hcju/isabelleps huggingface dataset:
matrix_vector_mult_hom: fixes A :: "((real, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "(real, 'd) Finite_Cartesian_Product.vec" shows "HMA_Connect.map_matrix of_real A *v map_vector of_real v = map_vector of_real (A *v v)"
transfer_implies_def: shows "transfer_implies \<equiv> (\<longrightarrow>)"
subset_emptyI: fixes A :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> False" shows "A \<subseteq> {}"
axioms_4: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vsv (\<CC>\<lparr>Cod\<rparr>)"
deterministic_RP_refutation: fixes N0 :: "(('a, nat) Term.term literal list \<times> nat) list"   and n0 :: "nat"   and R :: "('a, nat) Term.term literal list list" assumes "deterministic_RP (RP.St0 N0 n0) = Some R" shows "(\<not> satisfiable (RP.grounded_N0 N0)) = ({#} \<in> RP.grounded_R R)"
less_eq_transition_ext_def: fixes t1 :: "'a transition_scheme"   and t2 :: "'a transition_scheme" shows "(t1 \<le> t2) = (t1 < t2 \<or> t1 = t2)"
init_exp_of_neq: fixes e :: "'a exp"   and e' :: "'a exp" assumes "init_exp_of e = \<lfloor>e'\<rfloor>" shows "e' \<noteq> e"
vdomain_dg_prod_Cod: shows "\<D>\<^sub>\<circ> (dg_prod I \<AA>\<lparr>Cod\<rparr>) = dg_prod I \<AA>\<lparr>Arr\<rparr>"
simps_5: fixes v :: "Core.term"   and va :: "Core.term"   and u :: "Core.term" shows "v $ va \<bullet> u = v $ va $ u"
uncurry_apply: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "uncurry f (a, b) = f a b"
sorted_list_of_set_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq {} = []"
geodesic_segment_union_1: fixes x :: "'b"   and z :: "'b"   and y :: "'b"   and G :: "'b set"   and H :: "'b set" assumes "dist x z = dist x y + dist y z"   and "geodesic_segment_between G x y"   and "geodesic_segment_between H y z" shows "geodesic_segment_between (G \<union> H) x z"
cases: fixes xa :: "'c list \<times> 'c set" obtains ws :: "'c list" and Z :: "'c set" where "xa = (ws, Z)"
axioms_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexFolding X f" shows "ChamberComplexRetraction X f"
square_free_factorizationD_4: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs)"   and "p = 0" shows "c = (0::'a) \<and> bs = []"
exhaust_sel: fixes rule :: "'c rule" obtains "rule = Rule (get_match rule) (get_action rule)"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "suffixes (x # xs) = suffixes xs @ [x # xs]"
simps_3: fixes x21 :: "'b"   and x22 :: "'b llist" shows "x21 ## x22 \<noteq> LNil"
assms_3: shows "J.mpt_factor proj2 M3 T3"
Int_atLeastAtMost: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b \<inter> ord.atLeastAtMost less_eq c d = ord.atLeastAtMost less_eq (ord.max less_eq a c) (ord.min less_eq b d)"
differing_vars_neg_intro: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "mem\<^sub>1 x = fst (mems ! i) x"   and "mem\<^sub>2 x = snd (mems ! i) x" shows "x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i"
listall_listsp_eq: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "listall P xs = listsp P xs"
While: fixes r :: "(state \<times> state) set"   and P :: "state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c :: "com" assumes "wf r"   and "\<forall>s'. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> b s \<and> s' = s} c {\<lambda>s. P s \<and> (s, s') \<in> r}" shows "\<turnstile>\<^sub>t {P} WHILE b DO c {\<lambda>s. P s \<and> \<not> b s}"
dom_char: fixes f :: "'a \<times> 'a" shows "VxV.dom f = (if arr (fst f) \<and> arr (snd f) then (local.dom (fst f), local.dom (snd f)) else (null, null))"
apsnd_def: fixes f :: "'c \<Rightarrow> 'd" shows "apsnd f = map_prod id f"
emp_act: fixes p :: "'b" shows "\<alpha> (\<Squnion>{}) p = \<bottom>"
igAbsCongS_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igAbsCongS MOD \<equiv> \<forall>xs x x' y s X X'. isInBar (xs, s) \<and> igWls MOD s X \<and> igWls MOD s X' \<longrightarrow> igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSwap MOD xs y x X = igSwap MOD xs y x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
inf_sup_aci_1: fixes x :: "'a"   and y :: "'a" shows "inf x y = inf y x"
comp_tfr\<^sub>s\<^sub>e\<^sub>t_def: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('a, 'b \<times> nat) Term.term list \<times> 'c"   and \<Gamma> :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('d, 'e) Term.term"   and M :: "('a, 'b \<times> nat) Term.term list" shows "comp_tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> M \<equiv> finite_SMP_representation arity Ana \<Gamma> M \<and> (let \<delta> = var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set M))) in \<forall>s\<in>set M. \<forall>t\<in>set M. is_Fun s \<and> is_Fun t \<and> \<Gamma> s \<noteq> \<Gamma> t \<longrightarrow> mgu s (t \<cdot> \<delta>) = None)"
A_ia_n: shows "A $ ia $ (LEAST n. A $ ia $ n \<noteq> (0::'a)) \<noteq> (0::'a)"
replaceEquivalentByEquivalent: fixes formula :: "Literal list list"   and formula' :: "Literal list list"   and formula1 :: "Literal list list"   and formula2 :: "Literal list list" assumes "equivalentFormulae formula formula'" shows "equivalentFormulae (formula1 @ formula @ formula2) (formula1 @ formula' @ formula2)"
split_face: shows "(f\<^sub>1, f\<^sub>2) = split_face f u v vs"
inEdges_correct: fixes g :: "'g"   and n :: "'node" assumes "invar g" shows "set (inEdges g n) = {(uu_, uu_, t). t = n} \<inter> \<alpha>e g"
skl1_implements_a0i_i: shows "implements med0sk1iai a0i skl1"
infinite_super: fixes S :: "'k set"   and T :: "'k set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
ring_tOp_commute: fixes R :: "('h, 'i) Ring_scheme"   and x :: "'h"   and y :: "'h" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y = y \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
kleene_lfp: fixes f :: "'g \<Rightarrow> 'g" assumes "cont f" shows "lfp f = (SUP i. (f ^^ i) bot)"
ball_UN: fixes B :: "'f \<Rightarrow> 'e set"   and A :: "'f set"   and P :: "'e \<Rightarrow> bool" shows "(\<forall>z\<in>\<Union> (B ` A). P z) = (\<forall>x\<in>A. \<forall>z\<in>B x. P z)"
span_add_eq: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "x \<in> pmdl S" shows "(x + y \<in> pmdl S) = (y \<in> pmdl S)"
sees_methods_decl_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "P \<turnstile> C sees_methods Mm" shows "\<exists>Mm' Mm\<^sub>2. P \<turnstile> C' sees_methods Mm' \<and> Mm' = Mm ++ Mm\<^sub>2 \<and> (\<forall>M m D. Mm\<^sub>2 M = \<lfloor>(m, D)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C)"
supp_some: fixes p :: "perm" shows "supp (SOME x. supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) \<subseteq> supp (p \<bullet> P)"
lprod_3_6: fixes y' :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set"   and x :: "'a"   and z :: "'a" assumes "(y', y) \<in> R" shows "([x, z, y'], [x, y, z]) \<in> lprod R"
prems_2: fixes lholed :: "Lholed" shows "\<not> Lfilled 0 lholed [$Return] (cs @ ($* es))"
elims: fixes x :: "real"   and xa :: "real"   and xb :: "real"   and xc :: "nat"   and y :: "bool stream \<Rightarrow> real" assumes "geom_rand_walk x xa xb xc = y" obtains u :: "real" and d :: "real" and v :: "real" where "x = ua__" and "xa = da__" and "xb = v__" and "xc = 0" and "y = (\<lambda>w. v__)"   | u :: "real" and d :: "real" and v :: "real" and n :: "nat" where "x = u" and "xa = d" and "xb = v" and "xc = Suc n" and "y = (\<lambda>w. (case w !! n of True \<Rightarrow> u | False \<Rightarrow> d) * geom_rand_walk u d v n w)"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "ShadowRootClass.known_ptr ptr = (DocumentClass.known_ptr ptr \<or> is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
beh_non_empty: fixes s :: "'e"   and b :: "'e list"   and T :: "('e, 'f) TS_scheme" assumes "s # b \<in> beh T" obtains "b = []" and "s \<in> init T"   | sa :: "'e" and ba :: "'e list" where "b = sa # ba" and "sa # ba \<in> beh T" and "(sa, s) \<in> TS.trans T"
gcd_of_int_poly: fixes f :: "int poly"   and g :: "int poly" shows "gcd (of_int_poly f) (of_int_poly g) = Polynomial.smult (inverse (of_int (Polynomial.lead_coeff (gcd f g)))) (of_int_poly (gcd f g))"
par_neq1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "A \<noteq> B"
lrev_work1_def: shows "lrev_work1 \<equiv> fix\<cdot>lrev_body1"
reachable_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v) = (\<exists>p. sG.awalk u p v)"
simps: fixes a1 :: "sequent list"   and a2 :: "sequent" shows "((a1, a2) \<in> provRules) = (\<exists>c F x A ps. a1 = ps \<and> a2 = c \<and> NominalSequents.mset c = \<LM> F \<nabla> [x].A  \<RM> \<and> ps \<noteq> [] \<and> x \<sharp> set_of_prem ( ps - A ))"
is_none_code_2: fixes x :: "'d" shows "Option.is_none (Some x) = False"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and exception :: "'b exception" shows "P (case exception of Fault \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = ((exception = Fault \<longrightarrow> P f1) \<and> (\<forall>x2. exception = OK x2 \<longrightarrow> P (f2 x2)))"
length_sublist: fixes r :: "nat"   and xs :: "'b list"   and l :: "nat" assumes "r \<le> length xs" shows "length (sublist l r xs) = r - l"
arr_inv: fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.arr (\<phi>.B.inv f)"
m_eq: shows "(map_option \<alpha> \<circ>\<circ> map_of) lc = m"
Collect_case_prod_in_rel_leI: fixes X :: "('a \<times> 'b) set"   and Y :: "('a \<times> 'b) set" assumes "X \<subseteq> Y" shows "X \<subseteq> {(x, y). in_rel Y x y}"
Iface_single_1: fixes \<beta> :: "'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and X :: "iface"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (Match (IIface X)) a p = match_iface X (p_iiface p)"
r_triangle: fixes n :: "nat" shows "eval r_triangle [n] \<down>= \<Sum> {0..n}"
assms_5: shows "(deriv ^^ n) f \<xi> \<noteq> 0"
\<pi>l: shows "\<pi> k = \<pi>l 0"
measurable_lborel2: fixes M :: "'d measure" shows "lborel \<rightarrow>\<^sub>M M = borel \<rightarrow>\<^sub>M M"
inv: shows "oalist_inv_raw (update_by_pair (k, v) xs)"
sub_instr_sub1_def: fixes instr_name :: "sparc_operation"   and result :: "word32"   and rs1_val :: "word32"   and operand2 :: "word32" shows "sub_instr_sub1 instr_name result rs1_val operand2 \<equiv> if instr_name \<in> {arith_type SUBcc, arith_type SUBXcc} then do psr_val \<leftarrow> gets (cpu_reg_val PSR); result_31 \<leftarrow> gets (\<lambda>s. ucast (result >> 31)); rs1_val_31 \<leftarrow> gets (\<lambda>s. ucast (rs1_val >> 31)); operand2_31 \<leftarrow> gets (\<lambda>s. ucast (operand2 >> 31)); new_n_val \<leftarrow> gets (\<lambda>s. result_31); new_z_val \<leftarrow> gets (\<lambda>s. if result = 0 then 1 else 0); new_v_val \<leftarrow> gets (\<lambda>s. rs1_val_31 AND NOT operand2_31 AND NOT result_31 OR NOT rs1_val_31 AND operand2_31 AND result_31); new_c_val \<leftarrow> gets (\<lambda>s. NOT rs1_val_31 AND operand2_31 OR result_31 AND (NOT rs1_val_31 OR operand2_31)); new_psr_val \<leftarrow> gets (\<lambda>s. update_PSR_icc new_n_val new_z_val new_v_val new_c_val psr_val); write_cpu new_psr_val PSR; return () od else return ()"
induct: fixes x1 :: "com"   and x2 :: "state"   and P :: "com \<Rightarrow> state \<Rightarrow> bool" assumes "x1 \<down> x2"   and "\<And>f s. f s \<noteq> {} \<Longrightarrow> P (Do f) s"   and "\<And>c1 s0 c2. \<lbrakk>c1 \<down> s0; P c1 s0; \<And>s1. s0 -c1\<rightarrow> s1 \<Longrightarrow> c2 \<down> s1; \<And>s1. s0 -c1\<rightarrow> s1 \<Longrightarrow> P c2 s1\<rbrakk> \<Longrightarrow> P (c1; c2) s0"   and "\<And>b s c1 c2. \<lbrakk>b s; c1 \<down> s; P c1 s\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) s"   and "\<And>b s c2 c1. \<lbrakk>\<not> b s; c2 \<down> s; P c2 s\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) s"   and "\<And>b s c. \<not> b s \<Longrightarrow> P (WHILE b DO c) s"   and "\<And>b s c. \<lbrakk>b s; c \<down> s; P c s; \<And>t. s -c\<rightarrow> t \<Longrightarrow> WHILE b DO c \<down> t; \<And>t. s -c\<rightarrow> t \<Longrightarrow> P (WHILE b DO c) t\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s"   and "\<And>s. \<lbrakk>body \<down> s; P body s\<rbrakk> \<Longrightarrow> P CALL s"   and "\<And>c f s g. \<lbrakk>c \<down> f s; P c (f s)\<rbrakk> \<Longrightarrow> P (LOCAL f; c; g) s" shows "P x1 x2"
generate_is_subgroup: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.group G"   and "H \<subseteq> carrier G" shows "subgroup (generate G H) G"
NatTransFtorDom: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTransP \<eta>" shows "NTCatDom \<eta> = CatDom (NTCod \<eta>)"
Rf_def: fixes N :: "'b literal multiset set" shows "standard_redundancy_criterion.Rf N = {C. \<exists>DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD \<longrightarrow> I \<Turnstile> C) \<and> (\<forall>D. D \<in># DD \<longrightarrow> D < C)}"
liftn_context_conv: fixes k :: "nat"   and k' :: "nat"   and t :: "dB"   and n :: "nat" assumes "k \<le> k'"   and "\<forall>i\<in>frees t. i < k \<or> k' \<le> i" shows "liftn n t k = liftn n t k'"
not_rt_fresh_asI2: fixes dip :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "\<not> rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1" shows "\<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
sort_append_swap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) (xs @ ys) = linorder.sort_key less_eq (\<lambda>x. x) (ys @ xs)"
deriv_zeta_2_bound: shows "- 1 < Re (deriv zeta 2)"
one_abs: fixes a :: "name"   and t :: "lam"   and t' :: "lam" assumes " Lam [a].t \<longrightarrow>\<^sub>1 t'" shows "\<exists>t''. t' = Lam [a].t'' \<and>  t \<longrightarrow>\<^sub>1 t''"
right_adjoint_is_ide: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.is_right_adjoint f" shows "S.ide f"
assms: shows "check \<C> es (Type tn) = TopType tm"
rm_vars_apply': fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and vs :: "'a set" assumes "subst_domain \<delta> \<inter> vs = {}" shows "rm_vars vs \<delta> = \<delta>"
STCal_step_2: fixes Source :: "'a processCalculus"   and Target :: "'b processCalculus"   and T :: "'b"   and P' :: "('a, 'b) Proc" shows "TargetTerm T \<longmapsto>STCal Source Target P' = (\<exists>T'. P' = TargetTerm T' \<and> T \<longmapsto>Target T')"
homotopic_paths_imp_subset: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q" shows "path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
c1: shows "\<forall>p\<in>set ps. extend S p \<in> set Ps"
quantale_hom_prop1: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" assumes "f \<in> quantale_homset" shows "radj f (f x \<rightarrow> y) = x \<rightarrow> radj f y"
received_messages_were_sent: fixes i :: "'a"   and j :: "nat" assumes "is_valid_state_id (i, j)" shows "make_set j (\<lambda>k m. \<exists>s. event_at (i, k) (Receive s m)) \<subseteq> {m. \<exists>i. event_at i (Send m)}"
A_FMT: shows "it' s (case_dres False False c) (\<lambda>(k, v) s. s \<bind> f k v) \<sigma> = foldli l (case_dres False False c) (\<lambda>(k, v) s. s \<bind> f k v) \<sigma>"
record_simps_1: fixes f :: "(char list, char list, 'b) namespace \<Rightarrow> (char list, char list, 'a) namespace"   and x :: "'b sem_env" shows "sem_env.v (update_v f x) = f (sem_env.v x)"
preorder_on_converse: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "preorder_on A (r\<inverse>) = preorder_on A r"
divide_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f div g = f * Inverse g"
iso_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and F :: "('e, 'f) monoid_scheme"   and H :: "'c set"   and \<phi> :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "Group.group F"   and "subgroup H G"   and "\<phi> \<in> Group.iso G F" shows "subgroup (\<phi> ` H) F"
idempotent_one_closed: shows "matrix_idempotent_semiring.idempotent mone"
inf_sup_aci_4: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (inf x (y::'b::type)) = inf x y"
\<Delta>_simps_4: shows "local.dom ((f \<star> tab\<^sub>0 f) \<cdot> f.tab) = tab\<^sub>1 f"
is_generalisation_of_derestricts_input: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" assumes "is_generalisation_of t' t i r" shows "\<forall>g\<in>set (Guards t'). \<not> gexp_constrains g (V (I i))"
Mp_inv_Mp_id: fixes m :: "int"   and f :: "int poly" assumes "poly_mod_2 m" shows "poly_mod.Mp m (poly_mod.inv_Mp m f) = poly_mod.Mp m f"
ih: shows "wstate_of_dstate (N, P, Q, n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate ([], P, Q, n)"
inner_prod_outer_prod: fixes a :: "'c vec"   and n :: "nat"   and b :: "'c vec"   and c :: "'c vec"   and m :: "nat"   and d :: "'c vec" assumes "a \<in> carrier_vec n"   and "b \<in> carrier_vec n"   and "c \<in> carrier_vec m"   and "d \<in> carrier_vec m" shows "inner_prod a (outer_prod b c *\<^sub>v d) = inner_prod a b * inner_prod c d"
unit_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (G.unit a) = G.map\<^sub>0 a"
modify_def: fixes l :: "'d \<Longrightarrow> 'c"   and f :: "'d \<Rightarrow> 'd" shows "modify l f = zoom l (State_Monad.update f)"
proj_FDOM_eff_subset_FDOM_valid_states: fixes p :: "('d, 'e) fmap"   and e :: "('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set"   and s :: "('d, 'g) fmap" assumes "(p, e) \<in> prob_proj PROB vs"   and "s \<in> valid_states PROB" shows "fmdom' e \<subseteq> fmdom' s"
cmult_le_self: fixes \<kappa> :: "V"   and \<alpha> :: "V" assumes "Card \<kappa>"   and "\<alpha> \<noteq> 0" shows "\<kappa> \<le> \<kappa> \<otimes> \<alpha>"
sum_slice2: fixes n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"   and "sum_SDG_slice2p n n'" shows "sum_SDG_slice2p n n''"
ptensor_encode1_encode2: shows "partial_state.encode1 st2.dims0 st2.vars1' = partial_state.encode2 st2.dims0 st2.vars2'"
simps_1: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" shows "TS M2 M1 \<Omega> V m 0 = {}"
conditionalize: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "finite A" shows "F.F g A = G (\<lambda>a. if a \<in> A then g a else \<^bold>1)"
ign_res_eq_subst_res: fixes P :: "'b assn"   and w :: "vals" shows "P\<down>=w\<leftarrow>w = P\<down>"
bij_face_cycle_succ: shows "bij G.face_cycle_succ"
nat_eq_iff: fixes w :: "int"   and m :: "nat" shows "(nat w = m) = (if 0 \<le> w then w = int m else m = 0)"
conf_fs_hext: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and M :: "char list"   and n :: "nat"   and T\<^sub>r :: "ty"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "conf_fs P h \<Phi> M n T\<^sub>r frs"   and "h \<unlhd> h'" shows "conf_fs P h' \<Phi> M n T\<^sub>r frs"
basic_1: shows "\<exists>!x. root_cond (p1, l1, r1) x"
f0: shows "i < 2 ^ (n + 1) \<and> j < 2 ^ (n + 1)"
\<iota>_nonneg: fixes s :: "'s"   and t :: "'s" shows "0 \<le> \<iota> s t"
valid_cases: fixes X :: "msg" assumes "X \<in> asym.valid" obtains A :: "agent" and B :: "agent" and M :: "msg" where "X = asym.implInsec Aa__ Ba__ Ma__" and "Ma__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = asym.implConfid Ab__ Bb__ Mb__" and "Mb__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = asym.implAuth Ac__ Bc__ Mc__" and "Mc__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = asym.implSecure A B M" and "M \<in> payload"
image_constant: fixes x :: "'a"   and A :: "'a set"   and c :: "'b" assumes "s2p A x" shows "\<P> (\<lambda>x. c) A = \<eta> c"
floor_one: shows "\<lfloor>1::'d\<rfloor> = 1"
monom_var_def: fixes l :: "linear_poly" shows "monom_var l = max_var l"
u_var: fixes a :: "name" shows "Var a \<mapsto> [],Var a"
size_hps_pass2: fixes hs :: "'b heap list" assumes "hs \<noteq> []"   and "no_Emptys hs" shows "no_Empty (pass\<^sub>2 hs) \<and> size_hps hs = size_hps (hps (pass\<^sub>2 hs)) + 1"
span_eq: fixes S :: "('t \<Rightarrow>\<^sub>0 'c) set"   and T :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "(pmdl S = pmdl T) = (S \<subseteq> pmdl T \<and> T \<subseteq> pmdl S)"
local_soundness: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and "structure" :: "'c set"   and sat :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a"   and s :: "'b"   and sl :: "'b fset" assumes "Soundness eff rules structure sat"   and "r \<in> i.R rules"   and "eff r s sl"   and "\<And>s'. s' |\<in>| sl \<Longrightarrow> \<forall>S\<in>structure. sat S s'" shows "\<forall>S\<in>structure. sat S s"
W_consts_2: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n true\<^sub>n"
k: shows "card (set_mset (\<Sum>b2\<in>#remove1_mset blv \<B>. mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2})) = nat \<k> choose 2"