hcju/isabelleps huggingface dataset:
m1_inv2i_serv_lemma: fixes t :: "'a m1r_state_scheme"   and Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Nb :: "fresh_t"   and Ra' :: "fid_t"   and P :: "bool" assumes "runs t Ra = Some (Init, [A, B], [aKey Kab, aNon Nb])"   and "runs t Ra' = Some (Init, [A, B], [aKey Kab])"   and "A \<notin> bad"   and "t \<in> m1_inv2i_serv" shows "P"
bind_lzero: fixes f :: "'b \<Rightarrow> 'a option" shows "None \<bind> f = None"
ivl_disj_un_two_touch_4: fixes l :: "'c"   and m :: "'c"   and u :: "'c" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m..u} = {l..u}"
Start_nsys_xcpts1_6: shows "Start \<noteq> NoSuchFieldError"
paths_hits_winning_strategy_is_winning: fixes v :: "'a" assumes "winning_strategy p \<sigma> v"   and "v \<in> lset P" shows "winning_path p P"
DynCom: fixes P :: "'e set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('e, 'f, 'g) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>s\<in>P. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c s Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P DynCom c Q,A"
space_PiM_empty: fixes M :: "'d \<Rightarrow> 'e measure" shows "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
fields_functional: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and Cf :: "varDef list"   and Cf' :: "varDef list" assumes "fields(CT,C) = Cf"   and "CT OK"   and "fields(CT,C) = Cf'" shows "Cf = Cf'"
assms: shows "\<forall>i\<in>fv\<^sub>s\<^sub>e\<^sub>t K. i < length T"
chamberD_simplex: fixes x :: "'a set" assumes "chamber x" shows "x \<in> X"
hyps_1: fixes i :: "nat"   and h :: "heap" assumes "k = j - i"   and "i \<le> j"   and "effect (for [i..<j] f) h h' u"   and "I i h"   and "\<And>k h h' r. \<lbrakk>i \<le> k; k < j; I k h; effect (f k) h h' r\<rbrakk> \<Longrightarrow> I (Suc k) h'" shows "I j h'"
n2: shows "nprv (insert (subst \<phi> (Var y) x) (insert \<phi> F)) (eql (Var y) (Var x))"
approx_eq: fixes t :: "char list \<Rightarrow> int option"   and c :: "com" shows "approx t \<Turnstile> c \<sim> Fold.fold c t"
assms_2: shows "r\<lparr>a\<rparr> = b"
homotopic_circlemaps_divide: fixes S :: "'a set"   and f :: "'a \<Rightarrow> complex"   and g :: "'a \<Rightarrow> complex" shows "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) f g = (continuous_on S f \<and> f ` S \<subseteq> sphere 0 1 \<and> continuous_on S g \<and> g ` S \<subseteq> sphere 0 1 \<and> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. f x / g x) (\<lambda>x. c)))"
trlt1: fixes n :: "nat" shows "trace (c \<cdot>\<^sub>m (f n - f 0)) < 1"
D_borel: shows "(\<lambda>x. if x \<in> S then \<bar>det (matrix (g' x))\<bar> * f (g x) else 0) \<in> borel_measurable (lebesgue_on UNIV)"
scaleR: fixes f :: "'c \<Rightarrow> 'd"   and r :: "real"   and b :: "'c" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
AssumeH_1: fixes A :: "fm"   and H :: "fm set" shows "insert A H \<turnstile> A"
always_def: fixes p :: "'a" shows "\<box> p = Inf (range ((!!) p))"
assms: shows "list_all (\<lambda>x. x = None) (map typeof xs)"
typeI_cube_explicit_spec: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" assumes "typeI_twoCube twoC" shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and> cubeImage twoC = {(x, y). x \<in> {a..b} \<and> y \<in> {g2 x..g1 x}} \<and> twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b))) \<and> g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b} \<and> (\<lambda>x. twoC (x, 0)) = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x))) \<and> (\<lambda>y. twoC (1, y)) = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b))) \<and> (\<lambda>x. twoC (x, 1)) = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x))) \<and> (\<lambda>y. twoC (0, y)) = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
neg_closed: fixes g :: "'v" assumes "g \<in> V" shows "- g \<in> V"
comlist_mk_ps_chain: fixes \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list" shows "comlist (mk_ps_chain \<Pi> \<pi>) = comlist \<Pi> @ \<pi>"
rep_eq: shows "blinfun_apply embed2_blinfun = Pair (0::'c)"
assms_2: shows "SubRel.subpath (red prb) rv1 res rv2 (subs prb)"
hcomp_simps_3: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "hseq \<nu> \<mu>" shows "local.dom (\<nu> \<star> \<mu>) = local.dom \<nu> \<star> local.dom \<mu>"
measurement_lowner_le_one1: fixes M0 :: "complex mat"   and dims :: "nat list"   and M1 :: "complex mat" assumes "M0 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "M1 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m (state_sig.d dims)" shows "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m (state_sig.d dims)"
moebius_rotation_plus: fixes \<phi>1 :: "real"   and \<phi>2 :: "real" shows "moebius_rotation \<phi>1 + moebius_rotation \<phi>2 = moebius_rotation (\<phi>1 + \<phi>2)"
segment_altdef: fixes b :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "b \<in> orbit f a" shows "segment f a b = (\<lambda>n. (f ^^ n) a) ` {1..<funpow_dist1 f a b}"
inner_ge_zero: fixes x :: "'a" shows "0 \<le> real_of (inner_field x x)"
A: shows "\<forall>i. countable (A i) \<and> Ball (A i) ((\<in>) ((x)\<^sub>F i)) \<and> Ball (A i) open \<and> (\<forall>S. open S \<and> (x)\<^sub>F i \<in> S \<longrightarrow> (\<exists>a\<in>A i. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A i \<longrightarrow> b \<in> A i \<longrightarrow> a \<inter> b \<in> A i)"
conj_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<and>. q) = (A \<turnstile> p \<and> A \<turnstile> q)"
ws_def_2: shows "hd ws = subst \<rho> w"
exists_two_distint: shows "\<exists>a b. a \<noteq> b"
Float_mantissa_exponent: fixes f :: "float" shows "Float (mantissa f) (exponent f) = f"
code: fixes xs :: "'c list" shows "cycle xs = hd xs ## cycle (tl xs @ [hd xs])"
integrable_sum: fixes I :: "'a set"   and M :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
subst_supportD: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and \<delta> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "\<theta> supports \<delta>" shows "\<theta> \<preceq>\<^sub>\<circ> \<delta>"
disjoint_demonotonicity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "PM"   and "P x y"   and "\<not> O z y" shows "\<not> O z x"
field_has_derivative_at: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b" shows "(f has_derivative (*) D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'b\<rightarrow> D"
add_0_right: fixes a :: "'a" shows "a \<squnion> bot = a"
cases: fixes x :: "real \<times> real \<times> real" obtains a :: "real" and b :: "real" and c :: "real" where "x = (a, b, c)"
\<beta>_mem_rmv_\<alpha>: shows "\<beta> \<in> set (remove1 \<alpha> xs)"
valid_path_cons: fixes v :: "'a"   and w :: "'a"   and Ps :: "'a llist" assumes "v \<in> V"   and "w \<in> V"   and "v \<rightarrow> w"   and "valid_path Ps"   and "\<not> lnull Ps"   and "lhd Ps = w" shows "valid_path (LCons v Ps)"
treehead_in_multiset: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "t \<in> set bq" shows "(BinomialHeapStruc.val t, BinomialHeapStruc.prio t) \<in># BinomialHeapStruc.queue_to_multiset bq"
refine: fixes Rm :: "('e \<times> 'f) set" shows "(pn_map_impl, pn_map) \<in> \<langle>\<langle>Rm, Id\<rangle>node_rel\<rangle>list_set_rel \<rightarrow> \<langle>\<langle>nat_rel, \<langle>Id\<rangle>list_set_rel \<times>\<^sub>r \<langle>Id\<rangle>list_set_rel\<rangle>iam_map_rel\<rangle>nres_rel"
iso_cyclic_groups_same_order: fixes H :: "('c, 'd) monoid_scheme"   and h :: "'c" assumes "Finite_And_Cyclic_Groups.cyclic_group H h"   and "order G = order H" shows "G \<cong> H"
bind_eq_Some_conv: fixes f :: "'b option"   and g :: "'b \<Rightarrow> 'a option"   and x :: "'a" shows "(Option.bind f g = \<lfloor>x\<rfloor>) = (\<exists>y. f = \<lfloor>y\<rfloor> \<and> g y = \<lfloor>x\<rfloor>)"
trg_inv: fixes \<mu> :: "'a" assumes "S.iso \<mu>" shows "S.trg (S.inv \<mu>) = S.trg \<mu>"
disj_idem: fixes P :: "'a upred" shows "(P \<or> P) = P"
simps_2: fixes a :: "'a"   and x :: "'a list" shows "mset (a # x) = add_mset a (mset x)"
set_spmf_of_set_finite: fixes A :: "'a set" assumes "finite A" shows "set_spmf (spmf_of_set A) = A"
assms_2: shows "x \<in>\<^sub>\<circ> B"
pair_by_index_def: shows "pair_by_index = (\<lambda>n1 n2. c_pair 5 (c_pair n1 n2))"
map_poly_1': fixes f :: "'c \<Rightarrow> 'b" assumes "f (1::'c) = (1::'b)" shows "map_poly f 1 = 1"
isomorphic_implies_equicomposable_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<sigma> :: "'b" assumes "weak_composition V H"   and "category.isomorphic V f g" shows "(H f \<sigma> \<noteq> partial_magma.null V) = (H g \<sigma> \<noteq> partial_magma.null V)"
alw2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "\<turnstile> F"   and "|~ F \<and> Unchanged v \<longrightarrow> \<circle>F" shows "\<turnstile> \<box>F"
word_div_mult_le: fixes a :: "'b word"   and b :: "'b word" shows "a div b * b \<le> a"
alw_HLD_smap: fixes S :: "'a set"   and \<omega> :: "'a stream"   and f :: "'a \<Rightarrow> 'b" assumes "alw (HLD S) \<omega>" shows "alw (HLD (f ` S)) (smap f \<omega>)"
init: shows "envObs a (tFirst t) = envObs a (tFirst t')"
Agii_mem: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and D :: "'e set"   and x :: "'c" assumes "aGroup A"   and "f \<in> carrier A \<rightarrow> D"   and "x \<in> carrier A"   and "bij_to f (carrier A) D" shows "Agii A f x \<in> carrier (Ag_ind A f)"
type_max_def: fixes t :: "'BExp set"   and mem :: "'Var \<Rightarrow> 'Val" shows "type_max t mem \<equiv> if \<forall>p\<in>t. ev\<^sub>B mem p then Low else High"
eventually_nhdsin: fixes P :: "'b \<Rightarrow> bool"   and X :: "'b topology"   and a :: "'b" shows "eventually P (nhdsin X a) = (a \<notin> topspace X \<or> (\<exists>S. openin X S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)))"
12: shows "r\<^sup>T * prim_W w v e\<^sup>\<star> * prim_EP w v e \<le> r\<^sup>T * prim_W w v e\<^sup>\<star>"
invar_insert: fixes s :: "'b tree"   and x :: "'b" assumes "splay.invar s" shows "splay.invar (insert x s)"
recurrent_subset_meas_2: fixes A :: "'a set" assumes "A \<in> sets M" shows "recurrent_subset_infty A \<in> sets M"
mk_path_induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>m n. m = n \<Longrightarrow> P (Suc m) (Suc n)"   and "\<And>m n. \<lbrakk>m < n; P (Suc m) (n - m)\<rbrakk> \<Longrightarrow> P (Suc m) (Suc n)"   and "\<And>m n. \<lbrakk>n < m; P (m - n) (Suc n)\<rbrakk> \<Longrightarrow> P (Suc m) (Suc n)"   and "\<And>uu_. P 0 uu_"   and "\<And>uv_. P uv_ 0" shows "P a0 a1"
hd_concat: fixes xs :: "'c list list" assumes "xs \<noteq> []"   and "hd xs \<noteq> []" shows "hd (concat xs) = hd (hd xs)"
induced_group_hom': fixes I :: "'a set" assumes "subgroup I G" shows "group_hom (G\<lparr>carrier := I\<rparr>) H h"
continuous_on_swap_args: fixes A :: "'a set"   and B :: "'b set"   and d :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "continuous_on (A \<times> B) (\<lambda>(x, y). d x y)" shows "continuous_on (B \<times> A) (\<lambda>(x, y). d y x)"
f_simps_1: fixes n :: "nat"   and x :: "'v" shows "f n (Some x) = (if x \<in> A_n n then enn2real (weight \<Gamma> x) else 0)"
FG: shows "(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. (c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice) \<sqinter> c ; x)"
insert_def: fixes x :: "'b"   and t :: "'b bro" shows "insert x t = tree (ins x t)"
bottom_neq_numeral_Integer: fixes k :: "num" shows "\<bottom> \<noteq> numeral k"
lub_is_either: fixes S :: "'a ttree set" shows "lub S = Either S"
finite_Sts0: shows "finite (N_of_wstate (lhd Sts) \<union> P_of_wstate (lhd Sts) \<union> Q_of_wstate (lhd Sts))"
scaleR_cancel_right: fixes a :: "real"   and x :: "'b"   and b :: "real" shows "(a *\<^sub>R x = b *\<^sub>R x) = (a = b \<or> x = (0::'b))"
emeasure_pmf: fixes M :: "'a pmf" shows "emeasure (measure_pmf M) (set_pmf M) = 1"
no_transition_after_FAIL: 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"   and io :: "'d \<times> 'e" assumes "productF A B FAIL AB" shows "succ AB io FAIL = {}"
generic_Assert_axioms: shows "generic_Assert (\<bind>) RETURN ASSERT ASSUME"
inject_11: fixes x141 :: "String.literal"   and x142 :: "nat"   and y141 :: "String.literal"   and y142 :: "nat" shows "(Invoke x141 x142 = Invoke y141 y142) = (x141 = y141 \<and> x142 = y142)"
rec_6: fixes k :: "'t"   and v :: "'t" assumes "k \<in> set ks"   and "v \<in> set vs" shows "v \<prec>\<^sub>t k"
arrE: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.arr f" obtains "f \<noteq> Cop_S.null" and "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Cop_S.Dom f) (Cop_S.Cod f) (Cop_S.Map f)"
assms: shows "\<forall>a\<in>set x. 1 \<le> a"
convergence: fixes xs :: "('a \<times> operation) list"   and ys :: "('a \<times> operation) list" assumes "set xs = set ys"   and "hb.concurrent_ops_commute xs"   and "hb.concurrent_ops_commute ys"   and "distinct xs"   and "distinct ys"   and "hb.hb_consistent xs"   and "hb.hb_consistent ys" shows "hb.apply_operations xs = hb.apply_operations ys"
bS: shows "\<forall>x\<in>S. x \<le> B"
set_bset_inverse: fixes x :: "'d bset" shows "Abs_bset (set_bset x) = x"
OclExcluding_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "x->excluding\<^sub>S\<^sub>e\<^sub>t(y) = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> - {y \<tau>}\<rfloor>\<rfloor> else \<bottom>)"
assms: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) set"   and m :: "nat"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) \<times> ('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and data :: "nat \<times> 'b"   and q_in_bs :: "bool"   and q :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and p :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "dickson_grading d"   and "set gs \<union> set bs \<union> set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "q_in_bs \<longrightarrow> q \<in> set gs \<union> set bs"   and "\<And>p' q'. \<lbrakk>(p', q') \<in>\<^sub>p snd ` set ps; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "\<And>p' q'. \<lbrakk>p' \<in> set gs \<union> set bs; q' \<in> set gs \<union> set bs; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "crit data gs bs hs q_in_bs ps p q" shows "crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)"
less_le: fixes x :: "'t"   and y :: "'t" shows "(x \<prec>\<^sub>t y) = (x \<preceq>\<^sub>t y \<and> x \<noteq> y)"
summandsZero: fixes P :: "pi" assumes "summands P = {}"   and "hnf P" shows "P = \<zero>"
flow_tendsto_compose: fixes x :: "'b \<Rightarrow> 'a"   and xs :: "'a"   and F :: "'b filter"   and t :: "'b \<Rightarrow> real"   and ts :: "real" assumes "(x \<longlongrightarrow> xs) F"   and "(t \<longlongrightarrow> ts) F"   and "ts \<in> existence_ivl0 xs" shows "((\<lambda>s. flow0 (x s) (t s)) \<longlongrightarrow> flow0 xs ts) F"
right_mult_is_bij: fixes h :: "'a" assumes "h \<in> carrier G" shows "restrict ((\<otimes>) h) (carrier G) \<in> Bij (carrier G)"
same_cardinality_post_swap_3: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and j :: "nat"   and i :: "nat"   and n :: "nat" assumes "computation channel trans send recv init final"   and "computation.prerecording_event channel trans send recv init t j"   and "computation.postrecording_event channel trans send recv init t i"   and "i < j"   and "j < length t"   and "computation.count_violations channel trans send recv init t = Suc n"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "distributed_system.trace channel trans send recv init t final" shows "{k \<in> {j + 1..<length t}. computation.prerecording_event channel trans send recv init t k} = {k \<in> {j + 1..<length (swap_events i j t)}. computation.prerecording_event channel trans send recv init (swap_events i j t) k}"
polyadd_eq_const_degreen: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and c :: "int \<times> int"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "p +\<^sub>p q = C c" shows "degreen p m = degreen q m"
ffold_fun_left_comm: fixes x :: "'a"   and z :: "'b"   and A :: "'a fset" shows "f x (ffold f z A) = ffold f (f x z) A"
non_empty_set_has_least: fixes S :: "'a set" assumes "S \<subseteq> T"   and "S \<noteq> {}" shows "least S \<in> S \<and> (\<forall>y\<in>S. least S \<noteq> y \<longrightarrow> \<not> y \<prec> least S)"
lookup_pair_update_by_fun_pair: fixes xs :: "('a \<times> 'b) list"   and k1 :: "'a"   and f :: "'b \<Rightarrow> 'b"   and k2 :: "'a" assumes "tc.oalist_inv_raw xs" shows "lookup_pair_tc (update_by_fun_pair_tc k1 f xs) k2 = (if k1 = k2 then f else id) (lookup_pair_tc xs k2)"
assms: shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
safe_formula_mmonitorable_exec: fixes \<phi> :: "Formula.formula" assumes "safe_formula \<phi>"   and "Formula.future_bounded \<phi>" shows "mmonitorable_exec \<phi>"
h1_var: shows "M_mat (get_signs (reduce_system p (qs, M_mat signs subsets, subsets, signs))) (get_subsets (reduce_system p (qs, M_mat signs subsets, subsets, signs))) = M_mat (take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) (take_indices subsets (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))))"
lisetD_1: fixes i :: "nat"   and A :: "'b set"   and w :: "'b llist" assumes "i \<in> liset A w" shows "enat i < llength w"
while_option_None_invD: fixes b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and s :: "'a"   and r :: "('a \<times> 'a) set"   and I :: "'a \<Rightarrow> bool" assumes "while_option b c s = None"   and "wf r"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s)"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> r" shows "False"
Limit_mk_doubleton_image_vsubset_Vset: fixes \<alpha> :: "V"   and X :: "V"   and Y :: "V" assumes "Limit \<alpha>"   and "X \<subseteq>\<^sub>\<circ> Y"   and "Y \<in>\<^sub>\<circ> Vset \<alpha>" shows "mk_doubleton_image X Y \<subseteq>\<^sub>\<circ> Vset \<alpha>"
PQ: fixes X :: "('a \<times> 'b) topology"   and X' :: "'a topology" assumes "X homeomorphic_space X'" shows "P X = Q X'"
leibniz_rule_holomorphic: fixes U :: "complex set"   and a :: "'a"   and b :: "'a"   and f :: "complex \<Rightarrow> 'a \<Rightarrow> complex"   and fx :: "complex \<Rightarrow> 'a \<Rightarrow> complex" assumes "\<And>x t. \<lbrakk>x \<in> U; t \<in> cbox a b\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"   and "\<And>x. x \<in> U \<Longrightarrow> f x integrable_on cbox a b"   and "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"   and "convex U" shows "(\<lambda>x. integral (cbox a b) (f x)) holomorphic_on U"
i_take_0: fixes f :: "nat \<Rightarrow> 'd" shows "f \<Down> 0 = []"
intros_2: fixes h :: "'e"   and H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
weak_pref_mix_with_my_U_weak_pref: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and p :: "'a pmf"   and q :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>"   and "p \<succeq>[\<R>] q" shows "mix_pmf (my_U outcomes \<R> p) (some_best outcomes \<R>) (some_worst outcomes \<R>) \<succeq>[\<R>] mix_pmf (my_U outcomes \<R> q) (some_best outcomes \<R>) (some_worst outcomes \<R>)"
prems: shows "Suc n \<le> mod_type_class.to_nat i"
rtrancl_join_join: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> A\<^sup>*"   and "(b, c) \<in> A\<^sup>\<down>" shows "(a, c) \<in> A\<^sup>\<down>"
1: fixes g :: "'a"   and h :: "'a" assumes "g \<in> G"   and "h \<in> G" shows "\<pp> (binop g h) = \<pp> g + \<pp> h"
z_def: shows "z \<equiv> getFr [y] [t] []"
perp_in_col_perp_in: fixes C :: "'p"   and E :: "'p"   and D :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" assumes "C \<noteq> E"   and "Col C D E"   and "P PerpAt A B C D " shows "P PerpAt A B C E "
Phi_Psi: fixes n :: "nat"   and l :: "real" shows "\<Phi> (n, l) = \<Psi> (l0 \<le> l / c) ai ad (f1' * l) (f2' * l) n"
split_beta: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and prod :: "'g \<times> 'h" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
simps_9: fixes v :: "'b"   and va :: "'b"   and vb :: "'c"   and vc :: "'b"   and vd :: "'b"   and ve :: "'c"   and l :: "'b set list" shows "smaller (AllowPortFromTo v va vb) (AllowPortFromTo vc vd ve) l = (AllowPortFromTo v va vb = AllowPortFromTo vc vd ve \<or> (if bothNet (AllowPortFromTo v va vb) = bothNet (AllowPortFromTo vc vd ve) then case AllowPortFromTo vc vd ve of DenyAllFromTo a b \<Rightarrow> AllowPortFromTo v va vb = DenyAllFromTo b a | _ \<Rightarrow> True else position (bothNet (AllowPortFromTo v va vb)) l \<le> position (bothNet (AllowPortFromTo vc vd ve)) l))"
vdiffD1: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A -\<^sub>\<circ> B" shows "x \<in>\<^sub>\<circ> A"
quotient_smaller: fixes a :: "nat"   and k :: "nat"   and b :: "nat" assumes "a \<noteq> 0"   and "a = k * b" shows "b \<le> a"
of_nat_of_integer: fixes k :: "integer" shows "of_nat (nat_of_integer k) = max 0 k"
openin_subopen: fixes U :: "'c topology"   and S :: "'c set" shows "openin U S = (\<forall>x\<in>S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"