hcju/isabelleps huggingface dataset:
max_input_Eq: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "GExp.max_input (Eq a1 a2) = max (AExp.max_input a1) (AExp.max_input a2)"
path_Entry_target_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as Entry" shows "as = []"
cipher_queries_def: fixes ose :: "'b \<Rightarrow> ('ekey \<times> 'dkey \<times> 'cipher list) option" shows "cipher_queries ose = (\<Union>(uu_, uu_, ciphers)\<in>ran ose. set ciphers)"
msg_ok_no_leq_no: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg"   and \<sigma>' :: "nat \<Rightarrow> state" assumes "msg_ok \<sigma> m"   and "\<forall>j. no (\<sigma> j) \<le> no (\<sigma>' j)" shows "msg_ok \<sigma>' m"
assms: shows "\<exists>p. gen_iapath (verts3 (with_proj G)) u p v"
of_nat_binomial_eq_mult_binomial_Suc: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "of_nat (n choose k) = of_nat (n + 1 - k) / of_nat (n + 1) * of_nat (Suc n choose k)"
connectedin_sing: fixes X :: "'b topology"   and a :: "'b" shows "connectedin X {a} = (a \<in> topspace X)"
share_append: fixes xs :: "'b memref list"   and ys :: "'b memref list"   and S :: "nat \<Rightarrow> bool option" shows "share (xs @ ys) S = share ys (share xs S)"
bsn: shows "(0::'a) \<le> b ^ Suc n"
that: fixes g :: "nat" assumes "g \<in> totatives n"   and "ord n g = Carmichael n" shows "thesis"
upd_cols_insert_rev: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and I :: "'d set"   and f :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_cols M (insert i I) f = upd_cols (upd_col M i (f i)) I f"
md1: shows "monomial_decomp (fst (split 0 X S))"
F1set1_JF1set: fixes c :: "'d"   and a :: "'d JF1" assumes "c \<in> F1set1 (dtor1 a)" shows "c \<in> JF1set a"
\<sigma>_def: fixes n :: "nat" shows "Presburger_Formula.\<sigma> n = List.n_lists n [True, False]"
irreducible\<^sub>d_m_def: fixes f :: "int poly" shows "q.irreducible\<^sub>d_m f \<equiv> 0 < q.degree_m f \<and> (\<forall>g h. q.degree_m g < q.degree_m f \<longrightarrow> q.degree_m h < q.degree_m f \<longrightarrow> q.Mp f \<noteq> q.Mp (g * h))"
less_length_mono: fixes s :: "'b list"   and t :: "'b list" assumes "s < t" shows "length s < length t"
Nml_implies_Arr: fixes t :: "'c local.term" assumes "Nml t" shows "Arr t"
redT_silent_move_preserves: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<tau>mredT s s'"   and "ts_ok P (thr s) (shr s)" shows "ts_ok P (thr s') (shr s')"
AndR: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and \<Delta> :: "'a formula multiset"   and G :: "'a formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "\<Gamma> \<Rightarrow> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<and> G, \<Delta>"
emeasure_Un_null_set: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> null_sets M" shows "emeasure M (A \<union> B) = emeasure M A"
iso_cancel_right: fixes g :: "'b"   and f :: "'b"   and f' :: "'b" assumes "D.iso g"   and "f \<cdot>\<^sub>D g = f' \<cdot>\<^sub>D g"   and "D.seq f g"   and "D.iso g" shows "f = f'"
normalize_def: fixes x :: "'c" shows "Projections.normalize x = (1 / norm x) *\<^sub>R x"
gd_powerprod_def: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "gd_powerprod ord ord_strict \<equiv> ordered_powerprod ord ord_strict"
monotone_def: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
log_floor_def: fixes b :: "int"   and x :: "int" shows "log_floor b x = fst (log_main (into_base b) x)"
comp_char: fixes g :: "'a"   and f :: "'a" shows "Right.comp g f = (if Right.arr f \<and> Right.arr g \<and> seq g f then g \<cdot> f else null)"
sup_id_star1: fixes x :: "'a" assumes "reflexive x" shows "x\<^sup>+ = x\<^sup>\<star>"
S5n_tc_rels_eq: fixes M :: "('b, 'c, 'd) KripkeStructure"   and w :: "'d"   and w' :: "'d"   and as :: "'b set" assumes "S5n M"   and "(w, w') \<in> (\<Union> (relations M ` as))\<^sup>+" shows "(\<Union> (relations M ` as))\<^sup>+ `` {w} = (\<Union> (relations M ` as))\<^sup>+ `` {w'}"
branchPath: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b" assumes "inherited subs P"   and "fans subs"   and "\<not> P (tree subs gamma)" shows "branch subs gamma (path subs gamma P)"
sel_16: fixes x71 :: "'a expr"   and x72 :: "call_kind"   and x73 :: "'a call_expr" shows "expr.kind (Call x71 x72 x73) = x72"
rep_eq: fixes x :: "'a mpoly"   and xa :: "'a mpoly" shows "mapping_of (x * xa) = mapping_of x * mapping_of xa"
remove_powers_def: fixes xs :: "(char list multiset \<times> int) multiset" shows "remove_powers xs = apfst remdups_mset `# xs"
wm_basic_def: fixes M :: "'a atom formula set" shows "wm_basic M \<equiv> \<forall>a\<in>M. is_predAtom a"
subgroup_finite: fixes H :: "'a set" assumes "subgroup H G"   and "finite (carrier G)" shows "finite H"
homeomorphism_cont1: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "homeomorphism S T f g" shows "continuous_on S f"
aid_def: fixes v :: "Value" shows "aid v = (case v of arrV T a \<Rightarrow> a)"
degle: fixes p :: "'b" assumes "p \<in> S - {q}" shows "degree (f p) < n"
UniqueSucStates_Status: fixes ST :: "('d, 'e, 'f) status" shows "UniqueSucStates (SAs (HA ST)) (CompFun (HA ST)) (Conf ST)"
higherPoly\<^sub>0_add: fixes x :: "nat"   and y :: "nat"   and p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "higherPoly\<^sub>0 x y (p + q) = higherPoly\<^sub>0 x y p + higherPoly\<^sub>0 x y q"
set_child_nodes_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_child_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
Pair_lv_rel_def: shows "Pair_lv_rel = Pair"
cases: fixes x :: "('b \<times> color) tree \<times> 'b \<times> ('b \<times> color) tree" obtains t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" where "x = (R t1 a__ t2__, b__, t3__)"   | a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" where "x = (\<langle>\<rangle>, aa__, B t2a__ ba__ t3a__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" where "x = (B v__ vc__ vb__, ab__, B t2b__ bb__ t3b__)"   | a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (\<langle>\<rangle>, ac__, R (B t2c__ bc__ t3c__) c__ t4__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (B va__ vca__ vba__, ad__, R (B t2 b t3) c t4)"   | a :: "'b" where "x = (\<langle>\<rangle>, ae__, \<langle>\<rangle>)"   | a :: "'b" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (\<langle>\<rangle>, af__, R \<langle>\<rangle> vcb__ vbb__)"   | a :: "'b" and va :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (\<langle>\<rangle>, ag__, R (R va vf__ ve__) vcc__ vbc__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" where "x = (B vd__ vcd__ vbd__, ah__, \<langle>\<rangle>)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (B vg__ vce__ vbe__, ai__, R \<langle>\<rangle> vfa__ vea__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and vd :: "('b \<times> color) tree" and vi :: "'b" and vh :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (B v vc vb, a, R (R vd vi vh) vf ve)"
prefix_imp_subseq: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs ys" shows "subseq xs ys"
Units_inv_inv: fixes x :: "'a" assumes "x \<in> Units R" shows "inv (inv x) = x"
interior_closure: fixes S :: "'a set" shows "interior S = - closure (- S)"
minus_def: fixes x :: "'b bit1" shows "- x = Abs_bit1 (- Rep_bit1 x mod int CARD('b bit1))"
sorted_annotated: fixes l :: "'b list" assumes "length l \<le> unat (- 1)" shows "sorted_descending (map fst (map (apfst word_of_nat) (annotate_rlen l)))"
atLeastLessThan_iff: fixes i :: "'c"   and l :: "'c"   and u :: "'c" shows "(i \<in> {l..<u}) = (l \<le> i \<and> i < u)"
ivl_disj_un_22: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m<..u} = {l..u}"
lm066: fixes R :: "('b \<times> 'a) set"   and X :: "'b set" shows "Range (Outside' X R) = R `` (Domain R - X)"
sum_with_cases: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "\<And>C. \<lbrakk>f ` S \<subseteq> C; comm_monoid_add_ow C pls z\<rbrakk> \<Longrightarrow> P (Finite_Set.fold (pls \<circ> f) z S)"   and "(\<And>C. comm_monoid_add_ow C pls z \<Longrightarrow> \<exists>s\<in>S. f s \<notin> C) \<Longrightarrow> P z" shows "P (sum_with pls z f S)"
simps_4: fixes r1 :: "'a rexp"   and r2 :: "'a rexp" shows "awidth (Plus r1 r2) = awidth r1 + awidth r2"
IH: fixes out :: "'a"   and c :: "('c, 'a, 'b) rpv"   and input :: "'b"   and s :: "'s" assumes "IO out c \<in> set_spmf gpv"   and "out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out"   and "I s" shows "lossless_spmf (exec_gpv callee (c input) s)"
Diagonalize_norm: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>\<^bold>\<parallel>t\<^bold>\<parallel>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
trg_hcomp: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.arr \<nu>"   and "Span.src \<nu> = Span.trg \<mu>" shows "Span.trg (\<nu> \<circ> \<mu>) = Span.trg \<nu>"
bl_of_nth_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "bl_of_nth n f = map f (rev [0..<n])"
gen_keys_simps_2: fixes k :: "'c"   and t :: "('c, 'd) rbt"   and kts :: "('c \<times> ('c, 'd) rbt) list" shows "RBT_Impl.gen_keys ((k, t) # kts) rbt.Empty = k # RBT_Impl.gen_keys kts t"
inv: fixes j :: "nat" shows "rb_aux_inv1 (seq' j)"
Tree\<^sub>\<alpha>_free_4: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and \<phi> :: "'f" shows "Not\<^sub>\<alpha> t\<^sub>\<alpha> \<noteq> Pred\<^sub>\<alpha> \<phi>"
desargues_config_3D_non_coplanar: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "4 \<le> rk {A, B, C, A', B', C', \<alpha>, \<beta>, \<gamma>}"
pointwise_def: fixes f :: "'a set \<Rightarrow> 'b set" shows "pointwise f = (\<forall>X. f X = \<Union> {f {x} |x. x \<in> X})"
edc_snd: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and e :: "expr"   and f :: "cexpr"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c e \<Rightarrow> f"   and "\<Gamma> \<turnstile> e : PRODUCT t t'" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Snd $$ e \<Rightarrow> \<integral>\<^sub>c (map_vars (case_nat 0 (\<lambda>x. x + 2)) f \<circ>\<^sub>c <CVar 0, CVar 1>\<^sub>c) \<partial>t"
sumIdLeft: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<oplus> \<zero> \<leadsto>[Rel] P"
c_redu_subst1: fixes M :: "trm"   and M' :: "trm"   and c :: "coname"   and y :: "name"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>c M'"   and "c \<sharp> M"   and "y \<sharp> P" shows "M{y:=<c>.P} \<longrightarrow>\<^sub>c M'{y:=<c>.P}"
mcont_inf2: fixes y :: "'a" shows "mcont Sup (\<le>) Sup (\<le>) (\<lambda>x. x \<sqinter> y)"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>src\<^sub>D f : src\<^sub>D a \<Rightarrow>\<^sub>D src\<^sub>D b\<guillemotright>"
extend_ensures: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "(extend h F \<in> extend_set h A ensures extend_set h B) = (F \<in> A ensures B)"
sc_bound: fixes P :: "'b \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "c * bound_of P = bound_of (\<lambda>x. c * P x)"
option_rel_Some1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b option" shows "rel_option A (pure x) y = (\<exists>y'. y = pure y' \<and> A x y')"
list_of_eucl_of_env: fixes xs :: "nat list"   and vs :: "real list" assumes "length xs = DIM('e)" shows "list_of_eucl (eucl_of_env xs vs) = map (\<lambda>i. vs ! (xs ! i)) [0..<DIM('e)]"
mod_add_cong: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and c :: "'d"   and a' :: "'d"   and b :: "'d"   and b' :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (plus a b) c = modulo (plus a' b') c"
simps_4: fixes a :: "'a"   and b :: "'b" shows "lderiv a (Atom b) = (if lookup b a then One else Zero)"
assms_2: fixes x :: "real" shows "(v has_real_derivative k) (at x)"
restrict_pp: fixes ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and ls :: "'c list" shows "ks \<langle>\<ominus> \<ominus> f\<rangle> ls = \<ominus> \<ominus> ks \<langle>f\<rangle> ls"
swapped_by_mtf2: fixes q :: "'a"   and xs :: "'a list"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "x < y in xs"   and "y < x in mtf2 n q xs" shows "y = q"
base_in_hom_2: shows "\<guillemotleft>r' : r' \<Rightarrow> r'\<guillemotright>"
fls_prpart_regpart: fixes f :: "'a fls" shows "fls_prpart (fls_regpart_as_fls f) = 0"
simps_5: shows "ext0.Top \<noteq> ext0.Bot"
span_subspace_induct: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and P :: "'b set" assumes "module scale"   and "x \<in> module.span scale S"   and "module.subspace scale P"   and "\<And>x. x \<in> S \<Longrightarrow> x \<in> P" shows "x \<in> P"
comm_neutral: fixes a :: "'f" shows "a + (0::'f) = a"
nat_pow_mult: fixes x :: "nat \<Rightarrow> int"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier Zp" shows "x [^] n \<otimes> x [^] m = x [^] (n + m)"
in_front: shows "s\<^sub>e < s\<^sub>o"
simps_1: fixes \<A> :: "'a ltl set" shows "\<A> \<Turnstile>\<^sub>P true = True"
resolution_decreases_selected_part: fixes atom_ordering :: "('a \<times> 'a) set"   and Sel :: "'a set"   and P1 :: "'a Literal set"   and P2 :: "'a Literal set"   and C :: "'a Literal set"   and A :: "'a"   and n :: "nat" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.ordered_sel_resolvent_upon atom_ordering Sel P1 P2 C A"   and "Neg A \<in> P2"   and "finite P1"   and "finite P2"   and "card (selected_part Sel P2) = Suc n" shows "card (selected_part Sel C) = n"
properfactor_divides: fixes a :: "'a"   and b :: "'a" assumes "properfactor (Ring_Divisibility.mult_of R) a b" shows "a divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> b"
word_unat_and_lt: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "unat x < n \<or> unat y < n" shows "unat (x AND y) < n"
Seq_decomp_Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p\<^sub>2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and p' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Seq Skip p\<^sub>2, s) \<rightarrow> (p', s')" shows "s' = s \<and> p' = p\<^sub>2"
\<Z>_sgrp_axioms_def: fixes \<SS> :: "V" shows "\<Z>_sgrp_axioms \<SS> \<equiv> \<forall>a b c. a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c)"
chamber_map: fixes C :: "'a set" assumes "domain.chamber C" shows "codomain.chamber (f ` C)"
simps_3: fixes l1 :: "value"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (L l1) (Plus v va) = True"
psi_conv_pre_mangoldt: fixes n :: "nat" shows "psi n = ln (real (prod pre_mangoldt {1..n}))"
ws_update_def: fixes E :: "('d \<times> 'c) set"   and u :: "'d"   and p :: "'d list"   and V :: "'c set"   and ws :: "'c \<Rightarrow> 'd list option" shows "ws_update E u p V ws \<equiv> RETURN (V \<union> E `` {u}, ws ++ (\<lambda>v. if (u, v) \<in> E \<and> v \<notin> V then Some (u # p) else None))"
remove_all_rev_distinct: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs" shows "distinct (x # remove_all_rev x xs)"
hyp: shows "P (\<Sum>i\<le>b. monom (g i) i) n"
mem_Sigma_iff: fixes a :: "'e"   and b :: "'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
simulation_reaches: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "A.reaches a a'"   and "equiv' a b" shows "\<exists>b'. B.reaches b b' \<and> equiv' a' b'"
new_AddrD2: fixes h :: "loc \<Rightarrow> obj option"   and a :: "loc" assumes "new_Addr h = Some a" shows "\<forall>b. h b \<noteq> None \<longrightarrow> b \<noteq> a"
vangle_scales: fixes c :: "real"   and v\<^sub>1 :: "'b"   and v\<^sub>2 :: "'b" assumes "0 < c" shows "vangle (c *\<^sub>R v\<^sub>1) v\<^sub>2 = vangle v\<^sub>1 v\<^sub>2"
elims: fixes x :: "int \<Rightarrow> int \<Rightarrow> int"   and xa :: "value option"   and xb :: "value option"   and y :: "value option" assumes "maybe_arith_int x xa xb = y" obtains f :: "int \<Rightarrow> int \<Rightarrow> int" and xc :: "int" and ya :: "int" where "x = f" and "xa = Some (Num xc)" and "xb = Some (Num ya)" and "y = Some (Num (f xc ya))"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and uw_ :: "value option" where "x = uu___" and "xa = None" and "xb = uw___" and "y = None"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and va :: "String.literal" and uw_ :: "value option" where "x = uua___" and "xa = Some (Str va__)" and "xb = uw_" and "y = None"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and uv_ :: "value option" where "x = uub___" and "xa = uv___" and "xb = None" and "y = None"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and uv_ :: "value option" and va :: "String.literal" where "x = uu_" and "xa = uv_" and "xb = Some (Str va)" and "y = None"
case_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and f4 :: "('c, 'd) Combinators \<Rightarrow> ('c, 'd) Combinators \<Rightarrow> 'b"   and x41 :: "('c, 'd) Combinators"   and x42 :: "('c, 'd) Combinators" shows "(case x41 \<oplus> x42 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f4 x41 x42"
good_item_simps_59: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xsa :: "'varSort"   and xa :: "'var" assumes "goodEnv rhoa" shows "rhoa &[xa \<and> xa]_xsa = rhoa"
Derives1_implies_derives1: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" assumes "Derives1 u i r v" shows "derives1 u v"
induct: fixes P :: "('n \<Rightarrow> 'o fundef option) \<Rightarrow> ('p, 'q, 'n, 'e, 'f, 'g, 'r, 's) instr \<Rightarrow> Unboxed.type option list \<Rightarrow> bool"   and a0 :: "'n \<Rightarrow> 'o fundef option"   and a1 :: "('p, 'q, 'n, 'e, 'f, 'g, 'r, 's) instr"   and a2 :: "Unboxed.type option list" assumes "\<And>uu_ uv_ \<Sigma>. P uu_ (IPush uv_) \<Sigma>"   and "\<And>uw_ ux_ \<Sigma>. P uw_ (IPushUbx1 ux_) \<Sigma>"   and "\<And>uy_ uz_ \<Sigma>. P uy_ (IPushUbx2 uz_) \<Sigma>"   and "\<And>va_ vb_ \<Sigma>. P va_ IPop (vb_ # \<Sigma>)"   and "\<And>vc_ vd_ \<Sigma>. P vc_ (ILoad vd_) (None # \<Sigma>)"   and "\<And>ve_ \<tau> vf_ \<Sigma>. P ve_ (ILoadUbx \<tau> vf_) (None # \<Sigma>)"   and "\<And>vg_ vh_ \<Sigma>. P vg_ (IStore vh_) (None # None # \<Sigma>)"   and "\<And>vi_ \<tau>\<^sub>1 vj_ \<tau>\<^sub>2 \<Sigma>. P vi_ (IStoreUbx \<tau>\<^sub>1 vj_) (None # Some \<tau>\<^sub>2 # \<Sigma>)"   and "\<And>vk_ op \<Sigma>. P vk_ (IOp op) \<Sigma>"   and "\<And>vl_ opinl \<Sigma>. P vl_ (IOpInl opinl) \<Sigma>"   and "\<And>vm_ opubx \<Sigma>. P vm_ (IOpUbx opubx) \<Sigma>"   and "\<And>F f \<Sigma>. P F (ICall f) \<Sigma>"   and "\<And>vn_. P vn_ IPop []"   and "\<And>vn_ v. P vn_ (ILoad v) []"   and "\<And>vn_ v vc vb. P vn_ (ILoad v) (Some vc # vb)"   and "\<And>vn_ v va. P vn_ (ILoadUbx v va) []"   and "\<And>vn_ v va vd vc. P vn_ (ILoadUbx v va) (Some vd # vc)"   and "\<And>vn_ v. P vn_ (IStore v) []"   and "\<And>vn_ v vc vb. P vn_ (IStore v) (Some vc # vb)"   and "\<And>vn_ v va. P vn_ (IStore v) [va]"   and "\<And>vn_ v va ve vd. P vn_ (IStore v) (va # Some ve # vd)"   and "\<And>vn_ v va. P vn_ (IStoreUbx v va) []"   and "\<And>vn_ v va vd vc. P vn_ (IStoreUbx v va) (Some vd # vc)"   and "\<And>vn_ v va vb. P vn_ (IStoreUbx v va) [vb]"   and "\<And>vn_ v va vb ve. P vn_ (IStoreUbx v va) (vb # None # ve)"   and "\<And>vn_ v vp_. P vn_ (ICJump v) vp_" shows "P a0 a1 a2"
cSUP_subset_mono: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> V set"   and g :: "'a \<Rightarrow> V set" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x" shows "\<Union> (f ` A) \<subseteq> \<Union> (g ` B)"
cons_exec_def: fixes e :: "'c \<times> 'd list"   and p :: "'d" shows "cons_exec e p \<equiv> (fst e, Cons p (snd e))"
that: fixes p :: "'a list"   and y :: "'a" assumes "path \<Gamma> x p y"   and "y \<in> B \<Gamma>"   and "distinct (x # p)"   and "\<And>z. z \<in> set p \<Longrightarrow> z \<notin> RF S" shows "thesis"
fd_subdist_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x\<rangle> (d y \<cdot> d z) \<le>  |x\<rangle> y"