x. \<lbrakk>a \<in> carrier R; b \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"   and "\<And>x. x \<in> carrier M \<Longrightarrow> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = x" shows "module_axioms R M"
assms_1: shows "dim_row A\<^sup>T = dim_vec c"
sem_5a: shows "\<forall>X. \<not> ob X \<bottom>"
eq: shows "(\<Inter>a\<in>\<V> F. ideal (range (\<lambda>x. monomial (1::'a) (monomial 1 x) - monomial (a x) 0))) = ideal F"
game_semi_forgetful_def: shows "game_semi_forgetful = seed_gen \<bind> (\<lambda>seed. exec_gpv (semi_forgetful_RO seed) \<A> (Map.empty, False) \<bind> (\<lambda>(b, rep). return_spmf (b, rep)))"
history_order_def: fixes x :: "'c"   and i :: "nat"   and z :: "'c" shows "trivial_node_histories.history_order x i z \<equiv> \<exists>xs ys zs. xs @ x # ys @ z # zs = []"