hcju/isabelleps huggingface dataset:
get_tag_name_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_tag_name element_ptr"
True_ltlp_def: shows "true\<^sub>p \<equiv> not\<^sub>p false\<^sub>p"
transversal_def: fixes p :: "'a set" shows "transversal p = (\<forall>x\<in>p. \<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> p)"
sameDom_lift_simp1: fixes inp :: "'a \<Rightarrow> 'b option"   and f :: "'d \<Rightarrow> 'c"   and inp' :: "'a \<Rightarrow> 'd option" shows "sameDom inp (lift f inp') = sameDom inp inp'"
FactGroup_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
assms: shows "matchers_map \<sigma> \<inter> matchers (set_mset P) \<noteq> {}"
dep_in_subI: fixes X :: "'a set"   and \<E>' :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>'"   and "\<not> indep_in \<E>' X" shows "\<not> indep_in \<E> X"
column_Gram_Schmidt_matrix: fixes k :: "'b"   and A :: "((real, 'b) vec, 'a) vec" shows "column k (Gram_Schmidt_matrix A) = column k A - (\<Sum>x\<in>{column i (Gram_Schmidt_matrix A) |i. i < k}. (x \<bullet> column k A / (x \<bullet> x)) *\<^sub>R x)"
FiniteSingleAgentEnvironment_axioms_def: fixes agent :: "'b" shows "FiniteSingleAgentEnvironment_axioms agent \<equiv> \<forall>a. a = agent"
not_le_imp_less: fixes ko :: "'a key_order"   and y :: "'a"   and x :: "'a" assumes "\<not> ko.le ko y x" shows "ko.lt ko x y"
append_eq_append_conv: fixes xs :: "'c list"   and ys :: "'c list"   and us :: "'c list"   and vs :: "'c list" assumes "|xs| = |ys| \<or> |us| = |vs|" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
termMOD_igVarIPresIGWls: shows "igVarIPresIGWls termMOD"
assms: fixes v :: "nat \<Rightarrow> 'a" shows "p \<lbrace> v \<rbrace> = q \<lbrace> v \<rbrace>"
distinct_23: fixes x3 :: "'b"   and x6 :: "'b rexp" shows "Atom x3 \<noteq> Star x6"
epair_def: fixes e :: "'b set" shows "epair e = (if card e = 2 then Some (SOME (u, v). e = {u, v}) else None)"
linear_inj_imp_surj: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f" shows "surj f"
enn2real_eq_0_iff: fixes x :: "ennreal" shows "(enn2real x = 0) = (x = 0 \<or> x = top)"
tllist_all2_tllist_of_llist: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'c"   and xs :: "'a llist"   and c :: "'d"   and ys :: "'b llist" shows "tllist_all2 A B (tllist_of_llist b xs) (tllist_of_llist c ys) = (llist_all2 A xs ys \<and> (lfinite xs \<longrightarrow> B b c))"
add_eq_0_iff_both_eq_0: fixes x :: "'c multiset"   and y :: "'c multiset" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
Anno_code_i_i_i_oE: fixes x :: "'b J_prog"   and xa :: "String.literal \<Rightarrow> ty option"   and xb :: "'b expr"   and xc :: "'b expr" assumes "pred.eval (Anno_code_i_i_i_o x xa xb) xc" obtains "x,xa \<turnstile> xb \<leadsto>' xc"
simps_13: fixes x2 :: "('a, 'b, 'c, 'd) Tree"   and x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree" shows "tNot x2 \<noteq> tAct x41 x42 x43"
cs_done_implies_same_snapshots: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "snd (cs (S t i) cid) = Done"   and "channel cid = Some (p, q)" shows "cs (S t i) cid = cs (S t j) cid"
5: shows "w\<^sub>1 !! i \<in> set (stake (Suc i) w\<^sub>1)"
subspace_def: fixes Sa :: "('a \<Rightarrow> real) set" assumes "Sa \<subseteq> diff_fun_space" shows "l.diff_fun_space.m1.subspace Sa = (0 \<in> Sa \<and> (\<forall>x\<in>Sa. \<forall>y\<in>Sa. x + y \<in> Sa) \<and> (\<forall>c. \<forall>x\<in>Sa. c *\<^sub>R x \<in> Sa))"
IH_2: shows "insertion f (insertion (\<lambda>v. monom 0 (f v)) p2) = insertion f (reduce_nested_mpoly p2)"
assms_3: shows "g2 \<in> o[F](g1)"
igSubstIGVar1STR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIGVar1STR MOD \<equiv> (\<forall>ys y y1 xs x. ys \<noteq> xs \<or> x \<noteq> y \<longrightarrow> igSubst MOD ys (igVar MOD ys y1) y (igVar MOD xs x) = igVar MOD xs x) \<and> (\<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x)"
fund_circuitI: fixes B :: "'a set"   and x :: "'a"   and C :: "'a set" assumes "\<E>.basis B"   and "x \<in> \<E> - B"   and "\<E>.circuit C"   and "C \<subseteq> insert x B" shows "\<E>.fund_circuit x B = C"
Domain_Un_eq: fixes A :: "('c \<times> 'd) set"   and B :: "('c \<times> 'd) set" shows "Domain (A \<union> B) = Domain A \<union> Domain B"
gauss_int_norm_eq_prime_squareD_2: fixes p :: "nat"   and z :: "gauss_int" assumes "prime p"   and "[p = 3] (mod 4)"   and "gauss_int_norm z = p\<^sup>2" shows "prime_elem z"
ConcAssoc: fixes A :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and B :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and D :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "Cp ((A \<oplus> B) \<oplus> D) = Cp (A \<oplus> B \<oplus> D)"
inorder_upd: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c"   and y :: "'d" assumes "sorted1 (inorder t)" shows "inorder (treeI (upd x y t)) = upd_list x y (inorder t)"
vsubset_reflexive: fixes A :: "V" shows "A \<subseteq>\<^sub>\<circ> A"
elims_3: fixes x :: "'edge list"   and xa :: "'edge list" assumes "\<not> valid_path_rev_aux x xa" obtains cs :: "'edge list" and as :: "'edge list" and a :: "'edge" where "x = cs" and "xa = as @ [a]" and "\<not> (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> valid_path_rev_aux [] as | c' # cs' \<Rightarrow> c' \<in> get_return_edges a \<and> valid_path_rev_aux cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> valid_path_rev_aux (a # cs) as | _ \<Rightarrow> valid_path_rev_aux cs as)"
sigma_algebra_Pow: fixes sp :: "'b set" shows "sigma_algebra sp (Pow sp)"
list_succ_nth: fixes xs :: "'b list"   and n :: "nat" assumes "distinct xs"   and "n < length xs" shows "list_succ xs (xs ! n) = xs ! (Suc n mod length xs)"
zero_le: fixes x :: "'a multiset" shows "\<Empt> \<subseteq># x"
and_eq_not_not_or: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "and a b = not (or (not a) (not b))"
dom_map_add: fixes m :: "'b \<Rightarrow> 'c option"   and n :: "'b \<Rightarrow> 'c option" shows "dom (m ++ n) = dom n \<union> dom m"
rcosets_inv_mult_group_eq: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and M :: "'c set" assumes "H \<lhd> G"   and "M \<in> rcosets\<^bsub>G\<^esub> H" shows "set_inv\<^bsub>G\<^esub> M <#>\<^bsub>G\<^esub> M = H"
INF_image: fixes Inf :: "'c set \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "Inf (g ` f ` A) = Inf ((g \<circ> f) ` A)"
hcomp_in_hhom: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b"   and \<nu> :: "'b"   and c :: "'b" assumes "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>\<nu> \<star> \<mu> : a \<rightarrow> c\<guillemotright>"
1: shows "action_proj a vs = (fmrestrict_set vs (fst a), fmrestrict_set vs (snd a))"
special_axiomsp_special_axioms_eq: shows "special_axiomsp = (\<lambda>x. x \<in> special_axioms)"
dist_vec_nth_le: fixes x :: "('c, 'd) vec"   and i :: "'d"   and y :: "('c, 'd) vec" shows "dist (x $ i) (y $ i) \<le> dist x y"
globally_sound_modes_compatible: fixes cms :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list"   and mem :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "abs.globally_sound_mode_use (cms, mem)" shows "abs.compatible_modes (map snd cms)"
ordLeq_csum1: fixes p1 :: "('d \<times> 'd) set"   and p2 :: "('e \<times> 'e) set" assumes "Card_order p1" shows "p1 \<le>o p1 +c p2"
ex_productE: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 'j" assumes "\<exists>a. J.has_as_product J D a" obtains a :: "'j" and \<pi> :: "'a \<Rightarrow> 'j" where "product_cone J (\<cdot>\<^sub>J) D a \<pi>"
mset_eq_length: fixes xs :: "'b list"   and ys :: "'b list" assumes "mset xs = mset ys" shows "length xs = length ys"
log_exp2_gt: fixes n :: "nat" shows "n < 2 * 2 ^ local.log n"
ucast_ipv6_piece: fixes n :: "nat"   and w :: "128 word" assumes "16 \<le> 128 - n" shows "UCAST(16 \<rightarrow> 128) (UCAST(128 \<rightarrow> 16) (w && (mask 16 << n) >> n)) << n = w && (mask 16 << n)"
joinable_sym: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'c) set" assumes "(x, y) \<in> joinable R" shows "(y, x) \<in> joinable R"
delete_dom_def: fixes p :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "delete_dom p i \<equiv> \<lambda>i'. p (insert_index i i')"
Inf_fold_inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A" shows "Inf A = Finite_Set.fold inf top A"
2: shows "summable (\<lambda>n. diffs c n * K ^ n)"
reachable_adjI: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>G \<restriction> {w. ua__ \<rightarrow>\<^sup>* w}\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {w. ua__ \<rightarrow>\<^sup>* w}\<^esub> v"
measure_mono_fmeasurable: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "A \<subseteq> B"   and "A \<in> sets M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M B"
index_map_vec_1: fixes i :: "nat"   and v :: "'b Matrix.vec"   and f :: "'b \<Rightarrow> 'c" assumes "i < dim_vec v" shows "map_vec f v $v i = f (v $v i)"
adjust_scope: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and body :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT \<pi> P body Q" shows "HT \<pi> (\<lambda>s. P <<>|s>) (SCOPE body) (\<lambda>s\<^sub>0 s. \<exists>l. Q <<>|s\<^sub>0> <l|s>)"
det_Q_eq: shows "Determinant.det Q = Q $$ (0, 0) * Q $$ (1, 1) + Q $$ (1, 0) * - Q $$ (0, 1)"
exhaust: fixes y :: "KS" obtains x1 :: "Keys" where "y = kKS x1"   | x2 :: "Secrets" where "y = sKS x2"
finite_Range: fixes r :: "('c \<times> 'd) set" assumes "finite r" shows "finite (Range r)"
spickI: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat"   and w :: "bool stream"   and v :: "bool" assumes "infinite_coin_toss_space p M" shows "stake n (spick w n v) = stake n w \<and> spick w n v !! n = v"
elims_3: fixes x :: "'b word \<times> nat"   and xa :: "'b word" assumes "\<not> simple_match_ip x xa" obtains base :: "'b word" and len :: "nat" and p_ip :: "'b word" where "x = (base, len)" and "xa = p_ip" and "p_ip \<notin> ipset_from_cidr base len"
0: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<guillemotleft>\<r> f : R f \<Rightarrow> local.map f\<guillemotright>"
assms: shows "\<langle>c #\<^bsub>n'\<^esub>inf_llist t\<rangle> = \<langle>c #\<^bsub>n\<^esub>inf_llist t\<rangle>"
ISpec_BInv_lemma: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "\<turnstile> BInit ic q oc \<and> \<box>[Nxt ic q oc]_(ic, q, oc) \<longrightarrow> \<box>BInv ic q oc"
smult_sum: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and p :: "'b poly" shows "Polynomial.smult (sum f S) p = (\<Sum>i\<in>S. Polynomial.smult (f i) p)"
Rep_Assertion_inverse: fixes x :: "'b Assertion" shows "Abs_Assertion (Rep_Assertion x) = x"
And_prop_entailment: fixes S :: "'b ltl set"   and xs :: "'b ltl list" shows "S \<Turnstile>\<^sub>P And xs = (\<forall>x\<in>set xs. S \<Turnstile>\<^sub>P x)"
ta_union_correct_aux1: fixes TA :: "('a, 'b) tree_automaton_rec"   and TA' :: "('a, 'b) tree_automaton_rec" assumes "tree_automaton TA"   and "tree_automaton TA'"   and "ta_rstates TA \<inter> ta_rstates TA' = {}" shows "ta_lang (ta_union TA TA') = ta_lang TA \<union> ta_lang TA'"
V_def: shows "V = {c_pair x y |x y. x \<in> A \<and> y = 0 \<or> x \<notin> A \<and> y = 1}"
catch_splits_2: fixes P :: "'a + 'b \<Rightarrow> bool"   and m :: "'c + 'b"   and f :: "'c \<Rightarrow> 'a + 'b" shows "P (try m catch f) = (\<not> ((\<exists>e. m = Inl e \<and> \<not> P (f e)) \<or> (\<exists>x. m = Inr x \<and> \<not> P (Inr x))))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G F \<tau>'"
MakeCat: fixes C :: "('g, 'h, 'i) Category_scheme" assumes "Category_axioms C" shows "Category (MakeCat C)"
exec_step_ind_Putstatic_NonStatic: fixes P :: "jvm_prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "P \<turnstile> C has F,NonStatic:t in D" shows "exec_step_ind (StepI (Putstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt IncompatibleClassChangeError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
MkIde_Cod': fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.ide (concrete_category.COMP Obj Hom Comp) a" shows "MkArr (Cod a) (Cod a) (Id (Cod a)) = a"
coinduct_strong: fixes R :: "('a, 'b, 'c, 'd) converter \<Rightarrow> ('a, 'b, 'c, 'd) converter \<Rightarrow> bool"   and converter :: "('a, 'b, 'c, 'd) converter"   and converter' :: "('a, 'b, 'c, 'd) converter" assumes "R converter converter'"   and "\<And>converter converter'. R converter converter' \<Longrightarrow> rel_fun (=) (rel_gpv (rel_prod (=) (\<lambda>converter converter'. R converter converter' \<or> converter = converter')) (=)) (run_converter converter) (run_converter converter')" shows "converter = converter'"
intyness_simps_2: fixes a :: "nat"   and b :: "nat" shows "real a * real b = real (a * b)"
l2product_rmult: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real"   and c :: "real" assumes "f square_integrable S"   and "g square_integrable S" shows "l2product S f (\<lambda>x. c * g x) = c * l2product S f g"
publicKey_eq_iff: fixes b :: "bool"   and A :: "agent"   and b' :: "bool"   and A' :: "agent" shows "(publicKey b A = publicKey b' A') = (b = b' \<and> A = A')"
abs_g_le: fixes m :: "nat" shows "\<bar>g m\<bar> \<le> 1"
simps: fixes a1 :: "'d set"   and a2 :: "'d set"   and a3 :: "'d set"   and a4 :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set" shows "(((a1, a2), a3, a4) \<in> br_step \<delta>) = (\<exists>q W Q' Q W'. a1 = Q \<and> a2 = W \<and> a3 = Q' \<and> a4 = W' \<and> q \<in> W \<and> Q' = Q \<union> br_dsq \<delta> q (Q, W) \<and> W' = W - {q} \<union> (br_dsq \<delta> q (Q, W) - Q))"
finite_res: fixes S :: "'b literal set set" assumes "\<forall>C\<in>S. finite C" shows "\<forall>C\<in>res S. finite C"
distinct_1: fixes x1 :: "nat"   and x21 :: "LambdaType.type"   and x22 :: "LambdaType.type" shows "Atom x1 \<noteq> x21 \<Rightarrow> x22"
indefinite_integral_continuous_left: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and c :: "real"   and e :: "real" assumes "f integrable_on {a..b}"   and "a < c"   and "c \<le> b"   and "0 < e" obtains d :: "real" where "0 < d" and "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e"
sup_inf_distrib1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.distrib_lattice inf less_eq less sup" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
mult_pCons_left: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "pCons a p * q = smult a q + pCons (0::'a) (p * q)"
restrictI: fixes G :: "('e, 'f) labeled_graph" assumes "edges G \<subseteq> {(l, v1, v2). v1 \<in> vertices G \<and> v2 \<in> vertices G}" shows "graph G"
R23s_defs_2: fixes implem :: "chan \<Rightarrow> msg"   and t :: "l3_state" assumes "pfslvl3 implem" shows "pfslvl3.med23s implem t \<equiv> \<lparr>ik = ik t \<inter> payload, secret = secret t, progress = progress t, signals = l1_state.signals t, chan = semivalid_implem.abs implem (ik t), bad = l3_state.bad t\<rparr>"
imp_forward: fixes P' :: "bool"   and Q' :: "bool"   and P :: "bool"   and Q :: "bool" assumes "P' \<longrightarrow> Q'"   and "P \<Longrightarrow> P'"   and "Q' \<Longrightarrow> Q" shows "P \<longrightarrow> Q"
induct: fixes P :: "'b \<Rightarrow> ('b, 'c) Kform \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "('b, 'c) Kform" assumes "\<And>a p. P a (Kprop p)"   and "\<And>a f. P a f \<Longrightarrow> P a (Knot f)"   and "\<And>a f g. \<lbrakk>P a f; P a g\<rbrakk> \<Longrightarrow> P a (Kand f g)"   and "\<And>a a' f. P a \<^bold>K\<^sub>a' f"   and "\<And>a as f. P a \<^bold>C\<^bsub>as\<^esub> f" shows "P a0 a1"
abs_eq: fixes x :: "ephkey" shows "EphK x = Ab (cEphK x)"
eq0: shows "(\<Prod>i = 1..k. a + (1::'a) - of_nat i) = (\<Prod>i = 0..h. a - of_nat i)"
simps_5: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'e \<Rightarrow> 'd"   and x1 :: "'e" shows "(case allow x1 of allow x \<Rightarrow> f1 x | deny x \<Rightarrow> f2 x) = f1 x1"
kmp2_refine: fixes s :: "'a list"   and t :: "'a list" shows "kmp2 s t \<le> kmp1 s t"
strongly_strategyproof_imp_not_manipulable: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "strongly_strategyproof_profile R i Ri'" shows "\<not> manipulable_profile R i Ri'"
select_convs_1: fixes Field1 :: "int \<times> day \<Rightarrow> int"   and Field2 :: "int"   and more :: "'a" shows "Field1 \<lparr>Field1 = Field1, Field2 = Field2, \<dots> = more\<rparr> = Field1"
oseq_groupcastTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>i\<^sub>p\<^sub>s :: "'e \<Rightarrow> nat set"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = groupcast (s\<^sub>i\<^sub>p\<^sub>s (\<sigma> i)) (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "q = p"
singleton_conv2: fixes a :: "'b" shows "{x. a = x} = {a}"
transpose_code: fixes A :: "(('c, 'b) vec, 'd) vec" shows "($) (Finite_Cartesian_Product.transpose A) = transpose_row A"
inpow_subset_nodes: fixes G :: "nat graph_rec"   and xs :: "nat list" assumes "cfg_doms G"   and "xs \<in> (rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G))" shows "set xs \<subseteq> set (cfg_doms.nodes G)"
const0_in_multiset: shows "(\<lambda>a. 0) \<in> multiset"
ts''a_def_1: shows "j < length (types_t \<C>)"
intros: fixes S :: "'procS" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<R>\<lbrakk>\<cdot>\<rbrakk>L SourceTerm S"
list_slice_def: fixes xs :: "'b list"   and k :: "nat" shows "list_slice xs k \<equiv> list_slice_aux xs k (length xs div k)"
if_SE_D1': fixes P :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d"   and B\<^sub>1 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and M :: "'d \<Rightarrow> (bool \<times> 'd) option" assumes "P \<sigma>" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M) = (\<sigma> \<Turnstile> B\<^sub>1 ;- M)"
no_trivial_inverse: fixes x :: "'a" assumes "x \<noteq> 0" shows "\<nexists>y. x + y = 0"
diff_eq_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.group_add minus plus zero uminus" shows "(minus a b = c) = (a = plus c b)"
inject: fixes x :: "proj2"   and ya :: "proj2" shows "(P2L x = P2L ya) = (x = ya)"
constant_functor_axioms: shows "constant_functor (\<cdot>\<^sub>J) (\<cdot>) a"
homI: fixes G :: "('c, 'd) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "\<And>x. x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" shows "h \<in> hom G H"
lossless_WT_gpv_induct: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and P :: "('c, 'a, 'b) gpv \<Rightarrow> bool" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>p. \<lbrakk>lossless_spmf p; \<And>out c. IO out c \<in> set_spmf p \<Longrightarrow> out \<in> outs_\<I> \<I>; \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> lossless_gpv \<I> (c input); \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> \<I> \<turnstile>g c input \<surd>; \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> P (c input)\<rbrakk> \<Longrightarrow> P (GPV p)" shows "P gpv"