hcju/isabelleps huggingface dataset:
homeomorphism_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" shows "homeomorphism S is_open S' is_open' f \<equiv> continuous_map S is_open S' is_open' f \<and> bijective_map f S S' \<and> continuous_map S' is_open' S is_open (inverse_map f S S')"
unlift_subst: fixes j :: "nat"   and i :: "nat"   and n :: "nat"   and t :: "'a itrm"   and s :: "dB" assumes "j \<le> i \<and> j \<le> n" shows "unlift' (Suc n) t (Suc i)[s/j] = unlift' n t i"
trans_nchotomy: fixes tval :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and aval :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "('a, 'b) com"   and s :: "'c" shows "(\<exists>c' s'. PL.transC tval aval (c, s) (c', s')) \<or> (\<exists>s'. PL.transT tval aval (c, s) s')"
state_nondeterminism_def: fixes og :: "nat"   and nt :: "(nat \<times> transition \<times> nat list) fset" shows "state_nondeterminism og nt = (if size nt < 2 then {||} else ffUnion ((\<lambda>x. let (dest, t) = x in (\<lambda>y. let (dest', t') = y in (og, (dest, dest'), t, t')) |`| (nt |-| {|x|})) |`| nt))"
Cons_1: assumes "distinct (map fst xs)" shows "lookup_dflt (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_dflt xs k))"
simps: fixes pat :: "[:'r:]" shows "matches2\<cdot>pat = smap\<cdot>cfst oo sfilter\<cdot>(snull oo csnd oo abs2 oo csnd) oo sscanl\<cdot>(\<Lambda> (n, x) y. (n + 1, op2\<cdot>pat\<cdot>x\<cdot>y))\<cdot>(0, root2\<cdot>pat)"
b_assn_triv: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "b_assn A (\<lambda>_. True) = A"
left_add_zero_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "0\<^sub>m nr nc + A = A"
closest_pair_rec_ys_eq: fixes xs :: "(int \<times> int) list"   and ys :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ys' :: "(int \<times> int) list"   and \<delta>' :: "int"   and c\<^sub>0' :: "int \<times> int"   and c\<^sub>1' :: "int \<times> int" assumes "1 < length xs"   and "(ys, c\<^sub>0, c\<^sub>1) = closest_pair_rec xs"   and "(ys', \<delta>', c\<^sub>0', c\<^sub>1') = closest_pair_rec_code xs" shows "ys = ys'"
next_trans: fixes c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and s :: "'b"   and s' :: "'b"   and c' :: "('a, 'b, 'c) configuration" shows "c \<turnstile> Trans p s s' \<mapsto> c' = (can_occur (Trans p s s') c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> (\<forall>i. msgs c' i = msgs c i) \<and> no_snapshot_change c c')"
L4_8: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set"   and \<sigma> :: "nat \<Rightarrow> 'b node"   and \<xi> :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltlr" assumes "create_gba_from_nodes_precond \<phi> qs"   and "gb_graph.is_acc_run (create_gba_from_nodes \<phi> qs) \<sigma>"   and "\<forall>i. gba_L (create_gba_from_nodes \<phi> qs) (\<sigma> i) (\<xi> i)"   and "\<psi> \<in> old (\<sigma> 0)" shows "\<xi> \<Turnstile>\<^sub>r \<psi>"
dim_def: fixes V :: "'a set" shows "local.dim V = (if \<exists>b\<subseteq>S. independent b \<and> ??.Real_Vector_Spaces.span b = ??.Real_Vector_Spaces.span V then card (SOME b. b \<subseteq> S \<and> independent b \<and> ??.Real_Vector_Spaces.span b = ??.Real_Vector_Spaces.span V) else 0)"
FAssReds2: fixes e :: "expr"   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 e' :: "expr"   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"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Val v\<bullet>F{Cs} := e,s\<rangle> \<rightarrow>* \<langle>Val v\<bullet>F{Cs} := e',s'\<rangle>"
ok_extend_iff: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and G :: "'c program" assumes "Extend h" shows "extend h F ok extend h G = F ok G"
pbij_lens_axioms_def: fixes x :: "'c \<Longrightarrow> 'd" shows "pbij_lens_axioms x \<equiv> \<forall>\<sigma> v \<rho>. put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v"
components_maximal: fixes c :: "'a set"   and s :: "'a set"   and t :: "'a set" assumes "c \<in> components s"   and "connected t"   and "t \<subseteq> s"   and "c \<inter> t \<noteq> {}" shows "t \<subseteq> c"
memb_equiv_2: fixes f :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "\<F>C.arr f"   and "t \<in> f"   and "u \<in> f" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"
type_less_x_Enum: fixes \<tau> :: "'a basic_type"   and \<E> :: "('a, String.literal) phantom" assumes "\<tau> < Enum \<E>" obtains "\<tau> = OclVoid"
pentagon: fixes f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "ide f"   and "ide g"   and "ide h"   and "ide k"   and "src f = trg g"   and "src g = trg h"   and "src h = trg k" shows "(f \<star> \<a>[g, h, k]) \<cdot> \<a>[f, g \<star> h, k] \<cdot> (\<a>[f, g, h] \<star> k) = \<a>[f, g, h \<star> k] \<cdot> \<a>[f \<star> g, h, k]"
map\<^sub>0_in_hhom: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<rho>\<^sub>0 a : src\<^sub>D (G a) \<rightarrow>\<^sub>D src\<^sub>D (H a)\<guillemotright>"
one_minus_z2_notin_nonpos_Reals: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "1 - z\<^sup>2 \<notin> \<real>\<^sub>\<le>\<^sub>0"
Im_express_cnj: fixes z :: "complex" shows "cor (Im z) = (z - cnj z) / (2 * \<i>)"
symp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "symp r = (\<forall>x y. r x y \<longrightarrow> r y x)"
abc_bcd_acd: shows "\<forall>a b c d. ord a b c \<and> ord b c d \<longrightarrow> ord a c d"
degree_v: shows "degree v < (\<Sum>i | i < n. degree (m i))"
mult_le_cancel_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times c a) (times c b) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
simps_131: fixes x9 :: "Term.var"   and x5 :: "val" shows "Acc x9 \<noteq> Lit x5"
poly_primes: fixes a :: "'a"   and p :: "'a list"   and q :: "'a list" shows "[a, 1::'a] divides (p *** q) = ([a, 1::'a] divides p \<or> [a, 1::'a] divides q)"
m: shows "r_parallel = Cn 3 r_both [m, recf.Id 3 0, recf.Id 3 1, recf.Id 3 2]"
disc_iff: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "\<not> lnull (iterates f x)"
vector_7_2: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x6 :: "'b"   and x7 :: "'b" shows "vector [x1, x2, x3, x4, x5, x6, x7] $ 2 = x2"
cnvalid_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and n :: "nat"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A \<equiv> (\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P Call p Q,A) \<longrightarrow> \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
eval_tm_graph_aux_def: shows "eval_tm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>eval_tm_aux e. x1 = (e, Zero) \<and> x2 = 0) \<or> (\<exists>eval_tm_aux e k. x1 = (e, Var k) \<and> x2 = finfun_apply e k) \<or> (\<exists>eval_tm_aux e t u. x1 = (e, Eats t u) \<and> x2 = ((\<lambda>uu_. uu_) \<circ> eval_tm_aux) (e, t) \<triangleleft> ((\<lambda>uu_. uu_) \<circ> eval_tm_aux) (e, u) \<and> p (e, u) (((\<lambda>uu_. uu_) \<circ> eval_tm_aux) (e, u)) \<and> p (e, t) (((\<lambda>uu_. uu_) \<circ> eval_tm_aux) (e, t))))"
termi_f: shows "terminate f (map (\<lambda>g. rec_exec g xs) gs)"
3: shows "contour_integral (reversepath (linepath b' a')) f = - contour_integral (linepath b' a') f"
pcr_fset_def: fixes T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "pcr_fset T \<equiv> rel_set T OO cr_fset"
Gromov_extension_isometry_iterates: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "isometry f" shows "Gromov_extension (f ^^ n) = Gromov_extension f ^^ n"
assms_2: shows "lambda C -quasi_isometry_on {a..b} f"
supp_5: fixes bool :: "bool" shows "supp (Bool bool) = supp bool"
assms_6: shows "Eb \<subseteq> Pow (space B)"
ilt_ex_notempty: fixes x :: "'b list"   and y :: "'b list" shows "(x \<prec> y) = (\<exists>xs. xs \<noteq> [] \<and> y \<in> xs \<otimes> x)"
hom_one: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
IH: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ts' :: "'a \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "thread_oks ts tas = thread_oks ts' tas"
length_power_polys: fixes mult_p :: "'a"   and u :: "'a"   and curr_p :: "'a"   and n :: "nat" shows "length (power_polys mult_p u curr_p n) = n"
map_smult_closed: fixes r :: "('f, 'g) aezfun"   and ms :: "'v list" assumes "r \<in> FG"   and "set ms \<subseteq> V" shows "set (map ((\<cdot>) r) ms) \<subseteq> V"
preserves_hom: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>E.map f : E.map a \<rightarrow>\<^sub>D E.map b\<guillemotright>"
rExt: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> X" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"
if_eq_is_pr2: fixes p1 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and p2 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and q1 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and q2 :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "p1 \<in> PrimRec2"   and "p2 \<in> PrimRec2"   and "q1 \<in> PrimRec2"   and "q2 \<in> PrimRec2" shows "(\<lambda>x y. if p1 x y = p2 x y then q1 x y else q2 x y) \<in> PrimRec2"
intros_3: fixes N :: "nat" shows "NONCE N \<sim> NONCE N"
eq\<rho>_mono: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" assumes "ae \<sqsubseteq> ae'"   and "eq\<rho> ae' \<rho>1 \<rho>2" shows "eq\<rho> ae \<rho>1 \<rho>2"
fds_truncate_def: fixes m :: "nat"   and f :: "'b fds" shows "fds_truncate m f = fds (\<lambda>n. if n \<le> m then fds_nth f n else (0::'b))"
is_zero_def: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "is_zero p = (p = 0)"
parComm: fixes P :: "ccs"   and Q :: "ccs" shows "P \<parallel> Q \<sim> Q \<parallel> P"
set_tag_name_get_child_nodes_is_l_set_tag_name_get_child_nodes: shows "l_set_tag_name_get_child_nodes ShadowRootClass.type_wf set_tag_name set_tag_name_locs ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs"
IH_2: assumes "q \<in> ctermsl p2"   and "r \<in> ctermsl q" shows "r \<in> ctermsl p2"
cnj_rw: shows "(0::'a) = cnj_class.cnj a"
simps_2: fixes n :: "nat"   and k :: "nat"   and \<Gamma> :: "binding list" shows "\<down>\<^sub>e (Suc n) k \<Gamma> = \<down>\<^sub>e n k (\<Gamma>[k \<mapsto>\<^sub>\<tau> Top]\<^sub>e)"
intros_7: fixes r :: "'b rexp" shows "[] \<in> Star r \<rightarrow> Stars []"
simps_2: shows "b\<^sub>c \<noteq> a\<^sub>c"
MWLsSteps_det_cases_2: fixes x :: "'id"   and \<iota> :: "nat"   and e :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (x:=\<^bsub>\<iota>\<^esub> e) m \<alpha> p m'" obtains "\<alpha> = []" and "p = None" and "m' = m(x := E e m)"
M2: shows "mat_of_cblinfun (- M) \<in> carrier_mat nB nA"
rank_invar_not_empty_hd: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" assumes "rank_invar (t # bq)"   and "bq \<noteq> []" shows "rank t < rank (hd bq)"
compl_sup_top: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup (uminus x) x = top"
both_assoc: fixes t :: "'a ttree"   and t' :: "'a ttree"   and t'' :: "'a ttree" shows "t \<otimes>\<otimes> (t' \<otimes>\<otimes> t'') = t \<otimes>\<otimes> t' \<otimes>\<otimes> t''"
Max_wrt_among_subset_1: fixes A :: "'a set" shows "Max_wrt_among le A \<subseteq> carrier"
quotient_ClassE: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "A \<in> additive.Partition"   and "\<And>a. a \<in> R \<Longrightarrow> P (additive.Class a)" shows "P A"
csubspace_INF: fixes A :: "'d set set" assumes "\<And>x. x \<in> A \<Longrightarrow> csubspace x" shows "csubspace (\<Inter> A)"
div_add: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
distinct_9: fixes x11 :: "'a"   and x12 :: "'b"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x11 \<leftarrow> x12 \<noteq> While x61 x62"
has_derivative_transform_eventually: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "(f has_derivative f') (at x within s)"   and "\<forall>\<^sub>F x' in at x within s. f x' = g x'"   and "f x = g x"   and "x \<in> s" shows "(g has_derivative f') (at x within s)"
f1: fixes g :: "'a \<Rightarrow>\<^sub>L 'b"   and r :: "real"   and x :: "'a" assumes "g \<noteq> 0"   and "0 < r" shows "\<xi> g x r \<in> ball x r \<and> real_of_rat 2 / 3 * r * norm g \<le> norm (g *\<^sub>v \<xi> g x r)"
intros_16: fixes A :: "triple set"   and C :: "triple set"   and c :: "triple" assumes "A |\<turnstile> C"   and "c \<in> C" shows "A |\<turnstile> {c}"
B_def: shows "B = cmod (P $$ (j, i - (m - 1))) / 2"
path_entry_loop: fixes p :: "nat list"   and n :: "nat" assumes "path_entry (g_E G) p n"   and "n \<in> set p" shows "\<exists>p'. path_entry (g_E G) p' n \<and> n \<notin> set p'"
relImage_mono: fixes R1 :: "('d \<times> 'd) set"   and R2 :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" assumes "R1 \<subseteq> R2" shows "BNF_Greatest_Fixpoint.relImage R1 f \<subseteq> BNF_Greatest_Fixpoint.relImage R2 f"
no_oif_matchD: fixes fw :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme"   and any :: "char list" assumes "no_oif_match fw" shows "simple_fw fw p = simple_fw fw (p\<lparr>p_oiface := any\<rparr>)"
trss_drop_context_s: fixes s :: "'f list"   and c :: "'f list multiset"   and ce :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c + ce), e, s', c' + ce) \<in> trss fg" shows "((s, c), e, s', c') \<in> trss fg \<and> mon_e fg e \<inter> mon_c fg ce = {}"
del_barrier1_locs_def: shows "mut_m.del_barrier1_locs = prefixed ''store_del_mo'' \<union> {mut_m.lop_store_ins}"
inj_on_index2: fixes I :: "'b set"   and xs :: "'b list" assumes "I \<subseteq> set xs" shows "inj_on (index xs) I"
case_2: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d llist" shows "(case LCons x21 x22 of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
r_cball: shows "cball z r \<subseteq> ball z e"
abs_venv_map_of: fixes l :: "(((nat \<times> char list) \<times> contour) \<times> d) list" shows "|map_of l| \<lessapprox> \<Union>.map (\<lambda>(v, k). |[v \<mapsto> k]|) l"
wtl_Nil: fixes c :: "'a list"   and pc :: "nat"   and s :: "'a" shows "wtl [] c pc s = s"
Le_le_LtD: fixes a :: "'b"   and b :: "'b" assumes "Le a \<le> Lt b" shows "a < b"
dependent_explicit: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b set" assumes "module scale" shows "module.dependent scale s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. scale (u v) v) = (0::'b) \<and> (\<exists>v\<in>t. u v \<noteq> (0::'a)))"
Proc_CFG_Call_nodes_eq': fixes prog :: "cmd"   and n :: "label"   and p :: "char list"   and es :: "expr list"   and rets :: "char list list"   and n' :: "label"   and n'' :: "label"   and p' :: "char list"   and es' :: "expr list"   and rets' :: "char list list" assumes "prog \<turnstile> n -CEdge (p, es, rets)\<rightarrow>\<^sub>p n'"   and "prog \<turnstile> n'' -CEdge (p', es', rets')\<rightarrow>\<^sub>p n'" shows "n = n'' \<and> p = p' \<and> es = es' \<and> rets = rets'"
p1_has_derivative: fixes x :: "real \<times> real" shows "((\<lambda>x. p1 (fst x) (snd x)) has_derivative p1d x) (at x)"
check_def: fixes b :: "bool"   and e :: "'a" shows "check b e = (if b then Inr () else Inl e)"
SSpec_derivable_empty: fixes c :: "IMP" shows "{} \<rhd> c : SSpec c"
focus_Int: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and Y :: "'b set"   and X :: "'b set" assumes "p \<in> P[Y]" shows "focus (X \<inter> Y) p = focus X p"
fs: shows "fs = map (to_int_poly_i ff_ops) fs'"
stable_on_blocking_onD: fixes ds :: "'b set"   and X :: "'a set"   and X'' :: "'a set"   and h :: "'c" assumes "stable_on ds X"   and "X'' = Ch h (X \<union> X'')"   and "X'' \<subseteq> CD_on ds (X \<union> X'')" shows "X'' = Ch h X"
index_update_vec2: fixes i' :: "nat"   and i :: "nat"   and v :: "'b vec"   and a :: "'b" assumes "i' \<noteq> i" shows "(v |\<^sub>v i \<mapsto> a) $ i' = v $ i'"
fold_mset_fusion: fixes g :: "'c \<times> 'd \<Rightarrow> 'e \<Rightarrow> 'e"   and h :: "'e \<Rightarrow> nat"   and w :: "'e"   and A :: "('c \<times> 'd) multiset" assumes "comp_fun_commute g"   and "\<And>x y. h (g x y) = ((+) \<circ> pair_size) x (h y)" shows "h (fold_mset g w A) = fold_mset ((+) \<circ> pair_size) (h w) A"
groupParts_same_fw_wi0: fixes V :: "'b wordinterval list"   and c :: "parts_connection"   and rs :: "'b simple_rule list" assumes "V \<in> set (groupWIs c rs)" shows "\<forall>w\<in>set (map wordinterval_to_set V). \<forall>a1\<in>w. \<forall>a2\<in>w. same_fw_behaviour_one a1 a2 c rs"
ladder_\<alpha>: shows "ladder_\<alpha> (\<delta> @ \<alpha>) D L (Suc index) = \<delta> @ ladder_\<alpha> \<alpha> D' L' (Suc index)"
map_sgas_def: fixes l :: "rat list" shows "map_sgas l = map (\<lambda>r. 1 - r\<^sup>2) l"
ae: shows "alw (ev (\<lambda>js. shd (stl js) \<noteq> shd js)) js"
invar_update: fixes m :: "'m"   and a :: "'a"   and b :: "'b" assumes "invar m" shows "invar (update a b m)"
orthogonal_matrix_id: shows "orthogonal_matrix (mat (1::'b))"
big_refl: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "f \<in> O[F](f)"
lim_ln_over_n: shows "(\<lambda>n. ln (real n) / real n) \<longlonglongrightarrow> 0"
assms_2: shows "\<not> (\<exists>n''>n'. n'' < n \<and> \<parallel>nid\<parallel>\<^bsub>t n'\<^esub>)"
shift1_induct: fixes args :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and P :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set \<Rightarrow> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set \<Rightarrow> bool" assumes "shift1_inv args"   and "\<And>qs B. \<lbrakk>shift1_inv (qs, B); card B \<le> 1\<rbrakk> \<Longrightarrow> P (qs, B) (qs, B)"   and "\<And>qs B h U x. \<lbrakk>shift1_inv (qs, B); 1 < card B; (h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m); x = (SOME y. y \<in> U); finite U; x \<in> U; card (U - {x}) = m; P (shift_list (h, U) x qs, B - {(h, U)}) (shift1 (shift_list (h, U) x qs, B - {(h, U)}))\<rbrakk> \<Longrightarrow> P (qs, B) (shift1 (shift_list (h, U) x qs, B - {(h, U)}))" shows "P args (shift1 args)"
lifting_inv_sconf_subject_ok: assumes "wf_J_prog P" shows "lifting_inv final_expr (\<lambda>t ((e, l), h) ta ((e', l'), h'). extTA2J P,P,t \<turnstile> \<langle>e,(h, l)\<rangle> -ta\<rightarrow> \<langle>e',(h', l')\<rangle>) sconf_type_ok"
c_exists_def: fixes C :: "nat set"   and R :: "'a set" shows "c_exists C R = (\<exists>c. sum w C = sum c (U - R) \<and> (\<forall>i. 0 \<le> c i) \<and> (\<forall>k\<in>{1..m}. sum c (S k \<inter> (U - R)) \<le> (\<Sum>j = card (S k \<inter> R) + 1..card (S k). inverse (real j)) * w k))"
op_bfs_def: fixes c :: "'ga"   and s :: "nat"   and t :: "nat" shows "op_bfs c s t \<equiv> Graph.bfs2 (absG c) (succ c) s t"
equiv_up_to_while_lemma_weak: fixes d :: "com"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com" assumes "(d, s) \<Rightarrow> s'"   and "P \<Turnstile> b <\<sim>> b'"   and "P \<Turnstile> c \<sim> c'"   and "\<And>s s'. \<lbrakk>(c, s) \<Rightarrow> s'; P s; bval b s\<rbrakk> \<Longrightarrow> P s'"   and "P s"   and "d = WHILE b DO c" shows "(WHILE b' DO c', s) \<Rightarrow> s'"
ta_hb_consistent_LCons: fixes P :: "'a prog"   and obs :: "('b \<times> ('c, 'b) obs_event action) list"   and ob :: "'b \<times> ('c, 'b) obs_event action"   and obs' :: "('b \<times> ('c, 'b) obs_event action) llist" shows "ta_hb_consistent P obs (LCons ob obs') = (ta_hb_consistent P (obs @ [ob]) obs' \<and> (case ob of (t, NormalAction (ReadMem ad al v)) \<Rightarrow> \<exists>w. w \<in> write_actions (llist_of (obs @ [ob])) \<and> (ad, al) \<in> action_loc P (llist_of (obs @ [ob])) w \<and> value_written P (llist_of (obs @ [ob])) w (ad, al) = v \<and> P,llist_of (obs @ [ob]) \<turnstile> w \<le>hb length obs \<and> (\<forall>w'\<in>write_actions (llist_of (obs @ [ob])). (ad, al) \<in> action_loc P (llist_of (obs @ [ob])) w' \<longrightarrow> (P,llist_of (obs @ [ob]) \<turnstile> w \<le>hb w' \<and> P,llist_of (obs @ [ob]) \<turnstile> w' \<le>hb length obs \<longrightarrow> w' = w) \<and> (is_volatile P al \<and> P,llist_of (obs @ [ob]) \<turnstile> w \<le>so w' \<and> P,llist_of (obs @ [ob]) \<turnstile> w' \<le>so length obs \<longrightarrow> w' = w)) | (t, NormalAction _) \<Rightarrow> True | (t, _) \<Rightarrow> True))"
leg1_simps_6: shows "cod r\<^sub>0s\<^sub>1.p\<^sub>1 = r\<^sub>0s\<^sub>1.p\<^sub>1"