hcju/isabelleps huggingface dataset:
new_pairs_naive_def: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "new_pairs_naive gs bs hs data = fold (add_pairs_single_naive data True gs bs) hs (pairs (add_pairs_single_naive data) False hs)"
wt_App_ge_fun: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "wt s \<le> wt (App s t)"
hb_broadcast_exists1: fixes m1 :: "'a"   and m2 :: "'a" assumes "hb m1 m2" shows "\<exists>i. Broadcast m1 \<in> set (history i)"
private_fun_deduct_in_ik: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term"   and f :: "'a"   and T :: "('a, 'b) Term.term list" assumes "intruder_model arity Ana"   and "intruder_model.intruder_deduct arity public Ana M t"   and "Fun f T \<sqsubseteq> t"   and "\<not> public f" shows "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M"
get_ancestors_si_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ancestor :: "(_) object_ptr"   and ancestor_ancestors :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"   and "ancestor \<in> set ancestors"   and "h \<turnstile> get_ancestors_si ancestor \<rightarrow>\<^sub>r ancestor_ancestors"   and "type_wf h"   and "known_ptrs h" shows "set ancestor_ancestors \<subseteq> set ancestors"
index_tensor_mat_with_vec2_row_cond: fixes i :: "nat"   and B :: "complex Matrix.mat"   and A :: "complex Matrix.mat" assumes "i < 2 * dim_row B"   and "dim_row B \<le> i"   and "0 < dim_col B"   and "dim_row A = 2"   and "dim_col A = 1" shows "(A \<Otimes> B) $$ (i, 0) = A $$ (1, 0) * B $$ (i - dim_row B, 0)"
is_closed_Fv: fixes idn :: "variable"   and \<tau> :: "typ" shows "is_closed (Fv idn \<tau>)"
assms_1: shows "BuildSeq B C s1 k1 y1"
lenlex_transI: fixes r :: "('b \<times> 'b) set" assumes "trans r" shows "trans (lenlex r)"
check_del_l_def: fixes spec :: "'f"   and A :: "'g"   and p :: "nat" shows "check_del_l spec A p = RETURN CSUCCESS"
map_idI: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in>. xs \<Longrightarrow> f x = x" shows "map f xs = xs"
r_def: shows "r \<equiv> \<lambda>q. state_rank q n"
Rep_hyp2_cases: fixes y :: "proj2" assumes "y \<in> hyp2" obtains x :: "hyp2" where "y = Rep_hyp2 x"
sum_distrib_left: fixes r :: "'a"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "r * comm_monoid_add.sum (\<squnion>) bot f A = comm_monoid_add.sum (\<squnion>) bot (\<lambda>n. r * f n) A"
ideI: fixes a :: "'a \<times> 'a" assumes "B.arr (fst a) \<and> B.arr (snd a) \<and> src\<^sub>B (fst a) = trg\<^sub>B (snd a)"   and "B.VxV.ide a" shows "B.VV.ide a"
my_U_def: fixes p :: "'a pmf" shows "local.my_U p = (SOME \<alpha>. \<alpha> \<in> {0..1} \<and> p \<approx>[\<R>] mix_pmf \<alpha> local.some_best local.some_worst)"
\<Delta>''_countable_network: shows "countable_network \<Delta>''"
valid_cfgD: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "cfg \<in> MDP.cfg_on (state cfg)"
mset_zip_take_Cons_drop_twice: fixes xs :: "'k list"   and ys :: "'l list"   and j :: "nat"   and x :: "'k"   and y :: "'l" assumes "length xs = length ys"   and "j \<le> length xs" shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) = add_mset (x, y) (mset (zip xs ys))"
Infinitesimal_add_not_zero: fixes h :: "'a star"   and x :: "'a" assumes "h \<in> Infinitesimal"   and "x \<noteq> (0::'a)" shows "star_of x + h \<noteq> 0"
open_Compl: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open"   and "topological_space.closed open S" shows "open (- S)"
positive_dim_eq: fixes A :: "complex mat" assumes "positive A" shows "dim_row A = dim_col A"
group_eq_subgrp_rcoset_un: fixes H :: "'g set"   and gs :: "'g list" assumes "Supgroup.Subgroup H"   and "Supgroup.is_rcoset_replist H gs" shows "H = (\<Union>g\<in>set gs. H + {g})"
isomorphic_reflexive: fixes f :: "'a \<times> 'a" assumes "S.VV.ide f" shows "S.VV.isomorphic f f"
has_domain_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "(partial_magma.domains (product_category.comp C1 C2) f \<noteq> {}) = (partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f))"
sinks_aux_single_dom: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" shows "sinks_aux I D {u} xs = insert u (sinks I D u xs)"
beukers_integral3_def: shows "local.beukers_integral3 = LBINT (w, x, y):D'. (x * (1 - x) * y * (1 - y) * w * (1 - w)) ^ n / (1 - (1 - x * y) * w) ^ (n + 1)"
1: shows "forest_components f \<le> - d\<^sup>T"
intro: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<sigma>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>1 :: "'c \<Rightarrow> 'd"   and \<rho>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<rho>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "composite_pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<rho>\<^sub>0 \<rho>\<^sub>1"   and "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<sigma>\<^sub>0 \<sigma>\<^sub>1"   and "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<rho>\<^sub>0 \<rho>\<^sub>1" shows "composite_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<rho>\<^sub>0 \<rho>\<^sub>1"
has_integral_def: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'c"   and s :: "'b set" shows "(f has_integral I) s = (if \<exists>a b. s = cbox a b then (sum (\<lambda>(x, k). content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter s) else \<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'b) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. (sum (\<lambda>(x, k). content k *\<^sub>R (if x \<in> s then f x else (0::'c))) \<longlongrightarrow> z) (division_filter (cbox a b)) \<and> norm (z - I) < e))"
Nat_atleast_mono: fixes s :: "state"   and a :: "loc"   and n :: "nat" assumes "s:get_field s a pred \<ge> n"   and "heap s a \<noteq> None" shows "s:Addr a \<ge> n"
Rep_comp_inverse: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b" assumes "Rep \<circ> f = g" shows "Abs \<circ> g = f"
lappend_code_1: fixes ys :: "'a llist" shows "[]\<^sub>l @\<^sub>l ys = ys"
cubic_root_radical_sqrt_rational: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" assumes "a \<in> \<rat>"   and "b \<in> \<rat>"   and "c \<in> \<rat>"   and "x \<in> radical_sqrt"   and "x ^ 3 + a * x\<^sup>2 + b * x + c = 0" shows "\<exists>x\<in>\<rat>. x ^ 3 + a * x\<^sup>2 + b * x + c = 0"
eq_iff: fixes x :: "'b"   and y :: "'b"   and ko :: "'b key_order" shows "(x = y) = (ko.le ko x y \<and> ko.le ko y x)"
T1: shows "[\<lambda>w. \<forall>x. (P x m\<rightarrow> \<diamond> (\<exists> x)) w]"
inverse_nonzero_iff_nonzero: fixes a :: "'a" shows "(inverse a = (0::'a)) = (a = (0::'a))"
FreeGroupD_transfer': fixes xs :: "'b list"   and S :: "'b set" assumes "Abs_freelist xs \<in> FreeGroup S" shows "xs \<in> lists S"
that: assumes "nop_v11 A (1\<^sub>\<nat>) f" shows "thesis"
convert_substInfinitesimalQuadratic: fixes var :: "nat"   and At :: "atom"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and At' :: "atomUni"   and a :: "real mpoly"   and A :: "real"   and b :: "real mpoly"   and B :: "real"   and c :: "real mpoly"   and C :: "real"   and d :: "real mpoly"   and D :: "real" assumes "convert_atom var At (xs' @ x # xs) = Some At'"   and "insertion (nth_default 0 (xs' @ x # xs)) a = A"   and "insertion (nth_default 0 (xs' @ x # xs)) b = B"   and "insertion (nth_default 0 (xs' @ x # xs)) c = C"   and "insertion (nth_default 0 (xs' @ x # xs)) d = D"   and "D \<noteq> 0"   and "0 \<le> C"   and "var \<notin> vars a"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "var \<notin> vars d"   and "length xs' = var" shows "eval (substInfinitesimalQuadratic var a b c d At) (xs' @ x # xs) = evalUni (substInfinitesimalQuadraticUni A B C D At') x"
assms_2: shows "recognises_execution e s' (evaluate_updates t (snd h) r) trace"
lock_ok_las_def: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions" shows "lock_ok_las ls t las \<equiv> \<forall>l. lock_actions_ok (ls $ l) t (las $ l)"
single_span: fixes A :: "'b set"   and z :: "'a"   and h :: "nat \<Rightarrow> 'a"   and n :: "nat"   and t :: "nat \<Rightarrow> 'b" assumes "ideal R A"   and "z \<in> carrier M"   and "h \<in> {j. j \<le> n} \<rightarrow> {z}"   and "t \<in> {j. j \<le> n} \<rightarrow> A" shows "\<exists>s\<in>{0} \<rightarrow> A. \<Sigma>\<^sub>e M (\<lambda>j. t j \<cdot>\<^sub>s h j) n = s 0 \<cdot>\<^sub>s z"
empty_offending_contra: fixes F :: "('c \<times> 'c) set"   and sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" assumes "F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP"   and "F = {}" shows "False"
simps_1: fixes uu :: "'a option" shows "(None \<le> uu) = True"
norm_bound_component_le_cart: fixes x :: "(real, 'b) vec"   and e :: "real"   and i :: "'b" assumes "\<parallel>x\<parallel> \<le> e" shows "\<bar>x $ i\<bar> \<le> e"
assms_5: shows "u'' \<turnstile>\<^bsub>v,n\<^esub> M"
empty_rep: fixes q :: "('a, 'b) SkewBinomialHeap" shows "(q = SkewBinomialHeap_loc.empty) = (Rep_SkewBinomialHeap q = BsSkewBinomialHeapStruc.bs_empty)"
Part_eqI: fixes a :: "'b"   and A :: "'b set"   and h :: "'c \<Rightarrow> 'b"   and b :: "'c" assumes "a \<in> A"   and "a = h b" shows "a \<in> Part A h"
constrains_def: fixes A :: "'b set"   and B :: "'b set" shows "A co B \<equiv> {F. \<forall>act\<in>Acts F. act `` A \<subseteq> B}"
elims_1: fixes x :: "'c arctic_delta"   and xa :: "'c arctic_delta"   and y :: "bool" assumes "(x \<le> xa) = y" obtains xb :: "'c arctic_delta" where "x = MinInfty_delta" and "xa = xb__" and "y"   | uu_ :: "'c" where "x = Num_arc_delta uu_" and "xa = MinInfty_delta" and "\<not> y"   | ya :: "'c" and xb :: "'c" where "x = Num_arc_delta ya" and "xa = Num_arc_delta xb" and "y = (ya \<le> xb)"
hyps_3: shows "\<turnstile>\<^sub>\<tau> t : \<tau> \<rightarrow> \<tau>'"
is_inf_meet: fixes x :: "'b"   and y :: "'b" shows "is_inf x y (x \<sqinter> y)"
nres_order_simps_3: fixes M :: "'c nres" shows "M \<le> FAIL"
IH: shows "set (interp_ins xs) \<subseteq> set (map fst xs)"
index_eq_nextl_le_states: shows "card (UNIV // eq_nextl) \<le> card (states M)"
cont_Pair: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'd" assumes "cont lub ord luba orda f"   and "cont lub ord lubb ordb g" shows "cont lub ord (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
S_init: fixes fg :: "('a, 'c, 'd, 'b, 'e) flowgraph_rec_scheme"   and p :: "'c"   and k :: "nat" shows "(entry fg p, {}, {#}) \<in> S_cs fg k"
Fadmit_And: fixes \<sigma> :: "('a, 'b, 'c) subst"   and \<phi> :: "('a, 'b, 'c) formula"   and \<psi> :: "('a, 'b, 'c) formula" assumes "Fadmit \<sigma> \<phi>"   and "Fadmit \<sigma> \<psi>" shows "Fadmit \<sigma> (\<phi> && \<psi>)"
combine_pairs_prop: fixes \<P> :: "(('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set) set"   and \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and w :: "nat \<Rightarrow> 'c" shows "(\<forall>P\<in>\<P>. accepting_pair\<^sub>R \<delta> q\<^sub>0 P w) = accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (combine_pairs \<P>) w"
well_def_h: shows "\<forall>x. List.member S x \<longrightarrow> x < length signs"
IQY: shows "SkewBinomialHeapStruc.invar (map bsmapt qy)"
mod0: shows "(\<Sum>i | i \<in> B \<and> j \<le> i. 2 ^ i) mod 2 ^ j = 0"
fmrestrict_set_insert_in: fixes xa :: "'c"   and N :: "('c, 'd) fmap"   and l1 :: "'c set" assumes "xa \<in> fset (fmdom N)" shows "fmrestrict_set (insert xa l1) N = fmupd xa (the (fmlookup N xa)) (fmrestrict_set l1 N)"
is_unused_elem_def: fixes dflt :: "'b"   and A :: "'d \<Rightarrow> 'b \<Rightarrow> assn" shows "is_unused_elem dflt A \<equiv> \<forall>x. A x dflt = false"
assms_2: shows "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>"
less_nd_fun_def: shows "(<) \<equiv> id (\<lambda>f g. f\<^sub>\<bullet> \<le> g\<^sub>\<bullet> \<and> f \<noteq> g)"
distinct_3: fixes x11 :: "tm"   and x12 :: "tm"   and x31 :: "fm"   and x32 :: "fm" shows "x11 IN x12 \<noteq> x31 OR x32"
map_replicate: fixes f :: "'b \<Rightarrow> 'a"   and n :: "nat"   and x :: "'b" shows "map f (replicate n x) = replicate n (f x)"
f_shrink_last_append_mod: fixes xs :: "'b list"   and k :: "nat"   and ys :: "'b list" assumes "length xs mod k = 0" shows "(xs @ ys) \<div>\<^sub>l k = xs \<div>\<^sub>l k @ ys \<div>\<^sub>l k"
v_2: shows "[v]\<^sub>\<R> = region X I' J' r'"
sep_add_disjD: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x + y ## z"   and "x ## y" shows "x ## z \<and> y ## z"
is_iso_Cons: fixes F\<^sub>1 :: "'a list"   and Fs\<^sub>1' :: "'a list list"   and Fs\<^sub>2 :: "'b list list"   and \<phi> :: "'a \<Rightarrow> 'b" assumes "distinct (F\<^sub>1 # Fs\<^sub>1')"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set (F\<^sub>1 # Fs\<^sub>1'))"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)" shows "is_pr_iso \<phi> (F\<^sub>1 # Fs\<^sub>1') Fs\<^sub>2 = (\<exists>F\<^sub>2\<in>set Fs\<^sub>2. length F\<^sub>1 = length F\<^sub>2 \<and> is_pr_iso \<phi> Fs\<^sub>1' (remove1 F\<^sub>2 Fs\<^sub>2) \<and> (\<exists>n. map \<phi> F\<^sub>1 = rotate n F\<^sub>2) \<and> inj_on \<phi> (set F\<^sub>1 \<union> \<Union> (set ` set Fs\<^sub>1')))"
eq_iff_3: fixes trm :: "trm"   and trma :: "trm" shows "(suc trm = suc trma) = (trm = trma)"
bounds_consistent_def: fixes s :: "('c, 'd) state" shows "\<diamond> s \<equiv> \<forall>x. if \<B>\<^sub>l s x = None \<or> \<B>\<^sub>u s x = None then True else the (\<B>\<^sub>l s x) \<le> the (\<B>\<^sub>u s x)"
med11_def: fixes t :: "skl1_state" shows "med11 t \<equiv> \<lparr>ik = ik t, secret = secret t, progress = med_progress (progress t), signalsInit = signalsInit2 t, signalsResp = signalsResp2 t\<rparr>"
single_eq_single_zmset: fixes a :: "'b"   and b :: "'b" shows "({#a#}\<^sub>z = {#b#}\<^sub>z) = (a = b)"
nth_idem: fixes f :: "nat \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.nth (list.map (lookup f) [0..<degree f]) = f"
left_adjoint_is_ide: fixes f :: "'a" assumes "C.is_left_adjoint f" shows "C.ide f"
locally_Int: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "locally P S"   and "locally P T"   and "\<And>S T. P S \<and> P T \<Longrightarrow> P (S \<inter> T)" shows "locally P (S \<inter> T)"
mem_samp_primes_prime: fixes x :: "nat" assumes "x \<in> set_spmf sample_primes" shows "prime x"
fbox_arden_whilei: fixes t :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and i :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" assumes "rel_nabla (rdom t ; x) = {}"   and "rdom i = (rdom t \<union> rdom q) ; wp (rdom t ; x) i" shows "rdom i = wp (rel_antidomain_kleene_algebra.whilei t i x) q"
extended_distance_triang_ineq: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c Bonk_Schramm_extension_unfolded"   and y :: "'c Bonk_Schramm_extension_unfolded"   and z :: "'c Bonk_Schramm_extension_unfolded" assumes "class.metric_space dist uniformity open"   and "x \<in> metric_space.extended_distance_set dist"   and "y \<in> metric_space.extended_distance_set dist"   and "z \<in> metric_space.extended_distance_set dist" shows "metric_space.extended_distance dist x z \<le> metric_space.extended_distance dist x y + metric_space.extended_distance dist y z"
a4': fixes x :: "'a"   and y :: "'a" shows "ad (x + y) = ad x \<cdot> ad y"
prems_4: shows "A\<^sub>P \<sharp>* P"
intro: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'e" assumes "diff k charts charts_eucl f" shows "diff_fun k charts f"
has_integral_restrict: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b" assumes "S \<subseteq> T" shows "((\<lambda>x. if x \<in> S then f x else (0::'b)) has_integral i) T = (f has_integral i) S"
len_lexext_rev_irrefl: fixes xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> len_lexext_rev gt xs xs"
ctermsSI: fixes p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and pn :: "'c" assumes "p \<in> sterms \<Gamma> (\<Gamma> pn)" shows "ctermsp \<Gamma> p"
permute_set_def: fixes p :: "perm"   and X :: "'b set" shows "p \<bullet> X = {p \<bullet> x |x. x \<in> X}"
dom_fun_upd: fixes f :: "'d \<Rightarrow> 'e option"   and x :: "'d"   and y :: "'e option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
n_ivl_narrow: fixes x :: "ivl"   and y :: "ivl" assumes "x \<triangle> y < x" shows "n_ivl (x \<triangle> y) < n_ivl x"
domintros_3: fixes c :: "nat"   and n :: "C"   and ns :: "C list"   and ms :: "C list" assumes "stepC_dom (c, C (funC (C (n # ns)) (Suc (Suc c)) @ ms))" shows "stepC_dom (c, C (C (n # ns) # ms))"
compactin_subtopology: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" shows "compactin (subtopology X S) T = (compactin X T \<and> T \<subseteq> S)"
rel_pmf_inf: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf" assumes "rel_pmf R p q"   and "rel_pmf R q p"   and "reflp R"   and "transp R" shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
pr_iso_test2_corr: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" assumes "\<forall>F\<in>set Fs\<^sub>1. distinct F"   and "\<forall>F\<in>set Fs\<^sub>2. distinct F"   and "[] \<notin> set Fs\<^sub>2"   and "distinct Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)" shows "pr_iso_test2 PlaneGraphIso.compat merge0 [] Fs\<^sub>1 Fs\<^sub>2 = (\<exists>\<phi>. is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2)"
rel_flip: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d pmf"   and b :: "'e pmf" shows "rel_pmf R\<inverse>\<inverse> a b = rel_pmf R b a"
Inf_to_Sup_var: fixes X :: "'a set" shows "\<Sqinter> X = - \<Squnion> (uminus ` X)"
g_bigo_self: shows "(\<lambda>n. g (real n) - g (real (Suc n))) \<in> O(\<lambda>n. h (real n))"
bind_coin_spmf_eq_const': fixes b :: "bool" shows "coin_spmf \<bind> (\<lambda>x. return_spmf (x = b)) = coin_spmf"
Abs_io_inverse: fixes y :: "\<^url> \<Rightarrow> 'c \<times> \<^url>" assumes "y \<in> UNIV" shows "Rep_io (Abs_io y) = y"
ordinal_times_div_le: fixes y :: "ordinal"   and x :: "ordinal" shows "y * (x div y) \<le> x"
simps_19: fixes f1 :: "i32 \<Rightarrow> 'a"   and f2 :: "i64 \<Rightarrow> 'a"   and f3 :: "f32 \<Rightarrow> 'a"   and f4 :: "f64 \<Rightarrow> 'a"   and x3 :: "f32" shows "(case ConstFloat32 x3 of ConstInt32 x \<Rightarrow> f1 x | ConstInt64 x \<Rightarrow> f2 x | ConstFloat32 x \<Rightarrow> f3 x | ConstFloat64 x \<Rightarrow> f4 x) = f3 x3"
new_configured_SecurityInvariant_sound: fixes sinvar :: "'a graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and defbot :: "'b"   and receiver_violation :: "bool"   and nP :: "'a \<Rightarrow> 'b"   and m :: "'a SecurityInvariant_configured" assumes "new_configured_SecurityInvariant (sinvar, defbot, receiver_violation, nP) = Some m" shows "configured_SecurityInvariant m"
hom_grading_varnum: fixes X :: "'c set" shows "hom_grading (varnum X)"
double_zero: fixes a :: "'b" shows "(a + a = (0::'b)) = (a = (0::'b))"
has_vector_derivative_within_subset: fixes f :: "real \<Rightarrow> 'c"   and f' :: "'c"   and x :: "real"   and S :: "real set"   and T :: "real set" assumes "(f has_vector_derivative f') (at x within S)"   and "T \<subseteq> S" shows "(f has_vector_derivative f') (at x within T)"
nonconstant_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a list" assumes "class.semiring_0 plus zero times" shows "semiring_0.nonconstant zero p = (semiring_0.pnormal zero p \<and> (\<forall>x. p \<noteq> [x]))"
par_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "A B Par C D \<and> A \<noteq> B \<and> C \<noteq> D"
antipar_1: shows "trg d\<^sub>1 = src e\<^sub>1"
assms_4: shows "(_Entry_) -as\<rightarrow>* (_Low_)"
I: shows "I = {a<..<b} \<or> I = {a<..} \<or> I = {..<b} \<or> I = UNIV"
functor_def: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" shows "Functor F : AA \<longrightarrow> BB \<equiv> two_cats AA BB \<and> functor_axioms AA BB F"
rank_ins2: fixes bq :: "('c, 'd) BinomialTree list"   and t :: "('c, 'd) BinomialTree" assumes "rank_invar bq" shows "rank t \<le> rank (hd (ins t bq)) \<or> rank (hd (ins t bq)) = rank (hd bq) \<and> bq \<noteq> []"
multiplicative_function_axioms: shows "multiplicative_function (dirichlet_prod f g)"
interior_halfspace_ge: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)" shows "interior {x. b \<le> a \<bullet> x} = {x. b < a \<bullet> x}"
poincare_between_def: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" shows "poincare_between u v w = (u = v \<or> v = w \<or> (let cr = cross_ratio u v w (inversion v) in is_real (to_complex cr) \<and> Re (to_complex cr) < 0))"
f2r_def: fixes f :: "'c \<Rightarrow> 'd set" shows "\<R> f = {(x, y). y \<in> f x}"
ffd_iffd_galois: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and f :: "'d \<Rightarrow> 'c set" assumes "Sup_pres \<phi>" shows "(\<phi> = fd\<^sub>\<F> f) = (f = fd\<^sup>-\<^sub>\<F> \<phi>)"
subset_mono: fixes N :: "('a, 'b) message set"   and M :: "('a, 'b) message set" assumes "N \<subseteq> M" shows "I ` insert_messages N \<subseteq> I ` insert_messages M"
W_mono: fixes b :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and r :: "((char list \<Rightarrow> int) \<times> (char list \<Rightarrow> int)) set" shows "mono (W b r)"